text:
\section*{Introduction}
In order to study the homotopy types of closed manifolds, Browder and Wall introduced the notion of Poincar{\'e} duality complexes. A Poincar{\'e} duality complex, or ${\rm PD}^n$--complex, is a ${\rm CW}$--complex, $X$, whose cohomology satisfies a certain algebraic condition. Equivalently, the chain complex, $\widehat C(X)$, of the universal cover of $X$ must satisfy a corresponding algebraic condition. Thus Poincar{\'e} complexes form a mixture of topological and algebraic data and it is an old quest to provide purely algebraic data determining the homotopy type of ${\rm PD}^n$--complexes. This has been achieved for $n=3$, but, for $n=4$, only partial results are available in the literature.

Homotopy types of $3$--manifolds and ${\rm PD}^3$--complexes were considered by Thomas \cite{Thomas}, Swarup \cite{Swarup} and Hendriks \cite{Hendriks}. The homotopy type of a ${\rm PD}^3$--complex, $X$, is determined by its \emph{fundamental triple}, consisting of the fundamental group, $\pi = \pi_1(X)$, the orientation character, $\omega$, and the image in ${\rm H}_3(\pi, \mathbb Z^{\omega})$ of the fundamental class, $[X]$. Turaev \cite{Turaev} provided an algebraic condition for a triple to be realizable by a ${\rm PD}^3$--complex. Thus, in dimension $3$, there are purely algebraic invariants which provide a complete classification.

Using primary cohomological invariants like the fundamental group, characteristic classes and intersection pairings, partial results were obtained for $n=4$ by imposing conditions on the fundamental group. For example, Hambleton, Kreck and Teichner classified ${\rm PD}^4$--complexes with finite fundamental group having periodic cohomology of dimension $4$ (see \cite{HamblKreck}, \cite{Teichner} and \cite{HamblKreckTeich}). Cavicchioli, Hegenbarth and Piccarreta studied ${\rm PD}^4$--complexes with free fundamental group (see \cite{CavHeg} and \cite{HegPic}), as did Hillman \cite{Hillman3}, who also considered ${\rm PD}^4$--complexes with fundamental group a ${\rm PD}^3$--group \cite{Hillman2}. Recently, Hillman \cite{Hillman1} considered homotopy types of ${\rm PD}^4$--complexes whose fundamental group has cohomological dimension $2$ and one end.

It is doubtful whether primary invariants are sufficient for the homotopy classification of ${\rm PD}^4$--complexes in general and we thus follow Ranicki's approach (\cite{Ranicki1} and \cite{Ranicki2}) who assigned to each ${\rm PD}^n$--complex, $X$, an \emph{algebraic Poincar{\'e} duality complex} given by the chain complex, $\widehat C(X)$, together with a \emph{symmetric} or \emph{quadratic structure}. However, Ranicki considered neither the realizability of such algebraic Poincar{\'e} duality complexes nor whether the homotopy type of a ${\rm PD}^n$--complex is determined by the homotopy type of its algebraic Poincar{\'e} duality complex.

This paper presents a structure on chain complexes which completely classifies ${\rm PD}^4$--complexes up to homotopy. The classification uses \emph{fundamental triples} of ${\rm PD}^4$--complexes, and, in fact, the chain complex model yields algebraic conditions for the realizability of fundamental triples.

Fundamental triples  of formal dimension $n \geq 3$ comprise an $(n-2)$--type $T$, a homomorphism $\omega: \pi_1(T) \rightarrow \mathbb Z / 2 \mathbb Z$ and a homology class $t \in {\rm H}_n(T, \mathbb Z^{\omega})$. There is a functor,
\begin{equation*}
\tau_+: {\rm \bf PD}^n_+ \longrightarrow {\rm\bf Trp}^n_+,
\end{equation*}
from the category ${\rm \bf PD}^n_+$ of ${\rm PD}^n$--complexes and maps of degree one to the category ${\rm\bf Trp}^n_+$ of triples and morphisms inducing surjections on fundamental groups. Our first main result is

\spec{Theorem \ref{mainthm}}{The functor $\tau_+$ reflects isomorphisms and is full  for $n \geq 3$.}

\spec{Corollary \ref{class}}{Take $n \geq 3$. Two closed $n$--dimensional manifolds or two ${\rm PD}^n$--complexes, respectively, are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic.}

\medskip

Corollary \ref{class} extends results of Thomas \cite{Thomas}, Swarup \cite{Swarup} and Hendriks \cite{Hendriks} for dimension $3$ to arbitrary dimension and establishes Turaev's conjecture \cite{Turaev} on ${\rm PD}^n$--complexes whose $(n-2)$--type is an Eilenberg--{Mac\,Lane} space $K(\pi_1X,1)$. Corollary \ref{class} is even of interest in the case of simply connected or highly connected manifolds.

Theorem \ref{mainthm} also yields a criterion for the existence of a map of degree one between ${\rm PD}^n$--complexes, recovering Swarup's result for maps between $3$--manifolds and Hendriks' result for maps between ${\rm PD}^3$--complexes.

In the oriented case, special cases of Corollary \ref{class} were proved by Hambleton and Kreck \cite{HamblKreck} and Cavicchioli and Spaggiari \cite{CavSpag}. In fact, in \cite{HamblKreck}, Corollary 3.2 is obtained under the condition that either the fundamental group is finite or the second rational homology of the 2--type is non--zero. Corresponding conditions were used in \cite{CavSpag} for oriented ${\rm PD}^{2n}$--complexes with $(n-1)$--connected universal covers, and Teichner extended the approach of \cite{HamblKreck} to the non--oriented case in his thesis \cite{Teichner}. Our result shows that the conditions on finiteness and rational homology used in these papers are not necessary.

It follows directly from Poincar{\'e} duality and Whitehead's Theorem that the functor $\tau_+$ reflects isomorphisms. To show that $\tau_+$ is full requires work. Given ${\rm PD}^n$--complexes $Y$ and $X$, $n \geq 3$, and a morphism $f: \tau_+ Y \rightarrow \tau_+ X$ in ${\rm\bf Trp}^n_+$, we first construct a chain map $\xi: \widehat C(Y) \rightarrow \widehat C(X)$  preserving fundamental classes, that is, $\xi_{\ast}[Y] = [X]$. Then we use the category ${\rm\bf H}^{k+1}_c$ of homotopy systems of order $(k+1)$ introduced in \cite{Baues1} to realize $\xi$ by a map $\overline f: Y \rightarrow X$ with $\tau_+(\overline f) = f$.

Our second main result describes algebraic models of homotopy types of ${\rm PD}^4$--complexes. We introduce the notion of \emph{${\rm PD}^n$--chain complex} and show that ${\rm PD}^3$--chain complexes are equivalent to ${\rm PD}^3$--complexes up to homotopy. In Section \ref{4chainrealize} we show that ${\rm PD}^4$--chain complexes classify homotopy types of ${\rm PD}^4$--complexes up to $2$--torsion. In particular, we obtain

\spec{Theorem \ref{finodd}}{The functor $\widehat C$ induces a 1--1 correspondence between homotopy types of ${\rm PD}^4$--complexes with finite fundamental group of odd order and homotoppy types of ${\rm PD}^4$--chain complexes with homotopy co--commutative diagonal and finite fundamental group of odd order.}

\medskip

To obtain a complete homotopy classification of ${\rm PD}^4$--complexes, we study the chain complex of a $2$--type in Section \ref{2typechains}. We compute this chain complex  up to dimension $4$  in terms of Peiffer commutators in pre--crossed modules. This allows us to introduce ${\rm PD}^4$--chain complexes together with a $\beta$--invariant, and we prove

\spec{Corollary \ref{PD4cor}}{The functor $\widehat C$ induces a 1--1 correspondence between homotopy types of ${\rm PD}^4$--complexes and homotopy types of $\beta$--${\rm PD}^4$--chain complexes.}

\medskip

Corollary \ref{PD4cor} highlights the crucial r\^{o}le of Peiffer commutators for the homotopy classification of $4$--manifolds.

The proofs of our results rely on the obstruction theory \cite{Baues1} for the realizability of chain maps which we recall in Section \ref{homsys}.
\section{Chain complexes}\label{chaincomp}
Let $X^n$ denote the $n$--skeleton of the CW--complex $X$. We call $X$ reduced if $X^0 = \ast$ is the base point. The objects of the category ${\rm\bf{CW}}_0$ are reduced CW--complexes $X$ with universal covering $p: \widehat X \rightarrow X$, such that $p(\widehat  \ast) = \ast$, where $\widehat \ast \in  \widehat X^0$ is the base point of $\widehat X$. Here the $n$--skeleton of $\widehat X$ is $\widehat X^n = p^{-1}(X^n)$. Morphisms in ${\rm\bf{CW}}_0$ are cellular maps $f: X \rightarrow Y$ and homotopies in $\rm\bf CW_0$ are base point preserving. A map $f: X \rightarrow Y$ in $\rm\bf CW_0$ induces a unique covering map $\widehat f: \widehat X \rightarrow \widehat Y$ with $\widehat f(\widehat \ast) = \widehat \ast$, which is equivariant with respect to $\varphi = \pi_1(f)$.

We consider pairs $(\pi, C)$, where $\pi$ is a group and $C$ a chain complex of left modules over the group ring $\mathbb Z [\pi]$. We write $\Lambda = \mathbb Z [\pi]$ and $C$ for $(\pi, C)$, whenever $\pi$ is understood. We call $(\pi, C)$ free if each $C_n, n \in \mathbb Z$, is a free $\Lambda$--module. Let $\rm aug: \Lambda \rightarrow \mathbb Z$ be the augmentation homomorphism, defined by ${\rm aug} (g) = 1$ for all $g \in \pi$. Every group homomorphism, $\varphi: \pi \rightarrow \pi'$, induces a ring homomorphism $\varphi_{\sharp}: \Lambda \rightarrow \Lambda'$, where $\Lambda' = \mathbb Z[\pi']$. A chain map is a pair $(\varphi, F): (\pi, C) \rightarrow (\pi', C')$, where $\varphi$ is a group homomorphism and $F: C \rightarrow C'$ a $\varphi$--equivariant chain map, that is a chain map of the underlying abelian chain complexes, such that $F(\lambda c) = \varphi_{\sharp}(\lambda) F(c)$ for $\lambda \in \Lambda$ and $c \in C$. Two such chain maps are homotopic, $(\varphi, F) \simeq (\psi, G)$ if $\varphi = \psi$ and if there is a $\varphi$--equivariant map $\alpha: C \rightarrow C'$ of degree $+1$ such that $G - F = d \alpha + \alpha d$.

A pair $(\pi, C)$ is a reduced chain complex if $C_0 = \Lambda$ with generator $\ast$, $C_i = 0$ for $i  < 0$ and ${\rm H}_0 C = \mathbb Z$ such that $C_0 = \Lambda \rightarrow {\rm H}_0 C = \mathbb Z$ is the augmentation of $\Lambda$. A chain map, $(\varphi, f): (\pi, C) \rightarrow (\pi', C')$, of reduced chain complexes, is reduced if $f_0$ is induced by $\varphi_{\sharp}$, and a chain homotopy $\alpha$ of reduced chain maps is reduced if $\alpha_0 = 0$. The objects of the category ${\rm\bf H}_0$ are reduced chain complexes and the morphisms are reduced chain maps. Homotopies in ${\rm\bf H}_0$ are reduced chain homotopies. Every chain complex $(\pi, C)$ in ${\rm\bf H}_0$ is equipped with the augmentation $\varepsilon: C \rightarrow \mathbb Z$ in ${\rm\bf H}_0$. The ring homomorphism $\mathbb Z \rightarrow \Lambda$ yields the co--augmentation $\iota: \mathbb Z \rightarrow C$, where we view $\mathbb Z = (0, \mathbb Z)$ as chain complex with trivial group $\pi = 0$ concentrated in degree $0$. Note that $\varepsilon \iota = {\rm id}_{\mathbb Z}$, and the composite $\iota \varepsilon: C \rightarrow C'$ is the \emph{trivial} map.

For an object $X$ in ${\rm\bf{CW}}_0$, the cellular chain complex $C(\widehat X)$ of the universal cover $\widehat X$ is given by $C_n(\widehat X) = {\rm H}_n(\widehat X^n, \widehat X^{n+1})$. The fundamental group $\pi = \pi_1(X)$ acts on $C(\widehat X)$,  and viewing $C(\widehat X)$ as a complex of left $\Lambda$--modules, we obtain the object $\widehat C(X) = (\pi, C(\widehat X))$ in ${\rm\bf H}_0$. Moreover, a morphism $f: X \rightarrow Y$ in ${\rm\bf CW}_0$ induces the homomorphism $\pi_1(f)$ on the fundamental groups and the $\pi_1(f)$--equivariant map $\widehat f: \widehat X \rightarrow \widehat Y$ which in turn induces the $\pi_1(f)$--equivariant chain map $\widehat f_{\ast}: C(\widehat X) \rightarrow C(\widehat Y)$ in ${\rm\bf H}_0$. As $\widehat f$ preserves base points, $\widehat C(f) = (\pi_1(f), \widehat f_{\ast})$ is a reduced chain map. We obtain the functor
\begin{equation}\label{Chat}
\widehat C: {\rm\bf CW}_0  \longrightarrow {\rm\bf H}_0.
\end{equation}
The chain complex $C$ in ${\rm\bf H}_0$ is \emph{$2$--realizable} if there is an object $X$ in ${\rm\bf CW}_0$ such that $\widehat C(X^2) \cong C_{\leq 2}$, that is, $\widehat C(X^2)$ is isomorphic to $C$ in degree $\leq 2$.
Given two objects $X$ and $Y$ in ${\rm\bf{CW}}_0$, their product again carries a cellular structure and we obtain the object $X \times Y$ in ${\rm\bf{CW}}_0$ with base point $(\ast, \ast)$ and universal cover $(X \times Y)\widehat \  = \widehat X \times \widehat Y$, so that
\begin{equation}
\widehat C(X \times Y) = (\pi \times \pi, C(\widehat X) \otimes_{\mathbb Z} C(\widehat Y)).
\end{equation}

For $i = 1, 2$, let $p_i: X \times X \rightarrow X$ be the projection onto the $i$--th factor. A \emph{diagonal} $\Delta: X \rightarrow X \times X$ in ${\rm\bf{CW}}_0$ is a cellular map with $p_i \Delta \simeq {\rm{id}}_X$ in ${\rm\bf{CW}}_0$ for $i = 1, 2$. A \emph{diagonal} on $(\pi, C)$ in ${\rm\bf H}_0$ is a chain map $(\delta, \Delta): (\pi, C) \rightarrow (\pi \times \pi, C \otimes_{\mathbb Z} C)$ in ${\rm\bf H}_0$ with $\delta: \pi \rightarrow \pi \times \pi, g \mapsto (g,g)$, such that $p_i \Delta \simeq {\rm id}_C$ for $i = 1, 2$, where $p_1 = {\rm id} \otimes \varepsilon$ and $p_2 = \varepsilon \otimes {\rm id}$.

The diagonal $(\delta, \Delta)$ in ${\rm\bf H}_0$ is homotopy co--associative if the diagram
\begin{equation*}
\xymatrix{
C \ar[rr]^-{\Delta} \ar[d]^{\Delta} && C \otimes _{\mathbb Z} C \ar[d]^-{\mbox{id} \otimes \Delta} \\
C \otimes _{\mathbb Z} C \ar[rr]^-{\Delta \otimes \mbox{id}} && C \otimes _{\mathbb Z} C \otimes _{\mathbb Z} C}
\end{equation*}
commutes up to chain homotopy in ${\rm\bf H}_0$. The diagonal $(\delta, \Delta)$ in ${\rm\bf H}_0$ is homotopy co--commutative if the diagram

\begin{equation*}
\xymatrix{
C \ar[rr]^-{\Delta} \ar[drr]_{\Delta} && C \otimes _{\mathbb Z} C \ar[d]^-{T} \\
&& C \otimes _{\mathbb Z} C}
\end{equation*}
commutes up to chain homotopy in ${\rm\bf H}_0$, where $T$ is given by $T(c \otimes d) = (-1)^{|c||d|} d \otimes c.$

By the cellular approximation theorem, there is a diagonal $\Delta: X \rightarrow X \times X$ in ${\rm\bf{CW}}_0$ for every object $X$ in ${\rm\bf{CW}}_0$. Applying the functor $\widehat C$ to such a diagonal, we obtain the diagonal $\widehat C(\Delta)$ in ${\rm\bf H}_0$. This raises the question of realizabilty, that is, given a diagonal $(\delta, \Delta): \widehat C (X) \rightarrow \widehat C(X) \otimes_{\mathbb Z} \widehat C(X)$ in ${\rm\bf H}_0$, is there a diagonal $\Delta$ in ${\rm\bf{CW}}_0$ with $\widehat C(\Delta) = (\delta, \Delta)$? As $\widehat C(\Delta)$ is homotopy co--associative and homotopy co--commutative for any diagonal $\Delta$ in ${\rm\bf{CW}}_0$, homotopy co--associativity and homotopy co--commutativity of $(\delta, \Delta)$ are necessary conditions for realizability.

To discuss questions of realizability for a functor $\lambda: {\rm\bf A} \rightarrow {\rm\bf B}$, we consider pairs $(A,b)$, where $b: \lambda A \cong B$ is an equivalence in $\rm\bf B$. Two such pairs are equivalent, $(A,b) \sim (A',b')$, if and only if there is an equivalence $g: A' \cong A$ in $\rm\bf A$ with $\lambda g = b^{-1} b'$. The classes of this equivalence relation form the class of $\lambda$--realizations of $B$,
\begin{equation}
{\rm Real}_{\lambda}(B) = \{ (A,b) \ | \ b: \lambda A \cong B \} / \sim.
\end{equation}
We say that $B$ is $\lambda$--realizable if ${\rm Real}_{\lambda}(B)$ is non--empty. The functor $\lambda: {\rm\bf A} \rightarrow {\rm\bf B}$ is \emph{representative} if all objects $B$ in $\rm\bf B$ are $\lambda$--realizable. Further, we say that $\lambda$ \emph{reflects isomorphisms}, if a morphism $f$ in $\rm\bf A$ is an equivalence whenever $\lambda(f)$ is an equivalence in $\rm\bf B$. The functor $\lambda$ is \emph{full} if, for every morphism $\overline f: \lambda(A) \rightarrow \lambda(A')$ in $\rm\bf B$, there is a morphism $f: A \rightarrow A'$ in $\rm\bf A$, such that $\lambda(f) = \overline f$, we then say $\overline f$ is \emph{$\lambda$--realizable}.

\section{$\rm PD$--chain complexes and $\rm PD$--complexes}\label{PDcomp}
We start this section with a description of the cap product on chain complexes. We fix a homomorphism $\omega: \pi \rightarrow {\mathbb{Z}} / 2 {\mathbb{Z}} = \{0,1\}$ which gives rise to the anti--isomorphism $\overline{\phantom{x}}: {{\Lambda}} \rightarrow {{\Lambda}}$ of rings defined by $\overline g = (-1)^{\omega(g)} g^{-1}$ for $g \in \pi$. To the left $\Lambda$--module $M$ we associate the right $\Lambda$--module $M^{\omega}$ with the same underlying abelian group and action given by $\lambda.a = a.\overline{\lambda}$ for $a \in A$ and $\lambda \in \Lambda$. Proceeding analogously for a right $\Lambda$--module $N$, we obtain a left $\Lambda$--module $^{\omega}N$. We put
\begin{equation*}
{\rm H}_n(C,M^{\omega}) = {\rm H}_n(M^{\omega} \otimes_{\Lambda} C); \quad
{\rm H}^k(C,M) = {\rm H}_{-k}(\mbox{Hom}_{\Lambda}(C,M)).\]
To define the $\omega$--twisted cap product $\cap$ for a chain complex $C$ in ${\rm\bf H}_0$ with diagonal $(\delta, \Delta)$, write $\Delta(c) = \sum_{i+j=n, \alpha} c_{i, \alpha}' \otimes c_{j, \alpha}''$ for $c \in C$. Then
\begin{eqnarray*}
\cap: \mbox{Hom}_{\Lambda}(C, M)_{-k} \otimes_{\mathbb Z}(\mathbb Z^{\omega} \otimes_{\Lambda} C)_{n} & \rightarrow & (M^{\omega} \otimes_{\Lambda} C)_{n-k} \\
\psi \otimes(z \otimes c) & \mapsto & \sum_{\alpha} z \psi(c'_{k, \alpha}) \otimes c''_{n-k, \alpha}
\end{eqnarray*}
for every left $\Lambda$--module $M$. Passing to homology and composing with
\begin{gather*}
{\rm H}^{\ast} (C, M) \otimes_{\mathbb Z} {\rm H}_{\ast} (C \otimes_{\mathbb Z} C, \mathbb Z^{\omega}) \rightarrow {\rm H}_{\ast} (\mbox{Hom}_{\Lambda}(C, M)\big) \otimes_{\mathbb Z} \big( \mathbb Z^{\omega} \otimes_{\Lambda} (C \otimes_{\mathbb Z} C) )), \\
[\psi] {\otimes}[y] \mapsto [\psi {\otimes}y],
\end{gather*}
we obtain
\begin{equation}
\cap: {\rm H}^k(C, M) \otimes_{\mathbb Z}{\rm H}_{n}(C,\mathbb Z^{\omega}) \rightarrow {\rm H}_{n-k}(C, M^{\omega}).
\end{equation}
A \emph{$\rm PD^n$--chain complex} $C = ((\pi, C), \omega, [C], \Delta)$ consists of a free chain complex $(\pi, C)$ in ${\rm\bf H}_0$ with $\pi$ finitely presented, a group homomorphism $\omega: \pi \rightarrow \mathbb Z / 2 \mathbb Z$, a fundamental class $[C] \in {\rm H}_n(C, \mathbb Z^{\omega})$ and a diagonal $\Delta: C \rightarrow C \otimes C$ in ${\rm\bf H}_0$, such that ${\rm H}_1C = 0$ and
\begin{equation}\label{capwithfund}
\cap [C]:   {\rm H}^{r}(C, M) \rightarrow {\rm H}_{n-r}(C, M^{\omega});
\quad \alpha \mapsto \alpha \cap [X]
\end{equation}
is an isomorphism of abelian groups for every $r \in \mathbb Z$ and every left $\Lambda$--module $M$. A morphism of $\rm PD^n$--chain complexes $f: ((\pi, C), \omega, [C], \Delta) \rightarrow ((\pi', C'), \omega', [C'], \Delta')$ is a morphism $(\varphi, f): (\pi, C) \rightarrow (\pi', C')$ in ${\rm\bf H}_0$ such that $\omega = \omega' \varphi$ and $(f \otimes f) \Delta \simeq \Delta' f$. The category ${\rm\bf PD}^n_{\ast}$ is the category of $\rm PD^n$--chain complexes and morphisms between them. Homotopies in ${\rm\bf PD}^n_{\ast}$ are reduced chain homotopies. The subcategory ${\rm\bf PD}^n_{\ast+}$ of ${\rm\bf PD}^n_{\ast}$ is the category consisting of ${\rm PD}^n$--chain complexes and \emph{oriented} or \emph{degree $1$} morphisms of ${\rm PD}^n$--chain complexes, that is, morphisms $f: C \rightarrow D$ with $f_{\ast}[C] = [D]$.
Wall \cite{Wall1} showed that it is enough to demand that (\ref{capwithfund}) is an isomorphism for $M = \Lambda$. If $1 \otimes x \in \mathbb Z^{\omega } \otimes_{\Lambda} C_n$ represents the fundamental class $[C]$, where $C_i$ is finitely generated for $i \in \mathbb Z$, then $\cap [C]$ in (\ref{capwithfund}) is an isomorphism if and only if
\begin{equation}
\cap 1 \otimes x: C^{\ast} = \ ^{\omega}{\rm Hom}_{\Lambda} (C, ^{\omega}\!\!\Lambda)
\rightarrow \Lambda \otimes_{\Lambda} C = C
\end{equation}
is a homotopy equivalence of chain complexes of degree $n$. Here finite generation implies that $C^{\ast}$ is a free chain complex.
\begin{lemma}\label{2real}
Every ${\rm PD}^n$--chain complex is homotopy equivalent in ${\rm\bf PD}^n_{\ast}$ to a $2$--realizable ${\rm PD}^n$--chain complex.
\end{lemma}
\begin{proof}
This follows from Proposition III 2.13 and Theorem III 2.12 in \cite{Baues1}.
\end{proof}
A \emph{$\rm PD^n$--complex} $X = (X, \omega, [X], \Delta)$ consists of an object $X$ in ${\rm\bf CW}_0$ with finitely presented fundamental group $\pi_1(X)$, a group homomorphism $\omega: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z$, a fundamental class $[X] \in {\rm H}_n(X, \mathbb Z^{\omega})$ and a diagonal $\Delta: X \rightarrow X \times X$ in ${\rm\bf CW}_0$, such that $(\widehat CX, \omega, [X], \widehat C \Delta)$ is a $\rm PD^n$--chain complex. A morphism of $\rm PD^n$--complexes $f: (X, \omega, [X], \Delta) \rightarrow (X', \omega', [X'], \Delta')$ is a morphism $f: X \rightarrow X'$ in ${\rm\bf CW}_0$ such that $\omega = \omega' \pi_1(f)$. The category ${\rm\bf PD}^n$ is the category of $\rm PD^n$--complexes and morphisms between them. Homotopies in ${\rm\bf PD}^n$ are homotopies in ${\rm\bf CW}_0$. The subcategory ${\rm\bf PD}^n_+$ of ${\rm\bf PD}^n$ is the category consisting of ${\rm PD}^n$--complexes and \emph{oriented} or \emph{degree $1$} morphisms of ${\rm PD}^n$--complexes, that is, morphisms $f: X \rightarrow Y$ with $f_{\ast}[X] = [Y]$.

\begin{remark}
Our ${\rm PD}^n$--complexes have finitely presented fundamental groups by definition and are thus finitely dominated by Propostion 1.1 in \cite{Wall3}.
\end{remark}

Let $X$ be a ${\rm PD}^n$--complex with $n \geq 3$. We say that $X$ is \emph{standard}, if $X$ is a $\rm CW$--complex which is $n$--dimensional and has exactly one $n$--cell $e^n$. We say that $X$ is \emph{weakly standard}, if $X$ has a subcomplex $X'$ with $X = X' \cup e^n$, where $X'$ is $n$--dimensional and satisfies ${\rm H}^n(X', B) = 0$ for all coefficient modules $B$. In this sense $X'$ is homologically $(n-1)$--dimensional. Of course standard implies weakly standard with $X' = X^{n-1}$.

\begin{remark*}
Every compact connected manifold $M$ of dimension $n$ has the homotopy type of a finite standard $\rm PD^n$--complex.
\end{remark*}

\begin{remark}\label{Wallres}
Wall's Theorem 2.4 in \cite{Wall1} and Theorem E in \cite{Wall2} imply that, for $n \geq 4$, every ${\rm PD}^n$--complex is homotopy equivalent to a standard ${\rm PD}^n$--complex and, for $n = 3$, every ${\rm PD}^3$--complex is homotopy equivalent to a weakly standard ${\rm PD}^3$--complex.
\end{remark}

Let $C$ be a ${\rm PD}^n$--chain complex with $n \geq 3$. We say that $C$ is \emph{standard}, if $C$ is $2$--realizable, $C_i = 0$ for $i > n$, and $C_n = \Lambda[e_n]$, where $[e_n] \in C_n$. We say that $C$ is \emph{weakly standard}, if $C$ is $2$--realizable and has a subcomplex $C'$ with $C = C' \oplus \Lambda[e_n]$, where $C'$ is $n$--dimensional and satisfies ${\rm H}^n(C', B) = 0$ for all coefficient modules $B$.

\begin{remark}\label{corr}
A ${\rm PD}^n$--complex, $X$, is homotopy equivalent to a finite standard, standard or weakly standard ${\rm PD}^n$--complex, respectively, if and only if the ${\rm PD}^n$--chain complex $\widehat CX$ is homotopy equivalent to a finite standard, standard  or weakly standard ${\rm PD}^n$--chain complex, respectively.
\end{remark}

\section{Fundamental triples}\label{triplesection}
Homotopy types of $3$--manifolds and ${\rm PD}^3$--complexes were considered by Thomas \cite{Thomas}, Swarup \cite{Swarup} and Hendriks \cite{Hendriks}. In particular, Hendriks and Swarup provided a criterion for the existence of degree $1$ maps between $3$--manifolds and ${\rm PD}^3$--complexes, respectively. In this section we generalize these results to manifolds and Poincar{\'e} duality complexes of arbitrary dimension $n$.

Let $k$--{\bf types} be the full subcategory of ${\rm\bf CW}_0 /\simeq$ consisting of $\rm CW$--complexes $X$ in ${\rm\bf CW}_0$ with $\pi_i(X) = 0$ for $i > k$. The $k$--th \emph{Postnikov functor}
\begin{equation*}
P_k: {\rm\bf CW}_0 \rightarrow k{\rm\bf-types}
\end{equation*}
is defined as follows. For $X$ in ${\rm\bf CW}_0$ we obtain $P_kX$ by ``killing homotopy groups'', that is, we choose a $\rm CW$--complex $P_kX$ with $(k+1)$--skeleton $(P_kX)^{k+1} = X^{k+1}$ and $\pi_i(P_kX) = 0$ for $i > k$. For a morphism $f: X \rightarrow Y$ in ${\rm\bf CW}_0$ we may choose a map $Pf: P_kX \rightarrow P_kY$ which extends the restriction $f^{k+1}: X^{k+1} \rightarrow Y^{k+1}$ as $\pi_i(P_kY) = 0$ for $i > k$. Then the functor $P_k$ assigns $P_kX$ to $X$ and the homotopy class of $Pf$ to $f$. Different choices for $P_kX$ yield canonically isomorphic functors $P_k$. The $\rm CW$--complex $P_1X = K(\pi_1X,1)$ is an Eilenberg--{Mac\,Lane} space and, as a functor, $P_1$ is equivalent to the functor $\pi_1$ of fundamental groups. There are natural maps
\begin{equation}\label{postnikov}
p_k: X \longrightarrow P_kX
\end{equation}
in ${\rm\bf CW}_0/\simeq$ extending the inclusion $X^{k+1} \subseteq P_kX$.

For $n \geq 3$, a \emph{fundamental triple} $T = (X, \omega, t)$ \emph{of formal dimension $n$} consists of an $(n-2)$--type $X$, a homomorphism $\omega: \pi_1X \rightarrow \mathbb Z / 2 \mathbb Z$ and an element $t \in {\rm H}_n(X, \mathbb Z^{\omega})$. A morphism $(X, \omega_X, t_X) \rightarrow (Y, \omega_Y, t_Y)$ between  fundamental triples is a homotopy class $\{f\}: X \rightarrow Y$ of maps of the $(n-2)$--types, such that $\omega_X = \omega_Y \pi_1(f)$ and $f_{\ast}(t_X) = t_Y$. We obtain the category ${\rm\bf Trp}^n$ of fundamental triples $T$ of formal dimension $n$ and the functor
\begin{equation*}
\tau: {\rm\bf PD}^n_+ /\simeq \ \longrightarrow {\rm\bf Trp}^n, \quad X \longmapsto (P_{n-2}X, \omega_X, p_{n-2 \ast}[X]).
\end{equation*}
Every degree $1$ morphism $Y \rightarrow X$ in ${\rm\bf PD}^n_+$ induces a surjection $\pi_1Y \rightarrow \pi_1X$ on fundamental groups, see for example \cite{Browder1}, and hence we introduce the subcategory ${\rm\bf Trp}^n_+ \subset {\rm\bf Trp}^n$ consisting of all morphisms inducing surjections on fundamental groups. Then the functor $\tau$ yields the functor
\begin{equation}\label{tauplus}
\tau_+: {\rm\bf PD}^n_+ /\simeq \ \longrightarrow {\rm\bf Trp}^n_+.
\end{equation}
As a main result in this section we show
\begin{theorem}\label{mainthm}
The functor $\tau_+$ reflects isomorphisms and is full  for $n \geq 3$.
\end{theorem}
As corollaries we mention
\begin{corollary}\label{class}
Take $n \geq 3$. Two $n$--dimensional manifolds, respectively, two ${\rm PD}^n$--complexes, are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic.
\end{corollary}
\begin{remark*}
For $n = 3$, Corollary \ref{class} yields the results by Thomas \cite{Thomas}, Swarup \cite{Swarup} and Hendriks \cite{Hendriks}. Turaev reproves Hendriks' result in the appendix of \cite{Turaev}, although the proof needs further explanation. We reprove the result again in a more algebraic way.
\end{remark*}
\begin{remark*}
Turaev conjectures in \cite{Turaev} that his proof for $n = 3$ has a generalization to ${\rm PD}^n$--complexes whose $(n-2)$--type is an Eilenberg--{Mac\,Lane} space $K(\pi,1)$. Corollary \ref{class} proves this conjecture.
\end{remark*}
Next consider ${\rm PD}^n$--complexes $X$ and $Y$ and a diagram
\begin{equation}\label{overfex}
\xymatrix{
Y \ar[r]^-{p_{n-2}} \ar@{..>}[d]_{\overline f} & P_{n-2}Y \ar[d]^f \\
X \ar[r]^-{p_{n-2}} & P_{n-2}X.}
\end{equation}
\begin{corollary}\label{degoneex}
For $n \geq 3$, there is a degree $1$ map $\overline f$ rendering Diagram (\ref{overfex}) homotopy commutative if and only if $f$ induces a surjection on fundamental groups, is compatible with the orientations $\omega_X$ and $\omega_Y$, that is, $\omega_X \pi_1(f) = \omega_Y$, and
\begin{equation*}
f_{\ast}p_{n-2 \ast}[Y] = p_{n-2 \ast}[X].
\end{equation*}
\end{corollary}
\begin{remark*}
Swarup \cite{Swarup} and Hendriks \cite{Hendriks} prove Corollary \ref{degoneex} for $3$--manifolds and ${\rm PD}^3$--complexes, respectively.
\end{remark*}
\begin{remark*}
For oriented ${\rm PD}^4$--complexes with finite fundamental group and $f$ a homotopy equivalence, the map $\overline f$ corresponds to the map $h$ in Lemma 1.3 \cite{HamblKreck} of Hambleton and Kreck. The reader is invited to compare our proof to that of Lemma 1.3 \cite{HamblKreck} which shows the existence of $h$ but not the fact that $h$ is of degree 1.
\end{remark*}
By Remark \ref{Wallres}, Theorem \ref{mainthm} is a consequence of the following Lemmata \ref{tauplusrefl} and \ref{mainlem}.
\begin{lemma}\label{tauplusrefl}
The functor $\tau_+$ reflects isomorphisms.
\end{lemma}
\begin{proof}
This is a consequence of Poincar{\'e} duality and Whitehead's Theorem.
\end{proof}
\begin{remark*}
For $n \geq 3$, let $[\frac{n}{2}]$ be the largest integer $\leq n$. Associating with a ${\rm PD}^n$--complex, $X$, the \emph{pre-fundamental triple} $(P_{[\frac{n}{2}]}X, \omega_X, p_{[\frac{n}{2}]\ast}[X])$, there is an analogue of  Lemma \ref{tauplusrefl}, namely, an orientation preserving map between ${\rm PD}^n$--complexes is a homotopy equivalence if and only if the induced map between pre-fundamental triples is an isomorphism. However, pre-fundamental triples do not determine the homotopy type of a ${\rm PD}^n$--complex as in Corollary \ref{class}, which is demonstrated by the fake products $X = (S^n \vee S^n) \cup_{\alpha} e^{2n}$, where $\alpha$ is the sum of the Whitehead product $[\iota_1 , \iota_2]$ and an element $\iota_1 \beta$ with $\beta \in \pi_{2n-1}(S^n)$ having trivial Hopf invariant. Pre-fundamental triples coincide with the fundamental triple for $n = 3$ and $n = 4$. It remains an open problem to enrich the structure of a pre-fundamental triple to obtain an analogue of Corollary \ref{class}.
\end{remark*}
\begin{lemma}\label{mainlem}
Let $X$ and $Y$ be standard ${\rm PD}^n$--complexes for $n \geq 4$ and weakly standard for $n = 3$ and let $f: \tau_+ Y \rightarrow \tau_+ X$ be a morphism in ${\rm\bf Trp}^n_+$.
Then $f$ is $\tau$--realizable by a map $\overline f: Y \rightarrow X$ in ${\rm PD}^n_+$ with $\tau \overline f = f$.
\end{lemma}
For the proof of Lemma \ref{mainlem}, we use
\begin{lemma}\label{abargenerates}
Let $X = X' \cup e^n$ be a weakly standard ${\rm PD}^n$--complex. Then there is a generator $[e] \in \widehat C_n(X)$, with $\widehat C_nX = \widehat C_nX' \oplus \Lambda[e]$, corresponding to the cell $e^n$, such that $1 \otimes [e] \in \mathbb Z^{\omega} \otimes_{\Lambda} \widehat C_n X$ is a cycle representing the fundamental class $[X]$. Let $\{ e_m \}_{m \in M}$ be a basis of $\widehat C_{n-1} X = \widehat C_{n-1} X'$. Then the coefficients $\{ a_m \}_{m \in M}, a_m \in \Lambda$ for $m \in M$, of the linear combination $d_n[e] = \sum a_m [e_m]$   generate $\overline{I(\pi_1X)}$ as a right $\Lambda$--module.
\end{lemma}
\begin{proof}
Poincar{\'e} duality implies ${\rm H}_n(X, \mathbb Z^{\omega}) \cong H^0(X, \mathbb Z) \cong \mathbb Z$. Hence $1 \otimes d$ maps a multiple of the generator $1 \otimes [e]$ of $\mathbb Z^{\omega} \otimes_{\Lambda} \widehat C_n(X) = \mathbb Z^{\omega} \otimes_{\Lambda} \Lambda [e] \cong \mathbb Z$ to zero, that is, there is an $n \in \mathbb N$ such that
\begin{eqnarray*}
0
& = & 1 \otimes d(n(1 \otimes [e])) = n (1 \otimes d[e])
   = n (1 \otimes \sum_{m \in M} a_m [e_m]) \\
& = & n \sum 1.a_{m} \otimes [e_m] = n \sum_{m \in M} {\rm aug}(\overline{a_m}) \otimes [e_m].
\end{eqnarray*}
Since $\mathbb Z^{\omega} \otimes_{\Lambda}D_{n-1} = \mathbb Z^{\omega} \otimes_{\Lambda}\bigoplus_{m \in M} \Lambda [e_{m}] \cong \bigoplus_{m \in M} \mathbb Z^{\omega} \otimes_{\Lambda}\Lambda [e_{m}] = \bigoplus_{m \in M} \mathbb Z$ is free as abelian group, this implies ${\rm aug}(\overline{a_{m}}) = 0$ and hence $\overline{a_m} \in I$ for every $m \in M$. Therefore
$1 \otimes d (1 \otimes [e]) = 0$ and $1 \otimes [e] \in \mathbb Z^{\omega} \otimes_{\Lambda}D_n$
is a cycle representing a generator of the group ${\rm H}_n(X, \mathbb Z^{\omega})$. Without loss of generality we may assume that $e$ is oriented such that $1 \otimes e$ represents the fundamental class $[X]$. Further, Poincar{\'e} duality implies ${\rm H}^n(X, ^{\omega}\!\! \Lambda) \cong \mathbb Z$ and hence $I(\pi) \cong {\rm im}(d^{\ast})[e]$. But, for every $\varphi \in ^{\omega}\!{\rm Hom}_{\Lambda}(C_{n-1}, ^{\omega}\! \Lambda)$,
\begin{equation*}
\big(d^{\ast} \varphi \big) [e] = \varphi (d[e]) = \varphi \big(\sum a_m [e_m] \big) = \sum a_m \varphi [e_m] = \big(\sum\overline{\varphi [e_m]} \overline a_m [e]^{\ast}\big)[e],
\end{equation*}
where $[e]^{\ast}: {{\Lambda}}[e] \rightarrow {{\Lambda}}, [e] \mapsto 1$. Thus $I(\pi)$ is generated by $\{ \overline a_m \}_{m \in M}$ as a left $\Lambda$--module and hence $\overline{I(\pi)}$ is generated by $\{ a_m \}_{m \in M}$ as a right $\Lambda$--module
\end{proof}
\begin{lemma}\label{split}
Let $\overline X = X' \cup_f e^3$ be a weakly standard ${\rm PD}^3$--complex. Then we can choose a homotopy $f \simeq g$ such that $X = X' \cup_g e^3$ admits a splitting $\widehat C_2X = S \oplus d_3(\widehat C_3X')$ as a direct sum of $\Lambda$--modules satisfying $d_3[e] \in S$.
\end{lemma}
\begin{proof}
As $X'$ is homologically $2$--dimensional, $\widehat C(\overline X)$ admits a splitting,
\begin{equation*}
\widehat C_2(\overline X) = {\rm im} d'_3 \oplus S,
\end{equation*}
as a direct sum of $\Lambda$--modules, where $d_3': \widehat C_3(X') \rightarrow \widehat C_2(X')$. Thus $d_3[e] \in \widehat C_2(\overline X) = {\rm im} d'_3 \oplus S$ decomposes as a sum $d_3[e] = \alpha + \beta$ with $\alpha \in {\rm im} d'_3$ and $\beta \in S$. Since $\alpha$, viewed as a map $S^2 \rightarrow X'$, is homotopically trivial in $X'$, there is a homotopy $f \simeq g$, where $g$ represents $\beta$, such that $X = X' \cup_g e^3$ has the stated properties.
\end{proof}

\begin{proof}[Proof of Lemma \ref{mainlem}]
Certain aspects of the proof for the case $n = 3$ differ from that for the case $n \geq 4$. Those parts of the proof pertaining to the case $n = 3$ appear in square brackets [ \ldots ]. [For $n = 3$ we assume that $X = X' \cup_g e^3$ is chosen as in Lemma \ref{split}.]

Given $X = X' \cup_g e^n$ and $Y = Y' \cup_{g'} {e'}^n$ and a morphism $\varphi = \{f\}: \tau(Y) = (P, \omega_Y, t_Y) \rightarrow \tau(X) = (Q, \omega_X, t_X)$ in ${\rm\bf Trp}^n$, the diagram
\begin{equation*}
\xymatrix{
\ \  \ \ \ \ X^{n-1} \subseteq X' \subset X \ar[r]^-p & P = P_{n-2}X \\
\ \ \ \ \ \ \ Y^{n-1} \subseteq Y' \ar[u]^{\overline \eta} \subset Y \ar[r]^-{p'} & Q = P_{n-2}Y, \ar[u]_f}
\end{equation*}
commutes in ${\rm\bf CW}_0$, where $p$ and $p'$ coincide with the identity morphisms on the $(n-1)$--skeleta, and where $\overline \eta$ is the restriction of $f$. For $n \geq 4$, we have $X' = X^{n-1}$ and $Y' = Y^{n-1}$. We obtain the following commutative diagram of chain complexes in ${\rm\bf H}_0$
\begin{equation*}
\xymatrix{
\ \ \ \ \ \ \ \widehat CX^{n-1} \subset \widehat CX \ar[r]^-{p_{\ast}} & \widehat CP \\
\ \ \ \ \ \ \ \widehat CY^{n-1} \ar[u]^{\overline \eta_{\ast}} \subset \widehat CY \ar[r]^-{p'_{\ast}} & \widehat CQ. \ar[u]_{f_{\ast}}}
\end{equation*}
For $n \geq 4$, we construct a morphism $(\xi, \eta): r(Y) \rightarrow r(X)$ in the category ${\rm\bf H}_{n-1}^c$ of homotopy systems of order $(n-1)$ (see Section \ref{homsys}), rendering the diagram
\begin{equation}\label{homcom}
\xymatrix{
r(X)  \ar[r]^-{r(p)} & r(P) \\
r(Y) \ar[r]^-{r(p')} \ar[u]^-{(\xi, \eta)} & r(Q) \ar[u]_-{r(f)}}
\end{equation}
homotopy commutative in ${\rm\bf H}_{n-1}^c$. Here $\xi: \widehat CY \rightarrow \widehat CX$ and $\eta: Y^{n-2} \rightarrow X^{n-2}$ is the restriction of $\overline \eta$ above.

[For $n = 3$, the map $\overline \eta$ itself need not extend to a map $Y' \rightarrow X'$. But, since $Y'$ is homologically $2$--dimensional, there is a map $\eta': Y' \rightarrow X'$ inducing $\pi_1\eta' = \pi_1\varphi$. Since we may assume that $Q$ is obtained from $Y$ by attaching cells of dimension $\geq 3$, we can choose $f$ representing $\varphi$ with $p\eta' = fp'$.]

We write $\pi = \pi_1X, \pi' = \pi_1Y,  \Lambda = \mathbb Z[\pi]$ and $\Lambda' = \mathbb Z[\pi']$ and let $[e'] \in \widehat C_nY$ and $[e] \in \widehat C_nX$ be the elements corresponding to the $n$--cells $e_n$ and $e'_n$, respectively, $n \geq 3$. Since $\{ f \}$ is a morphism in ${\rm\bf Trp}^n$, we obtain $f_{\ast} p'_{\ast} [Y] = p_{\ast}[X]$ in ${\rm H}_n(P, \mathbb Z^{\omega})$ and hence
\begin{equation*}
f_{\ast}p'_{\ast} [e'] - p_{\ast} [e] \in {\rm im}(d: \widehat C_{n+1}P \rightarrow \widehat C_nP) + \overline{I(\pi)}\widehat C_nP.
\end{equation*}
Thus there are elements $x \in \widehat C_{n+1}P$ and $y \in \overline{I(\pi)}\widehat C_nP$ with
\begin{equation}\label{xydfn}
f_{\ast}p'_{\ast} [e'] - p_{\ast} [e] = dx + y.
\end{equation}
Let $\{ e'_m\}_{m \in M}$ be a basis of $\widehat C_{n-1}Y$. By Lemma \ref{abargenerates},
\begin{equation}\label{de'}
d [e'] = \sum a_m [e'_m],
\end{equation}
for some $a_m \in \Lambda', m \in M$, where $\{ a_m \}_{m \in M}$ generate $\overline{I(\pi')}$ as right $\Lambda'$--module. Since $\varphi = f_{\ast}$ is surjective, $\overline{I(\pi)}$ is generated by $\{ \varphi(a_m) \}_{m \in M}$ as right $\Lambda$--module, and we may write
\begin{equation}\label{zmdfn}
y = \sum_{m \in M} \varphi(a_m) z_m,
\end{equation}
for some $z_m \in \widehat C_nP, m \in M$, since there is a surjection $\bigoplus_{m \in M} \Lambda [m] \twoheadrightarrow \overline{I(\pi)}$ of right $\Lambda$--modules which maps the generator $[m]$ to $\varphi(a_m)$. Then (\ref{xydfn}) implies $d( f_{\ast}p'_{\ast} [e'] - p_{\ast} [e])  = d y = \sum_{m \in M} \varphi(a_m) d z_m$, and hence
\begin{equation}\label{pastde}
p_{\ast} d [e] = \sum_{m \in M} \varphi(a_m) f_{\ast} p'_{\ast} [e_m'] - \sum_{m \in M} \varphi(a_m) d z_m.
\end{equation}
We define the $\varphi$--equivariant homomorphism
\begin{equation}\label{overlinealpha}
\overline \alpha_n: \widehat C_{n-1}Y \rightarrow \widehat C_nP \quad {\rm by} \quad
\overline\alpha_n([e_m']) = -z_m.
\end{equation}
For $n \geq 4$, we define $\xi: \widehat CY \rightarrow \widehat CX$ by $\xi[e'] = [e]$ and
\begin{equation}\label{xidfn}
\xi_i =
\begin{cases}
\widehat C_{n-1}(\overline \eta) + d \overline \alpha_n & {\rm for} \ i = n-1, \\
\widehat C_i(\eta) & {\rm for} \ i < n-1. \\
\end{cases}
\end{equation}
[For $n = 3$ we use the splitting $\widehat C_2Y = S \oplus d_3 \widehat C_3Y'$ in Lemma \ref{split} and define $\xi_i: \widehat C_iY \rightarrow \widehat C_iX$ by $\xi_3[e'] = [e], \xi_3 | \widehat C_3Y' = \widehat C_3 \eta'$, and
\begin{eqnarray*}
&& \xi_2 | S = (\widehat C_2 \eta' + d \overline \alpha_3) | S, \\
&& \xi_2 | d_3 \widehat C_3Y' = \widehat C_2\eta' | d_3 \widehat C_3Y', \\
&&\xi_i = \widehat C_i \eta \quad {\rm for} \ i <2.]
\end{eqnarray*}

To ensure that $\xi$ is a chain map, it is now enough to show that $d\xi[e'] = \xi d[e']$. But, for the injection $\widehat C(p) = p_{\ast}$, we obtain
\begin{eqnarray*}
p_{\ast} \xi d [e']
& = &  p_{\ast} ( \widehat C_{n-1} (\eta') + d \overline \alpha_n) d[e']\\
& = & (p \circ \eta')_{n-1}d[e']+ p_{\ast}\big(d \overline \alpha_n (\sum_{m \in M} a_m [e'_m])\big) \\
& = & (f \circ p')_{n-1} d[e']  + p_{\ast} \sum_{m \in M} \varphi(a_m) d \overline \alpha_n [e'_m] \\
& = & \sum_{m \in M} \varphi(a_m) f_{\ast}p'_{\ast} [e'_m] - p_{\ast} \sum_{m \in M} \varphi(a_m) d z_m \\
& = & p_{\ast} d [e] = p_{\ast} d \xi [e'], \quad {\rm by} \ (\ref{pastde}).
\end{eqnarray*}

[For $n = 3$, Theorem \ref{pd3funcrefisofull} now implies that there is a map $\overline f: Y \rightarrow X$ such that $\widehat C(\overline f) = \xi$. Then $\tau(\overline f) = f$, $\overline f$ is a degree $1$ map and the proof is complete for $n = 3$.]

Now let $n \geq 4$. To check that $(\xi, \eta)$ is a morphism in ${\rm \bf H}_{n-1}^c$, note that the attaching map satisfies the cocycle condition and hence, by its definition, the map $\xi_{n-1}$ commutes with attaching maps in $r(X)$ and $r(Y)$, since $\widehat C_{n-1}\overline \eta$ has this property. We must show that Diagram (\ref{homcom}) is homotopy commutative. But $r(f) = (f_{\ast}, \eta)$ and $r(p) = (p_{\ast}, j), r(p') = (p'_{\ast}, j')$, where $j$ and $j'$ are the identity morphisms on $X^{n-2} = P^{n-2}$ and $Y^{n-2} = Q^{n-2}$, respectively. Hence we must find a homotopy $\alpha: (p_{\ast} \xi, \eta) \simeq (f_{\ast} p'_{\ast}, \eta)$ in ${\rm\bf H}_{n-1}^c$, that is, $\varphi$--equivariant maps
\begin{equation*}
\alpha_{i+1}: \widehat C_iY \rightarrow \widehat C_{i+1}P, \ i \geq n-1,
\end{equation*}
such that
\begin{eqnarray}
&& \{ \eta \} + g_{n-1}\alpha_{n-1} = \{ \eta \}, \label{alphafirstcond}\\
&& (p_{\ast} \xi)_i - (f \circ p')_i = \alpha_i d + d \alpha_{i+1} \quad {\rm for} \quad i \geq n-1, \label{alphaseccond}
\end{eqnarray}
where $g_{n-1}$ is the attaching map of $(n-1)$--cells in $P$. Define $\alpha$ by $\alpha_{n+1}[e'] = - x$, see (\ref{xydfn}), and
\begin{equation}\label{alphadfn}
\alpha_i =
\begin{cases}
\overline \alpha_n & {\rm for} \ i= n, \\
0  & {\rm for} \ i < n.
\end{cases}
\end{equation}
Then $\alpha$ satisfies (\ref{alphafirstcond}) trivially. For $i = n -1$, we obtain
\begin{eqnarray*}
(p_{\ast} \xi)_{n-1} - (f \circ p')_{n-1}
& = & \xi_{n-1} - \widehat C_{n-1}(f) \\
& = & \xi_{n-1} - \widehat C_{n-1}(\overline \eta) \\
& =  & d \alpha_n, \quad {\rm by \ (\ref{xidfn}) \ and \ (\ref{alphadfn}).}
\end{eqnarray*}
For $i = n$, we evaluate (\ref{alphaseccond}) on $[e']$. By (\ref{xydfn}),
\begin{equation*}
(p_{\ast} \xi - f_{\ast}p'_{\ast})[e']
= p_{\ast} [e] - f_{\ast}p'_{\ast}[e'] =  - dx - y.
\end{equation*}
On the other hand,
\begin{eqnarray*}
(d \alpha_{n+1} + \alpha_n d)[e']
& = & d \alpha_{n+1}[e'] + \alpha_n \sum_{m \in M} a_m [e'_m], \quad {\rm by} \ (\ref{de'}), \\
& = & - dx - \sum_{m \in M} \varphi(a_m) z_m, \quad {\rm by} \ (\ref{alphadfn}) \ {\rm and} \ (\ref{overlinealpha}), \\
& = & - dx - y \quad {\rm by} \ (\ref{zmdfn}).
\end{eqnarray*}
Hence $\alpha$ satisfies (\ref{alphaseccond}) and Diagram (\ref{homcom}) is homotopy commutative.

To construct a morphism $\overline f: Y \rightarrow X$ in ${\rm\bf PD}^n_+$ with $\tau(\overline f) = f$, consider the obstruction $\mathcal O(\xi, \eta) \in {\rm H}^n(Y, \Gamma_{n-1}X)$ (see Section \ref{homsys}) and note that $p$ induces an isomorphism $p_{\ast}: \Gamma_{n-1}X \rightarrow \Gamma_{n-1}P$, see II.4.8 \cite{Baues1}. Hence the obstruction for the composite $r(p) (\xi, \eta)$ coincides with $p_{\ast} \mathcal O(\xi, \eta)$, where $p_{\ast}$ is an isomorphism. On the other hand, the obstruction for $r(f) r(p')$ vanishes, since this map is $\lambda$--realizable. Thus, by the homotopy commutativity of (\ref{homcom}), $p_{\ast} \mathcal O(\xi, \eta) = \mathcal O (r(f) r(p')) = 0$, so that $\mathcal O(\xi, \eta) = 0$ and there is a $\lambda$--realization $(\xi, \tilde \eta')$ of $(\xi, \eta)$ in ${\rm\bf H}_n^c$. Since ${\rm H}^{n+1}(Y, \Gamma_nX) = 0$, there is a $\lambda$--realization $(\xi, \overline f)$ of $(\xi, \tilde \eta')$ in ${\rm\bf H}_{n+1}^c$. As $Y = Y^n, X = X^n$ and $\xi$ is compatible with fundamental classes by construction, $\overline f: Y \rightarrow X$ is a degree $1$ map in ${\rm\bf PD}^n_+$ realizing the map $f$ in ${\rm \bf Trp}^n$.

\end{proof}

\section{$\rm PD^3$--complexes}\label{pd3section}
The fundamental triple of a ${\rm PD}^3$--complex consists of a group $\pi$, an orientation $\omega$ and an element $t \in {\rm H}_3(\pi, \mathbb Z^{\omega})$. Here we use the standard fact that the homology of a group $\pi$ coincides with the homology of the corresponding Eilenberg--{Mac\,Lane} space $K(\pi,1)$. In general, it is a difficult problem to actually compute ${\rm H}_3(\pi, \mathbb Z^{\omega})$. The homotopy type of a ${\rm PD}^3$--complex is characterized by its fundamental triple, but not every fundamental triple occurs as the fundamental triple of a ${\rm PD}^3$--complex. Via the invariant $\nu_C(t)$ Turaev \cite{Turaev} characterizes those fundamental triples which are realizable by a ${\rm PD}^3$--complex. Let ${\rm\bf Trp}^3_{+,\nu}$ be the full subcategory of ${\rm\bf Trp}^3_+$ consisting of fundamental triples satisfying Turaev's realization condition. Then Theorem \ref{mainthm} implies

\begin{theorem}\label{tauplus3}
The functor
\begin{equation*}
\tau_+: {\rm\bf PD}^3_+ / \simeq \rightarrow {\rm\bf Trp}^3_{+, \nu}
\end{equation*}
reflects isomorphisms and is representative and full.
\end{theorem}

\begin{remark*}
Turaev does not mention that the functor $\tau_+$ is actually full and thus only proves the first part of the following corollary which is one of the consequences of Theorem \ref{tauplus3}.
\end{remark*}

\begin{corollary}
The functor $\tau_+$ yields a 1--1 correspondence between oriented homotopy types of ${\rm PD}^3$--complexes and isomorphism types of fundamental triples satisfying Turaev's realization condition. Moreover, for every ${\rm PD}^3$--complex $X$, there is a surjection of groups
\begin{equation*}
\tau_+: {\rm Aut}_+(X) \rightarrow {\rm Aut}(\tau(X)),
\end{equation*}
where ${\rm Aut}_+(X)$ is the group of oriented homotopy equivalences of $X$ in ${\rm\bf PD}^3_+ / \simeq$ and ${\rm Aut}(\tau(X))$ is the group of automorphisms of the triple $\tau(X)$ in ${\rm\bf Trp}^3_+$ which is a subgroup of ${\rm Aut}(\pi_1X)$.
\end{corollary}

As every $3$--manifold has the homotopy type of a finite standard ${\rm PD}^3$--complex, the question arises which fundamental triples in ${\rm\bf Trp}^3_+$ correspond to finite standard ${\rm PD}^3$--complexes. While Turaev does not discuss this question, we use the concept of ${\rm PD}^3$--chain complexes (see Section \ref{PDcomp}) in the category ${\rm\bf PD}_{\ast}^3$ to do so.

\begin{theorem}\label{pd3funcrefisofull}
The functor $\widehat C: {\rm\bf PD}^3/\simeq \ \longrightarrow {\rm\bf PD}^3_{\ast}/\simeq$ reflects isomorphisms and is representative and full.
\end{theorem}

\begin{proof}
This follows from Theorems \ref{pdnfuncequ} and \ref{pdnfuncdet} in Section \ref{pdhomsys}.
\end{proof}

\begin{corollary}\label{onetoonepd3}
The functor $\widehat C$ yields a 1--1 correspondence between homotopy types of ${\rm PD}^3$--complexes and homotopy types of ${\rm PD}^3$--chain complexes. Moreover, for every ${\rm PD}^3$--complex $X$ there is a surjection of groups
\begin{equation*}
\widehat C: {\rm Aut}(X) \longrightarrow {\rm Aut}(\widehat C(X)).
\end{equation*}
\end{corollary}

\begin{remark}
Corollary \ref{onetoonepd3} implies that the diagonal of every ${\rm PD}^3$--chain complex is, in fact, homotopy co--associative and homotopy co--commutative.
\end{remark}

Connecting the functor $\widehat C$ and the functor $\tau_+$, we obtain the diagram
\begin{equation*}
\xymatrix{
{\rm\bf PD}^3_+ / \simeq \ar[rr]^{\widehat C} \ar[dr]_{\tau_+} &&
{\rm\bf PD}^3_{\ast+} / \simeq \ar[dl]^{\tau_{\ast}} \\
& {\rm\bf Trp}^3_{+, \nu} &}
\end{equation*}
where $\tau_+$ determines $\tau_{\ast}$ together with a natural isomorphism $\tau_{\ast} \widehat C \cong \tau_+$.

\begin{corollary}
All of the functors $\widehat C, \tau_+$ and $\tau_{\ast}$ reflect isomorphisms and are full and representative.
\end{corollary}

By Remark \ref{corr}, the functor $\widehat C$ yields a  1--1 correspondence between homotopy types of finite standard ${\rm PD}^3$--complexes and finite standard ${\rm PD}^3$--chain complexes, respectively.

\section{Realizability of $\rm PD^4$--chain complexes}\label{4chainrealize}
Given a ${\rm PD}^4$--chain complex $C$, we define an invariant $\mathcal O (C)$ which vanishes if and only if $C$ is realizable by a ${\rm PD}^4$--complex. To this end we recall the \emph{quadratic functor $\Gamma$} (see also (4.1) p. 13 in \cite{Baues1}). A function $f: A \rightarrow B$ between abelian groups is called a \emph{quadratic map} if $f(-a) = f(a)$, for $a \in A$, and if the function $A \times A \rightarrow B, (a,b) \mapsto f(a+b) - f(a) - f(b)$ is bilinear. There is a \emph{universal quadratic map}
\begin{equation*}
\gamma: A \rightarrow \Gamma(A),
\end{equation*}
such that for all quadratic maps $f: A \rightarrow B$ there is a unique homomorphism $f^{\square}: \Gamma(A) \rightarrow B$ satisfying $f^{\square} \gamma = f$. Using the cross effect of $\gamma$, we obtain the \emph{Whitehead product map}
\begin{eqnarray*}
P: A \otimes A & \longrightarrow & \Gamma(A), \\
a \otimes b & \longmapsto & [a,b] = \gamma(a+b) - \gamma(a) - \gamma(b).
\end{eqnarray*}
The \emph{exterior product $\Lambda^2A$} of the abelian group $A$ is defined so that we obtain the natural exact sequence
\begin{equation}\label{themapH}
\Gamma(A) \stackrel{H}{\longrightarrow} A \otimes A \longrightarrow \Lambda^2 A \longrightarrow 0,
\end{equation}
where $H$ maps $\gamma(a)$ to $a \otimes a$ for $a \in A$ (see also p.14 in \cite{Baues1}). The composite $P H: \Gamma(A) \rightarrow \Gamma(A)$ coincides with $2{\rm id}_{\Gamma(A)}$, in fact, $P H$ maps $\gamma(a)$ to $[a,a] = 2\gamma(a)$. Given a ${\rm CW}$--complex $X$, there is a natural isomorphism $\Gamma_3(X) \cong \Gamma(\pi_2X)$, by an old result of J.H.C. Whitehead \cite{Whitehead2}, where $\Gamma_3$ is \emph{Whitehead's functor} in \emph{A Certain Exact Sequence} \cite{Whitehead2}.

\begin{theorem}\label{pd4real}
Let $C = ( (\pi, C), \omega, [C], \Delta)$ be a ${\rm PD}^4$--chain complex with homology module ${\rm H}_2(C, \Lambda) = H_2$. Then there is an invariant
\begin{equation*}
\mathcal O(C) \in {\rm H}_0(\pi, \Lambda^2H_2^{\omega})
\end{equation*}
with $\mathcal O(C) = 0$ if and only if there is a ${\rm PD}^4$--complex $X$ such that $\widehat C(X)$ is isomorphic to $C$ in ${\rm\bf PD}^4_{\ast} / \simeq$. Moreover, if $\mathcal O(C) = 0$, the group
\begin{equation*}
\ker \big(H_{\ast}: {\rm H}_0(\pi, \Gamma(H_2^{\omega}) \longrightarrow {\rm H}_0(\pi, H_2^{\omega} \otimes H_2^{\omega}) \big)
\end{equation*}
acts transitively and effectively on the set ${\rm Real}_{\widehat C}(C)$ of realizations of $C$ in ${\rm\bf PD}^4 / \simeq$. Here $\ker H_{\ast}$ is $2$--torsion.
\end{theorem}

\begin{proof}
First note that
\begin{equation}\label{isos}
{\rm H}^4(C, \Lambda^2 H_2) \cong {\rm H}_0(C, \Lambda^2 H_2^{\omega}) \cong {\rm H}_0(\pi, \Lambda^2 H_2^{\omega}).
\end{equation}
By Lemma \ref{2real}, we may assume that $C$ is $2$--realizable. By Proposition \ref{objrealize}, there is thus a $4$--dimensional ${\rm CW}$--complex $X$ together with an isomorphism $\widehat CX \cong (\pi, C)$. The ${\rm CW}$--complex $X$ yields the homotopy systems $\overline{\overline X}$ in ${\rm\bf H}_3^c$ and $\overline X$ in ${\rm\bf H}_4^c$ with $\overline X = r(X)$ and $\overline{\overline X} = \lambda X$. By Theorem \ref{pdnfuncequ}, we may choose a diagonal $\overline{\overline \Delta}: \overline{\overline X} \rightarrow \overline{\overline X} \otimes \overline{\overline X}$ inducing $\Delta: C \rightarrow C \otimes C$, whose homotopy class is determined by $\Delta$. However, $\overline{\overline \Delta}$ need not be $\lambda$--realizable. Lemma \ref{diagreal0} shows that there is an obstruction
\begin{equation}
\mathcal O' = \mathcal O_{\overline X, \overline X \otimes \overline X} (\overline{\overline \Delta})\in {\rm H}^4(C, \Gamma_3(\overline X \otimes \overline X))
\end{equation}
which vanishes if and only if there is a diagonal $\overline \Delta: \overline X \rightarrow \overline X \otimes \overline X$ realizing $\overline{\overline \Delta}$. Note that $\mathcal O'$ is determined by the diagonal $\Delta$ on $C$, since the obstruction depends on the homotopy class of $\overline{\overline \Delta}$ only. By Theorem \ref{pdnfuncdet}, the existence of  $\overline \Delta$ realizing $\overline{\overline \Delta}$ also implies the existence of $\Delta: X \rightarrow X \times X$ realizing $\overline \Delta$. But
\begin{eqnarray*}
\Gamma_3(\overline X \otimes \overline X)
& \cong & \Gamma(\pi_2(\overline X \otimes \overline X)) \\
& \cong & \Gamma(\pi_2(X \times X)) \\
& \cong & \Gamma(\pi_2 \oplus \pi_2) \quad {\rm where} \ \pi_2 = \pi_2X.
\end{eqnarray*}
Applying Lemma \ref{diagreal1} (1), we see that
\begin{equation*}
\mathcal O' \in \ker p_{i\ast} \quad {\rm for} \ i = 1, 2,
\end{equation*}
where $p_i: \pi_2 \oplus \pi_2 \rightarrow \pi_2$ is the $i$--th projection. Now
\begin{equation*}
\Gamma(\pi_2 \oplus \pi_2) = \Gamma(\pi_2) \oplus \pi_2 \otimes \pi_2 \oplus \Gamma(\pi_2)
\end{equation*}
and hence $\mathcal O'$ yields $\mathcal O'' \in {\rm H}^4(C, \pi_2 \otimes \pi_2)$. While the homotopy type of $\overline{\overline X}$ is determined by $C$, the homtopy type of $\overline X$ is an element of ${\rm Real}_{\lambda}(\overline {\overline X})$ and the group ${\rm H}^4(C, \Gamma(\pi_2))$ acts transitively and effectively on this set of realizations. To describe the behaviour of the obstruction under this action using Lemma \ref{obsact}, we first consider the homomorphism
\begin{equation*}
\nabla = \Delta_{\ast} - \iota_{1\ast} - \iota_{2\ast}: \Gamma(\pi_2) \longrightarrow \Gamma(\pi_2 \oplus \pi_2),
\end{equation*}
where $\Delta: \pi_2 \rightarrow \pi_2 \oplus \pi_2$ maps $x \in \pi_2$ to $\iota_1(x) + \iota_2(x)$. We obtain, for $x \in \pi_2$,
\begin{eqnarray*}
\nabla(\gamma(x))
& = & \gamma(\iota_1(x) + \iota_2(x)) - \gamma(\iota_1(x)) - \gamma(\iota_2(x)) \\
& = & [ \iota_1(x) , \iota_2(x) ] \\
& = & x \otimes x \in \pi_2 \otimes \pi_2 \subset \Gamma(\pi_2 \oplus \pi_2),
\end{eqnarray*}
showing that $\nabla$ coincides with $H: \Gamma(\pi_2) \rightarrow \pi_2 \otimes \pi_2.$ Given $\alpha \in {\rm H}^4(C, \Gamma(\pi_2))$, the obstruction $\mathcal O''_{\alpha} = \mathcal O_{\overline Y, \overline Y \otimes \overline Y}(\overline{\overline \Delta})$ with $\overline Y = \overline X + \alpha$ satisfies
\begin{equation*}
\mathcal O''_{\alpha} = \mathcal O'' + H_{\ast}\alpha,
\end{equation*}
by Lemma \ref{obsact}. The exact sequence
\begin{equation*}
{\rm H}^4(C, \Gamma(\pi_2)) \longrightarrow {\rm H}^4(C, \pi_2 \otimes \pi_2) \longrightarrow {\rm H}^4(C, \Lambda^2\pi_2) \longrightarrow 0
\end{equation*}
allows us to identify the coset of ${\rm im} H_{\ast}$ represented by $\mathcal O''$ with an element
\begin{equation*}
\mathcal O \in {\rm H}^4(C, \Lambda^2H_2),
\end{equation*}
where $H_2 = {\rm H}_2(C, \Lambda) \cong \pi_2$. By the isomorphisms (\ref{isos}), this element yields the invariant
\begin{equation*}
\mathcal O \in {\rm H}_0(\pi, \Lambda^2H_2^{\omega})
\end{equation*}
with the properties stated. Given that $\mathcal O''$ vanishes, the obstruction $\mathcal O_{\alpha}''$ vanishes if and only if $\alpha \in \ker H_{\ast}$, and Proposition \ref{objrealize} yields the result on ${\rm Real}_{\widehat C}(C)$. We observe that $\ker H_{\ast}$ is $2$--torsion as $H_{\ast}(x) = 0$ implies $2x = P_{\ast}H_{\ast} x = 0$.
\end{proof}

\begin{theorem}
Let $C = ((\pi, C), \omega, [C], \Delta)$ be a ${\rm PD}^4$--chain complex for which $\Delta$ is homotopy co-commutative. Then the obstruction $\mathcal O(C)$ is $2$--torsion, that is, $2 \mathcal O(C) = 0$.
\end{theorem}

\begin{proof}
Lemma \ref{diagreal1} (2) states
\begin{equation*}
\mathcal O' \in \ker ({\rm id}_{\ast} - T_{\ast})_{\ast},
\end{equation*}
where ${\rm id}$ is the identity on $\pi_2 \oplus \pi_2$ and $T$ is the interchange map on $\pi_2 \oplus \pi_2$ with $T \iota_1 = \iota_2$ and $T \iota_2 = \iota_1$. Thus $T$ induces the map $-{\rm id}$ on $\Lambda^2 \pi_2$ and the result follows.
\end{proof}

\begin{remark*}
Lemma \ref{diagreal1} (3) concerning homotopy associativity of the diagonal does not yield a restriction of the invariant $\mathcal O(C)$.
\end{remark*}

\begin{theorem}\label{finodd}
The functor $\widehat C$ induces a 1--1 correspondence between homotopy types of ${\rm PD}^4$--complexes with finite fundamental group of odd order and homotoppy types of ${\rm PD}^4$--chain complexes with homotopy co--commutative diagonal and finite fundamental group of odd order.
\end{theorem}

\begin{proof}
Since $\pi$ is of odd order, the cohomology ${\rm H}^0(\pi, M)$ is odd torsion and the result follows from Theorem \ref{pd4real}.
\end{proof}

\begin{remark*}
By Theorem \ref{finodd}, every ${\rm PD}^4$--chain complex with homotopy co--com-mutative diagonal and odd fundamental group has a homotopy co--associative diagonal.
\end{remark*}

Up to $2$--torsion, Theorem \ref{pd4real} yields a correspondence between homotopy types of ${\rm PD}^4$--complexes and homotopy types of ${\rm PD}^4$--chain complexes. In Section \ref{algmodPD4} below we provide a precise condition for a ${\rm PD}^4$--chain complex to be realizable by a ${\rm PD}^4$--complex.

\section{The chains of a $2$--type}\label{2typechains}
The fundamental triple of a ${\rm PD}^4$--complex $X$ comprises its $2$--type $T = P_2X$ and an element of the homology ${\rm H}_4(T, \mathbb Z^{\omega})$. To compute ${\rm H}_4(T, \mathbb Z^{\omega})$, we construct a chain complex $P(T)$ which approximates the chain complex $\widehat C(T)$ up to dimension $4$. Our construction uses a presentation of the fundamental group as well as the concepts of \emph{pre--crossed module} and \emph{Peiffer commutator}. To introduce these concepts, we work with right group actions as in \cite{Baues1}, and define $P(T)$ as a chain complex of right $\Lambda$--modules. With any left $\Lambda$--module $M$ we associate a right $\Lambda$--module in the usual way by setting $x.\alpha = \alpha^{-1}.x $, for $\alpha \in \pi$ and $x \in M$, and vice versa.

A \emph{pre--crossed module} is a group homomorphism $\partial: \rho_2 \rightarrow \rho_1$ together with a right action of $\rho_1$ on $\rho_2$, such that
\begin{equation*}
\partial(x^{\alpha}) = - \alpha + \partial x + \alpha \quad {\rm for} \ x \in \rho_2, \alpha \in \rho_1,
\end{equation*}
where we use additive notation for the group law in $\rho_1$ and $\rho_2$, as in \cite{Baues1}. For $x, y \in \rho_2$, the \emph{Peiffer commutator} is given by
\begin{equation*}
\langle x, y \rangle = -x - y + x + y^{\partial x}.
\end{equation*}
A pre--crossed module is a \emph{crossed module}, if all Peiffer commutators vanish. A \emph{map of pre--crossed modules}, $(m,n): \partial \rightarrow \partial'$ is given by a commutative diagram
\begin{equation*}
\xymatrix{
\rho_2 \ar[rr]^m \ar[d]_{\partial} && \rho_2' \ar[d]^{\partial'} \\
\rho_1 \ar[rr]^n && \rho_1'}
\end{equation*}
in the category of groups, where $m$ is $n$--equivariant. Let ${\rm\bf cross}$ be the category of crossed modules and such morphisms. A \emph{weak equivalence} in ${\rm\bf cross}$ is a map $(m,n): \partial \rightarrow \partial'$, which induces isomorphisms ${\rm coker}\partial \cong {\rm coker}\partial'$ and $\ker\partial \cong \ker \partial'$, and we denote the localization of ${\rm\bf cross}$ with respect to weak equivalences by ${\rm\bf Ho(cross)}$. By an old result of Whitehead--{Mac\,Lane}, there is an equivalence of categories
\begin{equation*}
\overline \rho: 2-{\rm\bf{types}} \longrightarrow {\rm\bf Ho(cross)},
\end{equation*}
compare Theorem III 8.2 in \cite{Baues1}. The functor $\overline\rho$ carries a $2$--type $T$ to the crossed module $\partial: \pi_2(T, T^1) \rightarrow \pi_1(T^1).$

A pre--crossed module is \emph{totally free}, if $\rho_1 = \langle E_1 \rangle$ is a free group generated by a set $E_1$ and $\rho_2 = \langle E_2 \times \rho_1 \rangle$ is a free group generated by a free $\rho_1$--set $E_2 \times \rho_1$ with the obvious right action of $\rho_1$. A function $f: E_2 \rightarrow \langle E_1 \rangle$ yields the \emph{associated totally free pre--crossed module} $\partial_f: \rho_2 \rightarrow \rho_1$ with $\partial_f(x) = f(x)$ for $x \in E_2$. Let $Pei_n(\partial_f) \subset \rho_2$ be the subgroup generated by $n$--fold Peiffer commutators and put $\overline \rho_2 = \rho_2 / Pei_2(\partial_f)$. Let ${\rm\bf cross}^=$ be the category whose objects are pairs $(\partial_f, B)$, where $\partial_f$ is a totally free pre--crossed module $\partial_f: \rho_2 \rightarrow \rho_1$ and $B$ is a submodule of $\ker(\partial: \overline\rho_2 \rightarrow \rho_1$). Further, a morphism $m: (\partial_f, B) \rightarrow (\partial_{f'}, B')$ in ${\rm\bf cross}^=$ is a map $\partial_f \rightarrow \partial_{f'}$ which maps $B$ into $B'$. Then there is a functor
\begin{equation*}
q: {\rm\bf cross}^= \longrightarrow {\rm\bf cross} \longrightarrow {\rm\bf Ho(cross)},
\end{equation*}
which assigns to $(\partial_f, B)$ the crossed module $\overline \rho_2 / B \rightarrow \rho_1$, and one can check that $q$ is full and representative. Given any map $g: T \rightarrow T'$ between $2$--types, we may choose a map $\overline{\overline g}: (\partial_f, B) \rightarrow (\partial_{f'}, B')$ in ${\rm\bf cross}^=$ representing the homotopy class of $g$ via the functor $q$ and the equivalence $\overline \rho$. We call $\overline{\overline g}$ a \emph{map associated with} $g$.

Given an action of the group $\pi$ on the group $M$ and a group homomorphism $\varphi: N \rightarrow \pi$, a \emph{$\varphi$--crossed homomorphism} $h: N \rightarrow M$ is a function satisfying
\begin{equation*}
h(x + y) = (h(x))^{\varphi(y)} + h(y) \quad {\rm for} \ x, y \in N.
\end{equation*}
By an old result of Whitehead \cite{Whitehead1}, the totally free crossed module $\overline \rho_2 \rightarrow \rho_1$ enjoys the following properties.

\begin{lemma}\label{whitehead}
Let $X^2$ be a $2$--dimensional ${\rm CW}$--complex in ${\rm\bf CW}_0$ with attaching map of $2$--cells $f: E_2 \rightarrow \langle E_1 \rangle = \pi_1(X^1)$. Then there is a commutative diagram
\begin{equation*}
\xymatrix{
\pi_2(X^2, X^1) \ar[rr]^{\partial} \ar@{=}[d] && \pi_1(X^1) \ar@{=}[d] \\
\overline \rho_2 \ar[rr]^{\partial_f} && \rho_1,}
\end{equation*}
identifying $\partial$ with the totally free crossed module $\partial_f$. Moreover, the abelianization of $\overline \rho_2$ coincides with $\widehat C_2(X^2)$, identifying the kernel of $\partial_f$ with the kernel of $d_2: \widehat C_2(X^2) \rightarrow \widehat C_1(X^2)$, and $\partial_f$ determines the boundary $d_2$ via the commutative diagram
\begin{equation*}
\xymatrix{
\overline \rho_2 \ar[rr]^{\partial} \ar[d]_{h_2} && \rho_1 \ar[d]^{h_1} \\
\widehat C_2(X^2) \ar[rr]^d && \widehat C_1(X^2).}
\end{equation*}
Here $h_2$ is the quotient map and $h_1$ is the $(q: \rho_1 \rightarrow \pi_1(X^2))$--crossed homomorphism which is the identity on the generating set $E_1$. Each map $\partial_f \rightarrow \partial_{f'}$ induces a chain map $\widehat C_2(X^2) \rightarrow \widehat C_2(X'^2)$ where $X^2$ and $X'^2$ are the $2$--dimensional ${\rm CW}$--complexes with attaching maps $f$ and $f'$, respectively.
\end{lemma}

In addition to Lemma \ref{whitehead}, we need the following result on Peiffer commutators, which was originally proved in IV (1.8) of \cite{Baues1} and generalized in a paper with Conduch{\'e} \cite{BauesConduche}.

\begin{lemma}\label{omega}
With the notation in Lemma \ref{whitehead}, there is a short exact sequence
\begin{equation*}
0 \longrightarrow \Gamma(K) \longrightarrow \widehat C_2(X^2) \otimes \widehat C_2(X^2) \stackrel{\omega}{\longrightarrow} Pei_2(\partial_f) / Pei_3(\partial_f) \longrightarrow 0,
\end{equation*}
where $K = \ker d_2 = \pi_2 X^2$ and $\omega$ maps $x \otimes y$ to the Peiffer commutator $\langle \xi, \eta \rangle$ with $\xi, \eta \in \rho_2$ representing $x$ and $y$, respectively.
\end{lemma}

\begin{definition}
Given a $2$--type $T$ in $2$--{\bf{types}}, we define the chain complex $P(T) = P(\partial_f, B)$ as follows. Let $f: E_2 \rightarrow  \langle E_1 \rangle$ be the attaching map of $2$--cells in $T$ and put $C_i = \widehat C_i(T)$. Then the $2$--skeleton of $P(T)$ coincides with $\widehat C(T^2)$, that is, $P_i(T) = C_i$ for $i \leq 2$, and $P_i(T) = 0$ for $i > 4$. To define $P_4(T)$, let $H$ be the map in (\ref{themapH}) and put $B = {\rm im}(d: C_3 \rightarrow C_2)$ and $\nabla_B = B \otimes B + H[B, C_2]$ as a submodule of $C_2 \otimes C_2$. Then $P_4(T)$ is given by the quotient
\begin{equation*}
P_4(T) = C_2 \otimes C_2 / \nabla_B.
\end{equation*}
To define $P_3(T)$, we use Lemma \ref{whitehead}, Lemma \ref{omega} and the identification $\pi_2T^2 = \ker(d: C_2 \rightarrow C_1)$ and put $\sigma_2 = \rho_2 / Pei_3(\partial_f)$. Then $P_3(T)$ is given by the pull--back diagram
\begin{equation*}
\xymatrix{
P_3(T) \ \ar@{>->}[rr] \ar[d]_{\overline d} && \sigma_2 / \omega \nabla_B \ar@{->>}[d]\\
B \ \ar@{>->}[r] & \pi_2T^2 \ \ar@{>->}[r] & \overline \rho_2.}
\end{equation*}
The chain complex $P(T)$ is determined  by the commutative diagram
\begin{equation*}
\xymatrix{
P_4(T) \ar[r]^d \ar@{=}[d] & P_3(T) \ar@{->>}[dr]^{\overline d} \ar[rr] \ar@{>->}[d]&&
P_2(T) \ar[r] \ar@{=}[d] & P_1(T) \ar@{=}[d] \ar[r] & P_0(T) \ar@{=}[d] \\
C_2 \otimes C_2 / \nabla_B \ar[r]^{-\omega} & \sigma_2 / \omega \nabla_B & B \ \ar@{>->}[r] &
C_2 \ar[r] & C_1 \ar[r] & C_0.}
\end{equation*}
\end{definition}

Clearly, $P(T) = P(\partial_f, B)$ depends on the pair $(\partial_f, B)$ only and yields a functor
\begin{equation*}
P: {\rm\bf cross}^= \longrightarrow {\rm\bf H}_0.
\end{equation*}
The homology of $P(T)$ is given by
\begin{equation*}
{\rm H}_i(P(T)) =
\begin{cases}
0 & {\rm for} \ i = 1 \ {\rm and} \ i=3, \\
{\rm H}_2 C = \pi_2T & {\rm for} \ i = 2, \\
\Gamma(\pi_2(T)) & {\rm for} \ i = 4.
\end{cases}
\end{equation*}

\begin{lemma}\label{betabar}
Given a $2$--type $T$, there is a chain map
\begin{equation*}
\overline \beta: \widehat C(T) \longrightarrow P(T)
\end{equation*}
inducing isomorphisms in homology in degree $\leq 4$. The map $\overline \beta$ is natural in $T$ up to homotopy, that is, a map $g: T \rightarrow T'$ between $2$--types yields a homotopy commutative diagram
\begin{equation*}
\xymatrix{
\widehat C(T) \ar[d]_{\overline \beta} \ar[rr]^{g_{\ast}} && \widehat C(T') \ar[d]^{\overline \beta} \\
P(T) \ar[rr]^{\overline{\overline g}_{\ast}} && P(T'),}
\end{equation*}
where $\overline{\overline g}_{\ast}$ is induced by a map $\overline{\overline g}: \partial_f \rightarrow \partial_{f'}$ associated with $g$.
\end{lemma}

For a proof of Lemma \ref{betabar}, we refer the reader to diagram (1.2) in Chapter V of \cite{Baues1}. In order to compute the fourth homology or cohomology of a $2$--type $T$ with coefficients, choose a pair $(\partial_f, B)$ representing $T$ and a free chain complex $C$ together with a weak equivalence of chain complexes
\begin{equation*}
C \stackrel{\sim}{\longrightarrow} P(\partial_f, B).
\end{equation*}
Then, for right $\Lambda$--modules $M$ and left $\Lambda$--modules $N$,
\begin{eqnarray*}
{\rm H}_4(T, M) & = & {\rm H}_4(C \otimes M), \\
{\rm H}^4(T, N) & = & {\rm H}^4({\rm Hom}_{\Lambda}(C, N)).
\end{eqnarray*}
This allows for the computation of ${\rm H}_4$ in terms of chain complexs only, as is the case for the computation of group homology in Section \ref{pd3section}. Of course, it is also possible to compute the homology of $T$ in terms of a spectral sequence associated with the fibration
\begin{equation*}
K(\pi_2(T), 2) \longrightarrow T \longrightarrow K(\pi_1(T),1).
\end{equation*}
However, in general, this yields non--trivial differentials, which may be related to the properties of the chain complex $P(\partial_f, B)$.

\section{Algebraic models of ${\rm PD}^4$--complexes}\label{algmodPD4}
Let $X$ be a $4$--dimensional ${\rm CW}$--complex and let
\begin{equation*}
p_2: X \longrightarrow P_2X = T
\end{equation*}
be the map to the $2$--type of $X$, as in (\ref{postnikov}). Then $p_2$ yields the chain map
\begin{equation*}
\beta: \widehat C(X)
\stackrel{p_{2\ast}}{\longrightarrow} \widehat C(T)
\stackrel{\overline \beta_{\ast}}{\longrightarrow} P(T) = P(\partial_f, B),
\end{equation*}
were $\partial_f$ is given by the attaching map of $2$--cells in $X$ and $B = {\rm im}(d_3: \widehat C_3(X) \rightarrow \widehat C_2(X))$. We call the chain map $\beta$ the \emph{cellular boundary invariant of $X$}.

\begin{lemma}\label{4realize}
Let $X$ and $X'$ be $4$--dimensional ${\rm CW}$--complexes. A chain map $\varphi: \widehat C(X) \rightarrow \widehat C(X')$ is realizable by a map $g: X \rightarrow X'$ in ${\rm\bf CW}_0$, that is, $\varphi = g_{\ast}$, if and only if the diagram
\begin{equation*}
\xymatrix{
\widehat C(X) \ar[rr]^{\varphi} \ar[d]_{\beta} && \widehat C(X') \ar[d]^{\beta'} \\
P(\partial_f, B) \ar[rr]^{\overline{\overline \varphi}} && P(\partial_{f'}, B')}
\end{equation*}
commutes up to homotopy. Here $\overline{\overline \varphi}: \partial_f \rightarrow \partial_{f'}$ is a map in ${\rm\bf cross}^=$ inducing $\varphi_{\leq 2}: \widehat C(X^2) \rightarrow \widehat C(X'^2)$ as in Lemma \ref{whitehead}.
\end{lemma}

\begin{proof}
By Lemma \ref{betabar}, the diagram
\begin{equation*}
\xymatrix{
\widehat C(X) \ar[rr]^{\varphi} \ar[d]_{p_{2\ast}} && \widehat C(X') \ar[d]^{p_{2\ast}} \\
\widehat C(T) \ar[rr]^{g_{\ast}} && \widehat C(T')}
\end{equation*}
is homotopy commutative, where $g$ is given by $q(\overline{\overline g})$ in ${\rm\bf Ho(cross)}$. Since $p_{2\ast}$ and $g_{\ast}$ are realizable, the obstruction $\mathcal O_{X, X'}(\varphi)$ vanishes.
\end{proof}

\begin{definition}
A $\beta$--${\rm PD}^4$--chain complex is a ${\rm PD}^4$--chain complex $((\pi, C), \omega, [C], \Delta)$ together with a totally free pre--crossed module $\partial_f$ inducing $d_2: C_2 \rightarrow C_1$ and a chain map
\begin{equation*}
\beta: C \longrightarrow P(\partial_f, B)
\end{equation*}
which is the identity in degree $\leq 2$. Here $B = {\rm im}(d_3: C_3 \rightarrow C_2)$, the diagram
\begin{equation*}
\xymatrix{
C \ar[rr]^{\Delta} \ar[d]_{\beta} && C \otimes C \ar[d]^{\beta^{\otimes}} \\
P(\partial_f, B) \ar[rr]^{\overline{\overline \Delta}_{\ast}} && P(\partial_{f\otimes f}, B^{\otimes})}
\end{equation*}
commutes up to homotopy and $\beta$ is the cellular boundary invariant $\beta_{\sigma}$ of a totally free quadratic chain complex $\sigma$ defined in V(1.8) of \cite{Baues1}. Further, $\beta^{\otimes}$ is the cellular boundary invariant of the quadratic chain complex $\sigma \otimes \sigma$ defined in Section IV 12 of \cite{Baues1}, and there is an explicit formula expressing $\beta^{\otimes}$ in terms of $\beta$, which we do not recall here. The function $f \otimes f$ is the attaching map of $2$--cells in the product $X^2 \times X^2$, where $X^2$ is given by $f$, and $B^{\otimes}$ is the image of $d_3$ in $C \otimes C$. The map $\overline{\overline \Delta}$ in ${\rm\bf cross}^=$ is chosen such that $\overline{\overline \Delta}$ induces $\Delta$ in degree $\leq 2$ as in Lemma \ref{4realize}.) Let ${\rm\bf PD}^4_{\ast,\beta}$ be the category whose objects are $\beta$--${\rm PD}^4$--chain complexes and whose morphisms are maps $\varphi$ in ${\rm\bf PD}^4_{\ast}$ such that the diagram
\begin{equation*}
\xymatrix{
C \ar[rr]^{\varphi} \ar[d]_{\beta} && C' \ar[d]^{\beta'} \\
P(\partial_f, B) \ar[rr]^{\overline{\overline \varphi}} && P(\partial_{f'}, B')}
\end{equation*}
is homotopy commutative, where $\overline{\overline \varphi}$ induces $\varphi_{\leq 2}$ as in Lemma \ref{4realize}.
\end{definition}

\begin{theorem}
The functor $\widehat C$ yields a functor
\begin{equation*}
\widehat C: {\rm\bf PD}^4 / \simeq \longrightarrow {\rm\bf PD}^4_{\ast,\beta} / \simeq
\end{equation*}
which reflects isomorphisms and is representative and full.
\end{theorem}

\begin{proof}
Since $C$ is $2$--realizable, there is a $4$--dimensional ${\rm CW}$--complex $X$ with $\widehat C(X) = C$ and cellular boundary invariant $\beta$. By Lemma \ref{4realize}, the diagonal $\Delta$ is realizable by a diagonal $X \rightarrow X \times X$, showing that $X$ is a ${\rm PD}^4$--complex. By Lemma \ref{4realize}, a map $\varphi$ is realizable by a map $X \rightarrow X'$.
\end{proof}

\begin{corollary}\label{PD4cor}
The functor $\widehat C$ induces a 1--1 correspondence between homotopy types of ${\rm PD}^4$--complexes and homotopy types of $\beta$--${\rm PD}^4$--chain complexes.
\end{corollary}

The functor $\tau$ in Section \ref{triplesection} yields the diagram of functors
\begin{equation}\label{taustar4}
\xymatrix{
{\rm\bf PD}^4_+ / \simeq \ar[rr]^{\widehat C} \ar[dr]_{\tau_+} &&
{\rm\bf PD}^4_{\ast +,\beta} / \simeq \ar[dl]^{\tau_{\ast}} \\
& {\rm\bf Trp}^4_+ &}
\end{equation}
where $\tau_+$ determines $\tau_{\ast}$ together with a natural isomorphism $\tau_{\ast} \widehat C \cong \tau_+$.

\begin{corollary}
The functor $\tau_{\ast}$ in (\ref{taustar4}) reflects isomorphisms and is full.
\end{corollary}

\section{Homotopy systems of order (k+1)}\label{homsys}
To investigate questions of realizability, we work in the category ${\rm\bf H}_c^{k+1}$ of homotopy systems of order $(k+1)$. Let ${\rm\bf CW}_0^k$ be the full subcategory of ${\rm\bf CW}_0$ consisting of $k$--dimensional CW--complexes. A $0$--homotopy $H$ in ${\rm\bf CW}_0$, denoted by $\simeq^0$, is a homotopy for which $H_t$ is cellular for each $t, 0 \leq t \leq 1$.

Let $k \geq 2$. A {\emph{homotopy system of order $(k+1)$}} is a triple $X = (C, f_{k+1}, X^k)$, where $X^k$ is an object in ${\rm\bf CW}_0^k$, $C$ is a chain complex of free $\pi_1(X^k)$--modules, which coincides with $\widehat C(X^k)$ in degree $\leq k$, and where $f_{k+1}$ is a homomorphism of left $\pi_1(X^k)$--modules such that
\begin{equation*}
\xymatrix{
C_{k+1} \ar[d]_d \ar[r]^{f_{k+1}} & \pi_k(X^k) \ar[d]^j \\
C_k & \pi_k( {X}^k,  {X}^{k-1}) \ar[l]_-{h_k}}
\end{equation*}
commutes. Here $d$ is the boundary in $C$,
\begin{equation*}
\xymatrix{
h_k: \pi_k({X}^k, {X}^{k-1}) \ar[r]^{p_{\ast}^{-1}}_{\cong} & \pi_k(\widehat{X}^k, \widehat{X}^{k-1}) \ar[r]^h_{\cong} & {\rm{H}}_k(\widehat{X}^k, \widehat{X}^{k-1}),}
\end{equation*}
given by the Hurewicz isomorphism $h$ and the inverse of the isomorphism on the relative homotopy groups induced by the universal covering $p: \widehat X \rightarrow X$. Moreover, $f_{k+1}$ satisfies the {\emph{cocycle condition}}
\begin{equation*}
f_{k+1} d (C_{k+2}) = 0.
\end{equation*}
For an object $X$ in ${\rm\bf CW}_0$, the triple $r(X) = (\widehat C(X), f_{k+1}, X^k)$ is a homotopy system of order $(k+1)$, where $X^k$ is the $k$--skeleton of $X$, and
\begin{equation*}
\xymatrix{
f_{k+1}: \widehat C_{k+1}(X) \cong \pi_{k+1}(X^{k+1}, X^k) \ar[r]^-{\partial} & \pi_k(X^k)}
\end{equation*}
is the attaching map of $(k+1)$--cells in $X$. A {\emph{morphism}} or {\emph{map}} between homotopy systems of order $(k+1)$ is a pair
\begin{equation*}
(\xi, \eta): (C, f_{k+1}, X^k) \rightarrow (C', g_{k+1}, Y^k),
\end{equation*}
where $\eta: X^k \rightarrow Y^k$ is a morphism in ${\rm\bf CW}_0 / \simeq^0$ and the $\pi_1(\eta)$--equivariant chain map $\xi: C \rightarrow C'$ coincides with $\widehat C_{\ast}(\eta)$ in degree $\leq k$ such that
\begin{equation*}
\xymatrix{
C_{k+1} \ar[d]^{f_{k+1}} \ar[r]^{\xi_{k+1}} & C_{k+1}' \ar[d]^{g_{k+1}} \\
\pi_k(X^k) \ar[r]_{\eta_{\ast}} & \pi_k(Y^k)}
\end{equation*}
commutes. We also write $\pi_1 X = \pi_1 (X^k)$ for an object $X = (C, f_{k+1}, X^k)$ in ${\rm\bf H}_{k+1}^c$.

To define the homotopy relation in ${\rm{\bf H}}_{k+1}^c$, we use the action (see ??? in \cite{Baues1})
\begin{equation}\label{indaction}
[X^k, Y]_{\varphi} \times {\widehat{\rm{H}}}^k(X^k, \varphi^{\ast} \pi_kY) \rightarrow [X^k, Y]_{\varphi}, \quad (F, \{\alpha\}) \mapsto F+\{\alpha\},
\end{equation}
where $[X^n, Y]_{\varphi}$ is the set of elements in $[X^n, Y]$ which induce $\varphi$ on the fundamental groups. Two morphisms
\begin{equation*}
(\xi, \eta), (\xi', \eta'): (C, f_{k+1}, X^k) \rightarrow (C', g_{k+1}, Y^k)
\end{equation*}
are {\emph{homotopy equivalent}} in ${\rm\bf H}_{k+1}^c$ if $\pi_1(\eta) = \pi_1(\eta') = \varphi$ and if there are $\varphi$--equivariant  homomorphisms $\alpha_{j+1}: C_j \rightarrow C'_{j+1}$ for $j \geq k$ such that
\begin{equation*}
\{\eta\} + g_{k+1} \alpha_{k+1} = \{\eta'\} \quad {\rm{and}} \\
\end{equation*}
\begin{equation*}
\xi'_i - \xi_i = \alpha_i d + d \alpha_{i+1}, \quad  i \geq k+1,
\end{equation*}
where $\{\eta\}$ denotes the homotopy class of $\eta$ in $[X^k, Y^k]$ and $+$ is the action (\ref{indaction}).

Given homotopy systems $X = (C, f_{k+1}, X^k)$ and $Y = (C', g_{k+1}, Y^k)$, consider
\begin{equation*}
X \otimes Y = (C \otimes_{\mathbb Z} C', h_{k+1}, (X^k \times Y^k)^k),
\end{equation*}
where we choose CW--complexes $X^{k+1}$ and $Y^{k+1}$ with attaching maps $f_{k+1}$ and $g_{k+1}$, respectively, and $h_{k+1}$ is given by the attaching maps of $(k+1)$--cells in $X^{k+1} \times Y^{k+1}$. Then $X \otimes Y$ is a homotopy system of order $(k+1)$, and
\begin{equation*}
\otimes: {\rm\bf H}_{k+1}^c \times {\rm\bf H}_{k+1}^c \rightarrow {\rm\bf H}_{k+1}^c
\end{equation*}
is a bi--functor, called the \emph{tensor product of homotopy systems}. The projections  $p_1 :  X \otimes Y \rightarrow X$ and $p_2:  X \otimes Y \rightarrow Y$ in ${\rm\bf H}_{k+1}^c$ are given by the projections of the tensor product and the product of $\rm CW$--complexes. Similarly, we obtain the inclusions $\iota_1:  X \rightarrow  X \otimes  Y$ and $\iota_2:  Y \rightarrow  X \otimes  Y$. Then $p_1  \iota_1 = {\rm id}_X$ and $p_2  \iota_2 = {\rm id}_Y$, while $p_1  \iota_2$ and $p_2 \iota_1$ yield the trivial maps.

There are functors
\begin{equation}\label{functors}
\xymatrix{
{\rm\bf CW}_0 \ar[r]^-r & {\rm\bf H}_{k+1}^c \ar[r]^-{\lambda} & {\rm\bf H}_{k}^c \ar[r]^C & {\rm\bf H}_0}
\end{equation}
for $k \geq 3$, with $r(X) = (\widehat C(X), f_{k+1}, X^k)$ such that $r = \lambda r$. We write $\lambda X = \overline X$ for objects $X$ in ${\rm\bf H}_{k+1}^c$. As $\overline{X \otimes Y} = \lambda(X \otimes Y) = \overline X \otimes \overline Y$, the functor $\lambda$, and also $r$ and $C$, is a monoidal functor between monoidal categories. There is a homotopy relation defined on the category ${\rm\bf H}_{k+1}^c$ such that these functors induce functors between homotopy categories
\begin{equation*}
\xymatrix{
{\rm\bf CW}_0 / \simeq \ar[r]^-r & {\rm{\bf H}}_{k+1}^c / \simeq \ar[r]^-{\lambda} & {\rm{\bf H}}_{k}^c / \simeq \ar[r]^C & {\rm\bf H}_0/\simeq.}
\end{equation*}
For $k \geq 3$, Whitehead's functor $\Gamma_k$ factors through the functor $r: {\rm\bf CW} \rightarrow {\rm\bf H}_k^c$, so that the cohomology $\widehat {\rm H}_m (\overline X, \varphi^{\ast} \Gamma_k (\overline Y)) =  {\rm H}^m (C, \varphi^{\ast} \Gamma_k (\overline Y))$ is defined, where $\varphi: \pi_1 \overline X \rightarrow \pi_1 \overline Y$ and $\overline X$ and $\overline Y$ are objects in ${\rm\bf H}_k^c$.

To describe the obstruction to realizing a map $f = (\xi, \eta): \overline X \rightarrow \overline Y$ in ${\rm{\bf H}}_k^c$, where $\overline X = \lambda X$ and $\overline Y = \lambda Y$, by a map $X \rightarrow Y$ in ${\rm{\bf H}}_{k+1}^c$ for objects $X = (C, f_{k+1}, X^k)$ and $Y = (C', g_{k+1}, Y^k)$, choose a map $F: X^k \rightarrow Y^k$ in ${\rm\bf CW} / {\simeq}^0$ extending $\eta: X^{k-1} \rightarrow Y^{k-1}$ and for which $\widehat C_{\ast} F$ coincides with $\xi$ in degree $\leq n$. Then
\begin{equation*}
\xymatrix{
C_{k+1} \ar[d]^{f_{k+1}} \ar[r]^{\xi_{k+1}} & C_{k+1}' \ar[d]^{g_{k+1}} \\
\pi_k(X^k) \ar[r]^{F_{\ast}} & \pi_k(Y^k)}
\end{equation*}
need not commute and the difference $\mathcal O(F) = -g_{k+1} \xi_{k+1} + F_{\ast} f_{k+1}$ is a cocycle in ${\rm{Hom}}_{\varphi}(C_{k+1}, \Gamma_k(\overline Y))$. Theorem II 3.3 in \cite{Baues1} implies
\begin{proposition}\label{realize}
The map $f = (\xi, \eta): \overline X \rightarrow \overline Y$ in ${\rm{\bf H}}_k^c$ can be realized by a map $f_0 = (\xi, \eta_0): X \rightarrow Y$ in ${\rm\bf H}_{k+1}^c$ if and only if $\mathcal O_{X, Y}(f) = \{ \mathcal O(F) \} \in \widehat{\rm H}^{k+1}(\overline X, \varphi^{\ast} \Gamma_k \overline Y)$ vanishes. The obstruction $\mathcal O$ is a derivation, that is, for $f: \overline X \rightarrow \overline Y$ and $g: \overline Y \rightarrow \overline Z$,
\begin{equation}
\mathcal O_{X, Z}(gf) = g_{\ast} \mathcal O_{X, Y}(f) + f^{\ast} \mathcal O_{Y, Z}(g),
\end{equation}
and $\mathcal O_{X, Y}(f)$ depends on the homotopy class of $f$ only.
\end{proposition}
Denoting the set of morphisms $X \rightarrow Y$ in ${\rm\bf H}_{k+1}^c / \simeq$ by $[X, Y]$, and the subset of morphisms inducing $\varphi$ on the fundamental groups by $[X, Y]_{\varphi} \subseteq [X, Y]$, there is a group action
\begin{equation*}
[X, Y]_{\varphi} \times \widehat{\rm H}^k(\overline X, \varphi^{\ast} \Gamma_k \overline Y) \stackrel{+}{\longrightarrow} [X, Y]_{\varphi},
\end{equation*}
where $\overline X = \lambda X$ and $\overline Y = \lambda Y$. Theorem II 3.3 in \cite{Baues1} implies
\begin{proposition}\label{actdist}
Given morphisms $f_0, f_0' \in [X, Y]_{\varphi}$, then $\lambda f_0 = \lambda f_0' = f$ if and only if there is an $\alpha \in \widehat{\rm{H}}^k(\overline X, \varphi^{\ast} \Gamma_k \overline Y)$ with $f_0' = f_0 + \alpha$. In other words, $\widehat{\rm{H}}^k(\overline X, \varphi^{\ast} \Gamma_k \overline Y)$ acts transitively on the set of realizations of $f$. Further, the action satisfies the linear distributivity law
\begin{equation}\label{distributivity}
(f_0 + \alpha )(g_0 + \beta) = f_0 g_0 + f_{\ast} \beta + g^{\ast} \alpha.
\end{equation}
\end{proposition}
For the functor $\lambda$ in (\ref{functors}), Theorem II 3.3 in \cite{Baues1} implies
\begin{proposition}\label{objrealize}
For all objects $X$ in ${\rm\bf H}_{k+1}^c$ and for all $\alpha \in \widehat{\rm H}^{k+1}(\overline X, \Gamma_k \overline X)$, there is an object $X'$ in ${\rm\bf H}_{k+1}^c$ with $\lambda(X') = \lambda(X) = \overline X$ and $\mathcal O_{X, X'}({\rm id}_{\overline X}) = \alpha$. We then write $X' = X + \alpha$.

Now let $Y$ be an object in ${\rm\bf H}_k^c$. Then the group $\widehat{\rm H}^{k+1}(Y, \Gamma_k Y)$ acts transitively and effectively on ${\rm Real}_{\lambda}(Y)$ via $+$, provided ${\rm Real}_{\lambda}(Y)$ is non--empty. Moreover, ${\rm Real}_{\lambda}(Y)$ is non--empty if and only if an obstruction $\mathcal O(Y) \in \widehat{\rm H}^{k+2}(Y, \Gamma_k Y)$ vanishes.
\end{proposition}
For objects $X$ and $Y$ in ${\rm\bf H}_{k+1}^c$ and a morphism $f: \overline X \rightarrow \overline Y$ in ${\rm\bf H}_k^c$, Propositions \ref{realize} and \ref{objrealize} yield
\begin{equation}
\mathcal O_{X + \alpha, Y + \beta}(f) =  \mathcal O_{X, Y}(f) - f_{\ast} \alpha + f^{\ast} \beta
\end{equation}
for all $\alpha \in \widehat{\rm H}^{k+1}(\overline X, \Gamma_k \overline X)$ and $\beta \in \widehat{\rm H}^{k+1}(\overline Y, \Gamma_k \overline Y)$. Given another object $Z$ in ${\rm\bf H}_{k+1}^c$ with $\lambda Z = \overline Z$,
\begin{eqnarray}
\mathcal O_{X \otimes Z, Y \otimes Z}(f \otimes {\rm id}_{\overline Z})
& = &  \overline \iota_{1\ast} \overline p_1^{\ast} \mathcal O_{X, Y}(f), \label{tens1} \\
\mathcal O_{Z \otimes X, Z \otimes Y}({\rm id}_{\overline Z} \otimes f)
& = & \overline \iota_{2\ast} \overline p_2^{\ast} \mathcal O_{X, Y}(f), \label{tens2}
\end{eqnarray}
where $\overline \iota_1: \overline X \rightarrow \overline X \otimes \overline Z$ and $\overline p_1: \overline X \otimes \overline Z \rightarrow \overline X$ are the inclusion of and projection onto the first factor and $\overline \iota_2$ and $\overline p_2$ are defined  analogously. We obtain
\begin{equation}\label{tensobjact}
(X + \alpha) \otimes (Y + \beta) = (X \otimes Y) + \overline \iota_{1\ast} \overline p_1^{\ast} \alpha + \overline \iota_{2\ast} \overline p_2^{\ast} \beta.
\end{equation}

\section{Obstructions to the diagonal}\label{diagobs}
Let $k \geq 2$. A \emph{diagonal} on $X = (C, f_{k+1}, X^k)$ in ${\rm\bf H}_{k+1}^c$ is a morphism, $\Delta: X \rightarrow X \otimes X$, such that, for $i = 1, 2$, the diagram
\begin{equation}
\xymatrix{
X \ar[rr]^-{\Delta} \ar[drr]_{\rm id} && X \otimes X \ar[d]^{p_i} \\
&& X}
\end{equation}
commutes up to homotopy in ${\rm\bf H}_{k+1}^c$.
Applying the functor $r: {\rm\bf CW}_0 \rightarrow {\rm\bf H}_{k}^c$ to a diagonal $\Delta: X \rightarrow X \times X$ in ${\rm\bf CW}_0$, we obtain the diagonal $r(\Delta): r(X) \rightarrow r(X) \otimes r(X)$
in ${\rm\bf H}_k^c$.
\begin{lemma}\label{diagreal0}
Let $X$ be an object in ${\rm\bf H}_{k+1}^c$. Then every $\lambda$--realizable diagonal $\overline \Delta: \overline X = \lambda X \rightarrow \overline X \otimes \overline X$ in ${\rm\bf H}_k^c/\simeq$ has a $\lambda$--realization $\Delta: X \rightarrow X \otimes X$ in ${\rm\bf H}_{k+1}^c/\simeq$ which is a diagonal in ${\rm\bf H}_{k+1}^c$.
\end{lemma}
\begin{proof}
Suppose $\Delta': X \rightarrow X \otimes X$ is a $\lambda$--realization of $\overline \Delta$ in ${\rm\bf H}_{k+1}^c$. The projection $p_{\ell}: X \rightarrow X \otimes X$ realizes the projection $\overline{p}_{\ell}: \overline X \rightarrow \overline X \otimes \overline X$ and hence $p_{\ell} \Delta'$ realizes $\overline p_{\ell} \overline \Delta$ for $\ell = 1, 2$. Now the identity on $X$ realizes the identity on $\overline X$ and $\overline p_{\ell} \Delta$ is homotopic to the identity on $\overline X$ by assumption. Hence $p_{\ell} \Delta'$ and the identity on $X$ realize the same homotopy class of maps for $\ell = 1, 2$. As the group ${\widehat{\rm H}}^k(\overline{X}, \Gamma_k \overline X)$ acts transitively on the set of realizations of this homotopy class by Proposition \ref{actdist}, there are elements $\alpha_{\ell} \in {\widehat{\rm H}}^k(\overline X, \Gamma_k \overline X)$ such that
\begin{equation*}
\{p_{\ell} \Delta' \} + \alpha_{\ell} = \{ {\rm id}_X \} \quad {\rm{for}} \ {\ell} = 1,2,
\end{equation*}
where $\{f\}$ denotes the homotopy class of the morphism $f$ in ${\rm\bf H}_{k+1}^c$. We put
\begin{equation*}
\{\Delta\} = \{\Delta'\} + \iota_1 \alpha_1 + \iota_2 \alpha_2.
\end{equation*}
By Proposition \ref{actdist},
\begin{eqnarray*}
\{p_{\ell} \Delta \} & = & \{p_{\ell}\}(\{\Delta'\} + \iota_1 \alpha_1 + \iota_2 \alpha_2) \\
& = & \{p_{\ell} \Delta'\} + \overline p_{\ell \ast} \iota_1 \alpha_1 + \overline p_{\ell \ast} \iota_2 \alpha_2 \\
& = & \{p_{\ell} \Delta' \} +  \alpha_{\ell}
=  \{ {\rm id}_X \}.
\end{eqnarray*}
\end{proof}
\begin{lemma}\label{diagreal1}
For $X$ in ${\rm\bf H}_{k+1}^c$, let $\Delta_{\overline X}: \overline X \rightarrow \overline X \otimes \overline X$ be a diagonal on $\overline X = \lambda X$ in ${\rm\bf H}_k^c$. Then we obtain, in ${\rm H}^{k+1}(\overline X, \Gamma_k(\overline X \otimes \overline X)$,
\begin{enumerate}
\item $O_{X, X \otimes X}(\Delta_{\overline X}) \in {\rm ker} \ \overline p_{i\ast}$ for $i = 1, 2$,
\item $\mathcal O_{X, X \otimes X}(\Delta_{\overline X}) \in {\rm ker}({\rm id}_{\overline X \ast} - T_{\ast})_{\ast}$ if $\Delta_{\overline X}$ is homotopy commutative and
\item $\mathcal O_{X, X \otimes X}(\Delta_{\overline X}) \in {\rm ker}\big(\overline \iota_{1,2 \ast} - \overline \iota_{2,3 \ast} + (\Delta_{\overline X} \otimes {\rm id}_{\overline X})_{\ast} - ({\rm id}_{\overline X} \otimes \Delta_{\overline X})_{\ast}\big)_{\ast}$ if $\Delta_{\overline X}$ is homotopy associative. \label{homass}
\end{enumerate}
\end{lemma}
\begin{proof}
By definition, $\overline p_i \Delta_{\overline X} \simeq {\rm id}_{\overline X}$ for $i = 1, 2$. As the identity on $\overline X$ is realized by the identity on $X$ and $\overline p_i: \overline X \otimes \overline X \rightarrow \overline X$ is realized by $p_i: X \otimes X \rightarrow X$, Proposition \ref{realize} implies $\mathcal O_{X, X \otimes X}({\rm id}_{\overline X}) = 0$ and $\mathcal O_{X \otimes X, X}(\overline p_i) = 0$. Since $\mathcal O$ is a derivation, we obtain
\begin{equation*}
0 = \mathcal O_{X, X}(\overline p_i \Delta_{\overline X}) = \overline p_{i\ast} \mathcal O_{X, X \otimes X}(\Delta_{\overline X}) + \Delta_{\overline X}^{\ast} \mathcal O_{X \otimes X, X}(\overline p_i) = \overline p_{i\ast} \mathcal O_{X, X \otimes X}(\Delta_{\overline X}),
\end{equation*}
and hence $O_{X, X \otimes X}(\Delta_{\overline X}) \in {\rm ker} \ \overline p_{i\ast}$ for $i = 1, 2$. If $\Delta_{\overline X}$ is homotopy commutative, then
\begin{equation*}
\mathcal O_{X, X \otimes X}(\Delta_{\overline X}) = \mathcal O_{X, X \otimes X}(T \Delta_{\overline X}) = T_{\ast} \mathcal O_{X, X \otimes X}(\Delta_{\overline X}),
\end{equation*}
since $\mathcal O_{X \otimes X, X \otimes X}(T) = 0$, as $T$ is $\lambda$--realizable. Hence $\mathcal O_{X, X \otimes X}(\Delta_{\overline X}) \in {\rm ker}({\rm id}_{\overline X \ast} - T_{\ast})_{\ast}$. For $1 \leq k < \ell, \leq 3$, let $\iota_{k,\ell}: X \otimes X \rightarrow X \otimes X \otimes X$ denote the inclusion of the $k$-th and $\ell$-th factors and suppose $\Delta_{\overline X}$ is a homotopy commutative diagonal in ${\rm\bf H}_k^c$. Then $\mathcal O_{X, X \otimes X \otimes X}((\Delta_{\overline X} \otimes {\rm id}_{\overline X}) \Delta_{\overline X}) = \mathcal O_{X, X \otimes X \otimes X}(({\rm id}_{\overline X} \otimes \Delta_{\overline X}) \Delta_{\overline X})$, as the obstruction depends on the homotopy class of a morphism only, and
\begin{eqnarray*}
\mathcal O_{X, X \otimes X \otimes X}(\Delta_{\overline X} \otimes {\rm id}_{\overline X})
& = & \overline \iota_{1,2 \ast} \overline p_1^{\ast} \mathcal O_{X, X \otimes X}(\Delta_{\overline X}) \\
\mathcal O_{X, X \otimes X \otimes X}({\rm id}_{\overline X} \otimes \Delta_{\overline X})
& = & \overline \iota_{2,3 \ast} \overline p_2^{\ast} \mathcal O_{X, X \otimes X}(\Delta_{\overline X},
\end{eqnarray*}
by (\ref{tens1}) and (\ref{tens2}). Omitting the objects in the notation for the obstruction, we obtain
\begin{eqnarray*}
\mathcal O((\Delta_{\overline X} \otimes {\rm id}_{\overline X}) \Delta_{\overline X})
& = &  \Delta_{\overline X}^\ast \mathcal O(\Delta_{\overline X} \otimes {\rm id}_{\overline X})
+ (\Delta_{\overline X} \otimes {\rm id}_{\overline X})_{\ast} \mathcal O(\Delta_{\overline X}) \\
& = & \Delta_{\overline X}^\ast \overline \iota_{1,2 \ast} \overline p_1^{\ast} \mathcal O(\Delta_{\overline X}) + (\Delta_{\overline X} \otimes {\rm id}_{\overline X})_{\ast} \mathcal O(\Delta_{\overline X}) \\
& = &  \overline \iota_{1,2 \ast} (\overline p_1\Delta_{\overline X})^{\ast} \mathcal O(\Delta_{\overline X}) + (\Delta_{\overline X} \otimes {\rm id}_{\overline X})_{\ast} \mathcal O(\Delta_{\overline X}) \\
& = & \overline \iota_{1,2 \ast} \mathcal O(\Delta_{\overline X})
+ (\Delta_{\overline X} \otimes {\rm id}_{\overline X})_{\ast} \mathcal O(\Delta_{\overline X}).
\end{eqnarray*}
Similarly, we obtain
\begin{equation*}
\mathcal O(({\rm id}_{\overline X} \otimes \Delta_{\overline X}) \Delta_{\overline X})
= \overline \iota_{2,3 \ast} \mathcal O(\Delta_{\overline X})
+ (\Delta_{\overline X} \otimes {\rm id}_{\overline X})_{\ast} \mathcal O(\Delta_{\overline X}),
\end{equation*}
which proves (\ref{homass}).
\end{proof}
\begin{question}
Given a $\lambda$--realizable object $\overline X$ with a diagonal $\Delta_{\overline X}: \overline X \rightarrow \overline X \otimes \overline X$ in ${\rm\bf H}_k^c$, is there an object $X$ with $\lambda X = \overline X$ and a diagonal $\Delta_X: X \rightarrow X \otimes X$ in ${\rm\bf H}_{k+1}^c$ such that $\lambda \Delta_X = \Delta_{\overline X}$?
\end{question}
Let $X$ in ${\rm\bf H}_{k+1}^c$ be a $\lambda$--realization of $\overline X$. By Proposition \ref{objrealize}, any $\lambda$--realization $X'$ of $\overline X$ is of the form $X' = X + \alpha$ for some $\alpha \in \widehat{\rm H}^{k+1}(\overline X, \Gamma_k \overline X)$. By (\ref{tensobjact}), $X' \otimes X' = (X \otimes X) + \overline \iota_{1\ast} \overline p_1^{\ast} \alpha + \overline \iota_{2\ast} \overline p_2^{\ast} \alpha$ and as the obstruction $\mathcal O$ is a derivation, we obtain
\begin{eqnarray*}
\mathcal O_{X', X' \otimes X'}(\Delta_{\overline X})
& = & \mathcal O_{X + \alpha, (X + \alpha) \otimes (X + \alpha)}(\Delta_{\overline X})\\
& = & \mathcal O_{X, X \otimes X}(\Delta_{\overline X}) - \Delta_{\overline X \ast} \alpha + \Delta_{\overline X}^{\ast} (\overline \iota_{1\ast} \overline p_1^{\ast} \alpha + \iota_{2\ast} \overline p_2^{\ast} \alpha)\\
& = & \mathcal O_{X, X \otimes X}(\Delta_{\overline X}) - (\Delta_{\overline X \ast} - \overline \iota_{1 \ast} - \overline \iota_{2 \ast}) \alpha,
\end{eqnarray*}
since $\Delta^{\ast} \overline \iota_{i\ast} \overline p_i^{\ast} = \overline \iota_{i\ast} (\overline p_i \Delta)^{\ast} = \overline \iota_{i\ast}$, for $i = 1, 2$.

\begin{lemma}\label{obsact}
For $X$ in ${\rm\bf H}_{k+1}^c$, let $\Delta_{\overline X}: \overline X \rightarrow \overline X \otimes \overline X$ be a diagonal on $\overline X = \lambda X$ in ${\rm\bf H}_k^c$ and let $X' = X + \alpha$ for some $\alpha \in \widehat{\rm H}^{k+1}(\overline X, \Gamma_k \overline X)$. Then we obtain, in ${\rm H}^{k+1}(\overline X, \Gamma_k(\overline X \otimes \overline X)$,
\begin{equation*}
\mathcal O_{X', X' \otimes X'}(\Delta_{\overline X}) = \mathcal O_{X, X \otimes X}(\Delta_{\overline X}) - (\Delta_{\overline X \ast} - \overline \iota_{1 \ast} - \overline \iota_{2 \ast}) \alpha.
\end{equation*}
\end{lemma}

\section{$\rm PD^n$--homotopy systems}\label{pdhomsys}
A \emph{$\rm PD^n$--homotopy system} $X = (X, \omega_X, [X], \Delta_X)$ of order $(k+1)$ consists of an object $X = (C, f_{k+1}, X^k)$ in ${\rm\bf H}_{k+1}^c$, a group homomorphism $\omega_X: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z$, a fundamental class $[X] \in {\rm H}_n(C, \mathbb Z^{\omega})$ and a diagonal $\Delta: X \rightarrow X \otimes X$ in ${\rm\bf H}_{k+1}^c$ such that $(C, \omega_X, [X], \Delta_X)$ is a ${\rm PD}^n$--chain complex. A map $f:  (X, \omega_X, [X], \Delta_X) \rightarrow (Y, \omega_Y, [Y], \Delta_Y)$ of $\rm PD^n$--homotopy systems of order $(k+1)$ is a morphism in ${\rm\bf H}_{k+1}^c$ such that $\omega_X = \omega_Y \pi_1(f)$ and $(f \otimes f) \Delta_X \simeq \Delta_Y f$,
and we thus obtain the category ${\rm\bf PD}^n_{[k+1]}$ of $\rm PD^n$--homotopy systems of order $(k+1)$. Homotopies in ${\rm\bf PD}^n_{[k+1]}$ are homotopies in ${\rm\bf H}_{k+1}^c$, and restricting the functors in (\ref{functors}), we obtain, for $k \geq 3$, the functors
\begin{equation}
\xymatrix{
{\rm\bf PD}^n \ar[r]^-r & {\rm\bf PD}^n_{[k+1]} \ar[r]^-{\lambda} & {\rm\bf PD}_{[k]}^n \ar[r]^C & {\rm\bf PD}^n_{\ast}.}
\end{equation}
These functors induce functors between homotopy categories
\begin{equation*}
\xymatrix{
{\rm\bf PD}^n / \simeq \ar[r]^-r & {\rm\bf PD}^n_{[k+1]} / \simeq \ar[r]^-{\lambda} & {\rm\bf PD}^n_{[k]} / \simeq \ar[r]^C & {\rm\bf PD}^n_{\ast}/\simeq.}
\end{equation*}
\begin{theorem}\label{pdnfuncequ}
The functor $C: {\rm\bf PD}_{[3]}^n/\simeq \ \longrightarrow  {\rm\bf PD}^n_{\ast}/\simeq$ is an equivalence of categories for $n \geq 3$.
\end{theorem}
\begin{proof}
The functor $C$ is full and faithful by Theorem III 2.9 and Theorem III 2.12 in \cite{Baues1}. By Lemma \ref{2real}, every ${\rm PD}^n$--chain complex, $\overline X = (D, \omega, [D], \overline\Delta)$, in ${\rm\bf{PD}}^n_{\ast}$ is $2$--realizable, that is, there is an object $X^2$ in ${\rm CW}^2_0$ such that $\widehat C(X^2) = D_{\leq 2}$, and we obtain the object $X = (D, f_3, X^2)$ in ${\rm\bf H}_3^c$. As $C$ is monoidal, full and faithful, the diagonal $\overline\Delta$ on $\overline X$ is realized by a diagonal $\Delta$ on $X$ and hence $(X, \omega, [D], \Delta)$ is an object in ${\rm\bf PD}_{[3]}^n$ with $C(X) = \overline X$.
\end{proof}
\begin{theorem}\label{pdnfuncdet}
For $n \geq 3$, the functor $r: {\rm\bf PD}^n/\simeq \ \longrightarrow {\rm\bf PD}^n_{[n]}/\simeq$ reflects isomorphisms, is representative and full.
\end{theorem}
\begin{proof}
That $r$ reflects isomorphisms follows from Whitehead's Theorem.

Poincar{\'e} duality implies $\widehat {\rm H}^{n+1}(Y, \Gamma_n Y) = \widehat {\rm H}^{n+2}(Y, \Gamma_n Y) = 0$, for every object $Y = (Y, \omega_Y, [Y], \Delta_Y)$ in ${\rm\bf PD}^n_{[n]}$. Hence, by Proposition \ref{objrealize}, $Y = \lambda(X)$ for some object $X$ in ${\rm\bf H}_{n+1}^c$, and, by Proposition \ref{realize}, the diagonal $\Delta_Y$ is $\lambda$--realizable. Thus Lemma \ref{diagreal0} guarantees the existence of a diagonal $\Delta_X: X \rightarrow X \otimes X$ in ${\rm\bf H}_{n+1}^c$ with $\lambda \Delta_X = \Delta_Y$. The homomorphism $\omega_Y$ and the fundamental class $[Y]$ determine a homomorphism $\omega_X: \pi_1 X \rightarrow \mathbb Z / 2 \mathbb Z$ and a fundamental class $[X] \in {\rm H}_n(C, \mathbb Z^{\omega})$, such that $X = (X, \omega_X, [X], \Delta_X)$ is an object in ${\rm\bf PD}^n_{[n+1]}$. Inductively, we obtain an object $(X_k, \omega_{X_k}, [X_k], \Delta_{X_k})$ realizing $(Y, \omega_Y, [Y], \Delta_Y)$ in ${\rm\bf PD}^n_{[k]}$ for $k > n$, and in the limit an object $X = (X, \omega_X, [X], \Delta_X)$ in ${\rm\bf PD}^n$ with $r(x) = Y$.

Proposition \ref{realize} together with the fact that, by Poincar{\'e} duality, $\widehat {\rm H}^k(X, B) = 0$ for $k > n$ and every $\Lambda$--module $B$, implies that $r$ is full.
\end{proof}

redpajama_set_name: RedPajamaArXiv


GPS coordinates to get you close by are: N42 39 6.677 and W82 37 44.548 Many waters sports are enjoyed in this fairly protected bay including water skiing and tubing with the family.
How did Munchies Bay get its name? In the 1990's there was a family restaurant overlooking this swim area called ... you guessed it, "Munchies!" The restaurant had boat dockage and was open 7 days a week - very popular for breakfast on Sunday.
The building was remodeled and changed its name to the Brisbay Lounge. This restaurant was opened for a short period as seen in the photo below. This building has since been torn down and and a new showroom was built, home to Rose Marine.


Rose's heart motif radiates outward, comprised of dazzling diamond pavé set in sterling silver or gold plated. A classic heart locket shape, this piece captures the very spirit of With You. The 9-inch chain is long enough to wear as an anklet and can be shortened to custom fit your wrist. Available in Gold Plated or Sterling Silver. Locket pendant with .06 diamond CTW. 9" bracelet chain. Locket measures 15 x 16mm diameter.
Our Rose heart locket bracelet's namesake is an incredible woman. She is a firm believer that actions speak louder than words. Be the good you want to see in the world. This is what Rose is about. She's about listening to those around her and being there when we need her most, without ever asking for help. Her intuition and sense of global decency guide her to where she can do the most good in the world. We wish we could be more like her but she's one in a million. This locket is named after her because of the heart and how it radiates outward and shine brilliantly. Whether it's preparing a meal for her family or for families who don't have meals, she is the outward heart in the world. These women should be cherished and reminded that even though they outshine us sometimes, we're there cheering them on and providing support when they need to take a breath.


text:
Questa voce raccoglie le informazioni riguardanti la Pallacanestro Trieste nelle competizioni ufficiali della stagione 2013-2014.

Roster

2013-2014
Stagione 2013-2014 delle squadre di pallacanestro

redpajama_set_name: RedPajamaWikipedia


The Hawaii Housing Finance & Development Corporation (HHFDC) will hold an informational briefing at the time and place listed below to discuss HHFDC's intended sale of the leased fee interest, to the current lessees, of the following listed properties, as required by section 171-64.7, Hawaii Revised Statutes, as amended.
(1) Briefing will be held on Tuesday, August 8, 2017, 6:00 - 6:45 p.m. at Aliamanu Middle School - Classroom #C-2, 3271 Salt Lake Boulevard, Honolulu, HI.
(2) Briefing will be held on Wednesday, August 9, 2017, 6:00 - 6:45 p.m. at Waimanalo Public and School Library Conference Room, 41-1320 Kalanianaole Hwy, Waimanalo.


The baby name Kaelyn of english origin means pure, keeper of the keys. The Huggies Baby Name Finder can help you create a list of popular baby names including Kaelyn and then let you share with family and friends. You might also consider similar names to Kaelyn which include Kaelee, Kaelah, Kaila, Kailan, Kailee. Keep in mind Kaelyn is more popular as a girl name, and similar names chosen by Huggies members to Kaelyn include Ayla, Brooklyn, Charmaine, Jenna.