text:
\section{Introduction}

The Hodge theorem and Hirzebruch signature theorem form an
important bridge between geometric and topological properties of
compact smooth manifolds.  There has been a great deal of work over the
past thirty years aimed at understanding how to generalize these theorems
to $L^2$ results in the noncompact and singular settings.
Early and important work was done by Atiyah, Patodi and Singer \cite{APS1}.
Their work concerned both manifolds with the simplest sort of singularities,
namely boundaries, and noncompact manifolds with cylindrical ends,
that is, manifolds which off a compact set are isometric to $(0, \infty) \times N$
for some compact manifold $N$.  They proved both a Hodge result and a signature result.
Their Hodge result says that the space of $L^2$ harmonic forms on a
manifold, $\hat{M}$, with cylindrical end is canonically isomorphic to the image
of relative cohomology of  $\hat{M}$ in its absolute cohomology, i.e.
\begin{equation}
\ensuremath{\mathcal{H}_{(2)}}^k(\hat{M},g) \cong \mbox{Im}(H^k(\hat{M}, N) \longrightarrow H^k(\hat{M})).
\label{eq:apshodge}
\end{equation}
Their signature result says that the topological signature of a manifold with
boundary, $\overline{M}$, is equal first of all to the signature of the pairing
on middle degree $L^2$ harmonic forms given by integration, and secondly that both satisfy
\begin{equation}
\sigma(M) = \int_M L(p(M, N)) - \eta(N)(0),
\label{eq:apssign}
\end{equation}
where $L$ is the Hirzebruch $L$-polynomial in the relative Pontrjagin classes on $M$ and
$\eta$ is a spectral invariant of the boundary, $N$, of $M$.

The connection between $L^2$ Hodge theorems and intersection cohomology
was made by Cheeger shortly after the dual intersection homology theoreies
had been defined by Goresky and MacPherson \cite{GM1}.  Cheeger showed
in  \cite{Ch} that for a pseudomanifold, $X$, with conical singularities and only even
codimensional strata, the space of $L^2$ harmonic forms on the regular set is isomorphic
to the middle perversity intersection cohomology, the unique intersection cohomology
that satisfies Poincar\'{e} duality, that is:
\begin{equation}
H^i_2(X^{reg}, g_c) \cong IH^i_{m}(X),
\label{eq:cheegerhodge}
\end{equation}

In this paper, Cheeger also noted the importance in $L^2$ Hodge theorems
for incomplete manifolds generally of considering
different "ideal boundary conditions".  That is, whereas in the complete case, there is a unique
closed extension of the exterior derivative, $d$, in the incomplete case, there may be
several different closed extensions.  These correspond to several different $L^2$ cohomologies and
several different closed self-adjoint extensions of the Laplace operator.
For instance, in the case of a manifold with boundary, $M$, the exterior derivative may have a number
of closed extensions interpolating between the so called maximal and minimal extensions.  The
cohomology corresponding to the maximal extension of $d$ is absolute
cohomology on $M$.  Its classes are naturally represented by $L^2$ harmonic forms satisfying Neumann boundary conditions.
The complex given by the minimal extension of $d$ generates relative cohomology on $M$.   Its
classes are naturally represented by harmonic $L^2$
forms satisfying Dirichlet boundary conditions.  The even codimension condition
on conical pseudomanifolds in Cheeger's paper avoids this complication.

In the 1990's, new work was done on the eta invariant defined
by Atiyah, Patodi and Singer.  A number of mathematicians began to study its behavior
in a family of fibre bundle metrics which become singular.  Bismut and Cheeger, [5],
Melrose and Mazzeo, [38], and Dai, [14], all studied the eta invariant under such
adiabatic limits. In 1990, Muller linked his work on signature theorems for manifolds with cusps of
rank 1 to this new work on the eta invariant, [49], and interpreted it in terms of middle
perversity intersection cohomology.  He showed that the $L^2$ signature for such a manifold
was equal to the $L^2$ signature for the manifold with boundary formed by cutting off the cusps
plus the $\tau$ invariant of the resulting boundary fibration, defined by Dai.

Recently, Dai and Cheeger followed up Cheeger's Hodge theorem with
a signature theorem for conical pseudomanifolds with one even codimensional
smooth singular stratum. They show in \cite{CD} that
the signature of the intersection pairing on middle degree $L^2$ forms for a
conical pseudomanifold, $X$, whose singular set, $B$, is a smooth compact even codimensional
manifold is given by the formula:
\begin{equation}
\sigma(X)= \mbox{sgn}\,
\mbox{Im}\,\big( H^*(X, B) \longrightarrow H^*(X^{reg}))\big) +  \tau.
\label{eq:conesign}
\end{equation}
where $\tau$ is an invariant of the fibre bundle over $B$ which forms the boundary of the normal
neighborhood of the singular stratum.   Recently, in \cite{HM}, the author and Mazzeo have extended Cheeger and Dai's results to conical pseudomanifolds with one singular stratum in the form of a smooth compact manifold which may be odd dimensional.

The similarity and relationships between the $L^2$ Hodge and signature results for manifolds
with boundary and manifolds with conical singularities leads to to the question of what happens as
we interpolate between these metrics on a given incomplete smooth manifold.  In this paper, we
generalize the techniques of \cite{CD} to answer this question.

Consider a manifold $\overline{M}$ with boundary $\ensuremath{\partial} \overline{M} = Y$ where
$Y \stackrel{\phi}{\rightarrow} B$ is a fibre bundle with fibre $F$.  Endow
$M=\overline{M} - \ensuremath{\partial} \overline{M}$ with a metric $g_c$ which is quasi-isometric near
the boundary to one of the form
\begin{equation}
ds_c^2 = dr^2 + r^{2c}\tilde{h} + \phi^*ds_B^2
\label{eq:metrics}
\end{equation}
\noindent
where $\tilde{h}$ is a two form which restricts to a metric on each fibre of $Y$ and $0 \leq c \leq 1$.
Note that these metrics interpolate between a cylindrical metric when $c=0$ and a cone bundle metric
when $c=1$.  There is a second natural compactification, $X$, for the manifold $M$, obtained by
collapsing the fibres of the boundary fibration and adding a compactifying copy of the base.

As mentioned before, on a complete manifold, the exterior derivative $d$ on compactly
supported smooth forms and its formal
adjoint $\delta$ each have a unique extension to $L^2$ forms, and the Hodge Laplacian given by these
extensions, $\overline{d}+\overline{\delta}$, is self-adjoint.  On an incomplete manifold, this is not
generally true.  There may be many closed extensions of $d$, and many self-adjoint extensions of the
Hodge Laplacian.  Perhaps the most natural of these are the minimal and maximal extensions of $d$
and $\delta$ and their associated self-adjoint Laplacians.  Recall that the minimal and maximal
extensions of the exterior derivative, $d$, on compact smooth forms to $L^2$ forms on $(M, g_c)$ are
defined by:

\begin{definition}  Let $\sigma \in L^2\Omega^k(M,g)$ for
any $k$.  If there exists an
$\eta \in L^2\Omega^{k+1}(M,g)$ such that for all $\phi
\in C^{\infty}_0\Omega^{k+1}(M,g)$ we have
\begin{equation}
\langle \sigma , \delta \phi \rangle_{L^2(M)} = \langle
\eta , \phi \rangle_{L^2(M)},
\label{eq:dmax}
\end{equation}
we say that $\sigma \in \mathcal{D}(d_{max})$ and we write
$d_{max} \sigma = \eta$.
\end{definition}

\begin{definition}
If $\sigma \in L^2\Omega^{k}(M,g)$ for any $k$, and if
there is a sequence $\{\sigma_n\} \subset
C^{\infty}_0\Omega^{k}(M,g)$ such that $\sigma =
\lim_{L^2} \sigma_n$ and $\zeta = \lim_{L^2} d
\sigma_n$, we say $\sigma \in \mathcal{D}(d_{min})$ and
$\zeta = d_{min}\sigma$.  In this case, it is
always true that $\sigma \in \mathcal{D}(d_{max})$ as
well, and $d_{max} \sigma = d_{min} \sigma$.
\end{definition}

The minimal and maximal extensions of $\delta$ are defined analogously.  Both
extensions of $d$ define cohomology groups on $M$, which we will denote by $H^*_{min}(M, g_c)$
and $H^*_{max}(M, g_c)$.  We define two related self-adjoint extensions of the Hodge Laplacian
$D= d+\delta$:
\begin{equation}
D_{m,M}=d_{min}+\delta_{max}
\label{eq:mMlaplacian}
\end{equation}
\begin{equation}
D_{M,m}=d_{max}+\delta_{min}.
\label{eq:Mmlaplacian}
\end{equation}
We will
denote the corresponding spaces of harmonic forms by $\mathcal{H}^*_{m,M}(M, g_c)$ and
$\mathcal{H}^*_{M,m}(M,g_c)$, respectively.

Our first theorem is a Hodge theorem for these cohomology groups:

\begin{theorem}
Let $\overline{M}$ be a manifold with boundary $\ensuremath{\partial} \overline{M} = Y$ where
$Y \stackrel{\phi}{\rightarrow} B$ is a fibre bundle with $f$-dimensional fibre $F$.  Endow
$M=\overline{M} - \ensuremath{\partial} \overline{M}$ with a metric $g_c$ which is quasi-isometric near
the boundary to one of the form
$$
ds_c^2 = dr^2 + r^{2c}\tilde{h} + \phi^*ds_B^2
$$
\noindent
where $h$ is a two form which restricts to a metric on each fibre of $Y$.
Let $X$ be the compactification of $M$ obtained by collapsing
the fibres of the boundary fibration.  Then
$$
H^*_{min}(M, g_c) \cong  \mathcal{H}^*_{m,M}(M, g_c) \cong
\left\{ \begin{array}{ll}
I\!H_{\underline{\mathfrak{m}}+[[1+ \frac{1}{2c}]]}^*(X, B) & f \mbox{ is even} \\
I\!H_{\underline{\mathfrak{m}}+[[\frac{1}{2} + \frac{1}{2c}]]}^*(X, B) & f \mbox{ is odd} \\
\end{array} \right.
$$
and
$$
H^*_{max}(M, g_c) \cong  \mathcal{H}^*_{M, m}(M, g_c) \cong
\left\{ \begin{array}{ll}
I\!H_{\overline{\mathfrak{m}}-[[1+ \frac{1}{2c}]]}^*(X, B) & f \mbox{ is even} \\
I\!H_{\overline{\mathfrak{m}}-[[\frac{1}{2} + \frac{1}{2c}]]}^*(X, B) & f \mbox{ is odd} \\
\end{array} \right. ,
$$
\noindent
where $[[x]]$ denotes the greatest integer strictly less than $x$ and $\underline{\mathfrak{m}}$ and
$\overline{\mathfrak{m}}$ are the two middle perversities for $X$.
\label{th:hodge}
\end{theorem}
Here we use the notation $I\!H_{\mathfrak{p}}^*(X, B)$ instead of  $I\!H_{\mathfrak{p}}^*(X)$ in order to
indicate a slightly more general definition of intersection cohomology than is
standard.  In particular, it allows us to include the case where the boundary fibration
fibre $F$ is trivial, so $X$ is our original manifold with boundary, $\overline{M}$.

Note that when $c=1$, we get $[[\frac{1}{2} + \frac{1}{2c}]] =0$, so this result reduces
to the result for manifolds with edges in \cite{HM}, and in the case that the fibre is even dimensional,
it reduces to Cheeger's result in \cite{Ch}.  In the extended definition
of intersection cohomology , for $c$ sufficiently close to $0$,
these spaces become relative and absolute cohomologies of $M$, respectively, thus reducing to
the known results for manifolds with boundary.  As $c$ goes from $0$ to $1$, the intersection
cohomology groups isomorphic to the maximal cohomology interpolate between middle perversity
and absolute cohomology, while the intersection cohomology groups isomorphic to the minimal
cohomology interpolate between middle perversity and relative perversity.

Using our Hodge theorem, we can also obtain a signature theorem for the manifolds $(M, g_c)$ through
a signature theorem for intersection cohomology.  If $p$ and $q$ are dual perversities with $p \geq q$,
then we can define an intersection form on spaces of the form
$$
\mbox{Im} \, \big(I\!H^{n/2}_{p}(X, B)
\longrightarrow I\!H^{n/2}_{q}(X, B) \big)
$$
\noindent
as follows.  Define a (degenerate) pairing on
$$
I\!H^{k}_{p}(X, B) \otimes I\!H^{n-k}_{q}(X, B)
$$
\noindent via the map induced by inclusion
\begin{equation}
I\!H^{k}_{p}(X, B) \longrightarrow I\!H^{k}_{q}(X, B)
\label{eq:perversityinclusion}
\end{equation}
\noindent
and the nondegenerate pairing
\begin{equation}
I\!H^{k}_{p}(X, B) \otimes I\!H^{n-k}_{q}(X, B) \longrightarrow \mathbb{R}.
\label{eq:pairing}
\end{equation}
\noindent
Call its signature $\sigma_p(X)$.

The $\tau$-invariant defined by Dai in \cite{Dai} is given by a sum $\tau = \sum_{i=2}^{\infty} \tau_i$,
where $\tau_i$ is the
signature of a form defined on the $E_i$ term of the Leray-Serre spectral sequence for the boundary
fibration of $M$.  Our signature theorem for metrics interpolating between finite cylindrical and conical is:

\begin{theorem}  If $p = \underline{\mathfrak{m}} + k$, then the signature of the intersection form on these
spaces is
given by:
$$
\sigma_p(X)= \mbox{sgn}\,
\mbox{Im}\,\big( H^*(M, \ensuremath{\partial} M) \longrightarrow H^*(M))\big) + \sum_{i=2+2k}^{\infty} \tau_i.
$$
\label{th:signature}
\end{theorem}
\noindent

Thus as the metric becomes less and less cylindrical and more and more conical,
the signature theorem picks up more and more of the $\tau_i$ terms, until when the
metric is close to conical, the signature includes all of $\tau$.

The author would like to thank Tam\'{a}s Hausel, Xianzhe Dai, and Rafe Mazzeo for
useful conversations relating to this paper.

\section{Background}
Before we begin the proof of Theorem \ref{th:hodge}, we will review briefly some definitions and
theorems we will use in our proofs.

\subsection{Intersection Cohomology}
We will prove $H^*_{max}(M, g_c)$ is isomorphic
to an intersection cohomology group via a sheaf argument using the following
theorem of Goresky and MacPherson:
\begin{proposition}[\cite{GM2}] Let $X$ be a stratified space
and let $(\mathcal{L}^*, d)$ be a complex of fine sheaves on $X$ with
cohomology $H^*(X,\mathcal L)$. Suppose that if $\mathcal U$ is a neighbourhood
in the principal (smooth) stratum of $X$, then $H^*(\mathcal U,\mathcal L) =
H^*(\mathcal U,\mathbb C)$, while if $q$ lies in a stratum of codimension
$\ell$, and $\mathcal U = \mathcal V \times C(L)$ where $\mathcal V$ is a neighborhood of $q$ in the
codimension $l$ stratum, then
\begin{equation}
H^k(\mathcal U,\mathcal L) \cong I\!H_\mathfrak{p}^k(\mathcal U) =
\left\{ \begin{array}{ll}
I\!H_\mathfrak{p}^k(L) & k \leq \ell-2-p(\ell) \\
0         & k \geq \ell-1-p(\ell)
\end{array} \right.
\label{eq:localcalc}
\end{equation}
Then there is a natural isomorphism between the hypercohomology
$\mathbb{H}^{\,*}(X,\mathcal{L}^*)$ associated to this complex
of sheaves and $I\!H^*_\mathfrak{p}(X)$, the intersection cohomology of
perversity $\mathfrak{p}$.
\label{pr:shchar}
\end{proposition}

For a pseudomanifold $X$, such as ours, which has only one singular
stratum, $B$, the local calculation for intersection cohomology in a neighborhood of a point $b$
on the singular stratum
$\mathcal U \cong \Delta \times C(F)$ for $b \in \Delta$ a disk in $B$ is:
\begin{equation}
H^i(\mathcal U,\mathcal L) \cong I\!H_\mathfrak{p}^i(\mathcal U) =
\left\{ \begin{array}{ll}
H^i(F) & k \leq f-1-p \\
0         & k \geq f-p
\end{array} \right.
\label{eq:localcalc2}
\end{equation}

Note that for larger values of $p$, the truncation in $H^i(F)$ occurs at a lower degree.  Thus if $p>q$,
then $I\!H^i_p(U)$ vanishes for more degrees than $I\!H^i_q(U)$.

We generalize this definition slightly as follows and use this definition in both theorems:
\begin{equation}
I\!H^*_{j}(X,B) = \left\{ \begin{array}{ll}
H^*(X-B) & j \leq -1, \\
I\!H_\mathfrak{p}^*(X) & 0 \leq j \leq f-1, \\
H^*(X, B) & j \geq f,
\end{array} \right. .
\label{eq:extih}
\end{equation}

There are two particular perversities on a pseudomanifold, $X$, called the {\em middle perversities}.
In the case that the strata of $X$ are all even dimensional, $X$ is a particular kind of pseudomanifold
called a Witt space.  In this case, the two middle perversity intersection cohomologies are identical, and
this single middle perversity intersection cohomology satisfies Poincar\'{e} duality.  If the strata are not
all even dimensional, the two middle perversities are not generally the same.  In this case, they are
Poincar\'{e} duals of each other. Perversities which lie at equal distances from these two middle
perversities, $\overline{\mathfrak{m}} - k$ and $\underline{\mathfrak{m}} + k$ are also dual perversities, that is,
their corresponding intersection cohomologies are Poincar\'{e} duals.

The local calculations near a point on the compactifying layer $B$ for the two middle perversities are:
\begin{equation}
H^k(\mathcal U,\mathcal L) \cong I\!H_{\underline{\mathfrak{m}}}^k(\mathcal U) =
\left\{ \begin{array}{ll}
H^k(F) & k \leq \frac{f}{2} - 1 \mbox{ if $f$ is even and }  \frac{f-1}{2}  \mbox{ if $f$ is odd} \\
0         & k \geq \frac{f}{2} \mbox{ if $f$ is even and }  \frac{f+1}{2}  \mbox{ if $f$ is odd}
\end{array} \right.
\end{equation}
for the lower middle perversity and
\begin{equation}
H^k(\mathcal U,\mathcal L) \cong I\!H_{\overline{\mathfrak{m}}}^k(\mathcal U) =
\left\{ \begin{array}{ll}
H^k(F) & k \leq \frac{f}{2}  \mbox{ if $f$ is even and }  \frac{f-1}{2}  \mbox{ if $f$ is odd} \\
0         & k \geq \frac{f}{2} + 1 \mbox{ if $f$ is even and }  \frac{f+1}{2}  \mbox{ if $f$ is odd}
\end{array} \right.
\end{equation}
for the upper middle perversity.  Note that if we allow for the possibility that the fibre is a point, that is, if
our manifold $(M, g_c)$ is a simple manifold with boundary, then these local calculations correspond to
$H^*_0(M)$ and $H^*(M)$, respectively.  Thus our extended definition of intersection cohomology
allows us to state the
results for the case where $F$ is trivial in the same manner as the cases where $F$ is nontrivial.

\subsection{Geometry of Fibre Bundles}
It is useful to think of the Leray-Serre spectral sequence of a fibration in terms of bidegree.
So recall from geometry of fibre bundles that we can define the bidegree of a form on the
total space of a fibration as follows.
Let $\phi:Y \to B$ be a fibration with fibre $F$, and suppose that it is endowed
with a metric $G$ of the form $\phi^*(h) + k$, where $h$ is a metric on $B$. We
assume furthermore that $\phi: (Y,G) \to (B,h)$ is a Riemannian submersion.
The tangent bundle $TY$ splits into a vertical and horizontal subbundle,
$T^V Y \oplus T^H Y$, where $T^V Y =\mbox{ker\,}(d\phi)$ and $T^H Y$ is its
orthogonal complement (and also the subbundle annihilated by $k$). This induces
a splitting of the form bundles on $Y$, and thus every differential form has a
(horizontal,vertical) bidegree, i.e.
\begin{equation}
\Omega^{p,q}(Y) = \Omega^p(B)\, \widehat{\otimes} \, \Omega^q(Y,T^VY).
\label{eq:decomp}
\end{equation}

\subsection{Hilbert Complexes}
We will use the concept of Hilbert complexes and results about them from \cite{BL} in our Hodge
result, so we summarize them briefly here.
Consider a complex of the form
\begin{equation}
0 \to L_0 \stackrel{D_0}{\longrightarrow} L_1
\stackrel{D_1}{\longrightarrow}L_2 \ldots \stackrel{D_{n-1}}{\longrightarrow} L_n \to 0,
\label{eq:hc}
\end{equation}
where each $L_i$ is a separable Hilbert space, $D_i:L_i \to L_{i+1}$ is
a closed operator with dense domain $\mathcal D(D_i)$ such that ${\mathrm{ran}}(D_i) \subseteq
\mathcal D(D_{i+1})$ and $D_{i+1}\circ D_i = 0$ for all $i$. Under these
conditions, (\ref{eq:hc}) is called a Hilbert complex, and is denoted by
$(L_*,D_*)$.

Many familiar constructions in Hodge-de Rham theory carry over
immediately to this setting.  In particular:
\begin{itemize}
\item[i)] There is a dual Hilbert complex
\[
0 \to L_0 \stackrel{D_0^*}{\longleftarrow} L_1
\stackrel{D_1^*}{\longleftarrow}L_2 \ldots \stackrel{D_{n-1}^*}{\longleftarrow} L_n \to 0
\]
defined using the Hilbert space adjoints of the differentials, $D_i^*: L_{i+1} \to L_i$;
\item[ii)] The Laplacian $\Delta_i = D_{i}^* D_i + D_{i-1}D_{i-1}^*$
is a self-adjoint operator on $L_i$ with domain
\[
\mathcal D(\Delta_i) = \{u \in \mathcal D(D_i) \cap \mathcal D(D_{i-1}^*): D_i u \in \mathcal D(D_i^*),
D_{i-1}^* u \in \mathcal D(D_{i-1})\}
\]
and nullspace
\[
\ker \Delta_i := \mathcal H^i(L_*,D_*) = \ker D_i \cap \ker D_{i-1}^*;
\]
\item[iii)] (\cite{BL}, Lemma 2.1) There is a weak Kodaira decomposition
\[
L_i = \mathcal H^i \oplus \overline{{\mathrm{ran}} D_{i-1}} \oplus \overline{{\mathrm{ran}}{D_i^*}};
\]
\item[iv)] (\cite{BL}, Corollary 2.5) The cohomology of $(L_*,D_*)$ is defined by
\[
H^i(L_*,D_*) = \ker D_i/{\mathrm{ran}} D_{i-1};
\]
if this space is finite dimensional, then ${\mathrm{ran}} D_{i-1}$ is necessarily
closed and $H^i(L_*,D_*) = \mathcal H^i(L_*,D_*)$.
\item[v)] (\cite{BL}, Corollary 2.6)  The cohomology of the complex $(L_*,D_*)$
is finite dimensional if an only if the cohomology of the dual complex $(L^*_*,D^*_*)$ is.
\end{itemize}

We will also simplify our calculations using the K\"unneth theorem:
\begin{proposition}[\cite{BL} Corollary 2.15]
Let $(L',D')$ and $(L'' D'') $ be two Hilbert complexes.  Form the
completed tensor product Hilbert complex $(L,D)$:
\[
L_j = \bigoplus_{i+\ell=j} L'_i \, \hat \otimes \, L''_\ell,
\]
\[
D_j = \bigoplus_{i+\ell=j} (D'_i \otimes \mbox{id}_{L''_\ell} + (-1)^i
\mbox{id}_{L'_i} \otimes D"_\ell).
\]
\noindent
Suppose that $D''$ has closed range in all degrees. Then
\[
H^j(L,D)= \bigoplus_{i+\ell=j} H^i(L',D') \otimes H^\ell(L''D'').
\]
\label{pr:kunneth}
\end{proposition}

The last result we will use says that we may compute these
cohomology groups using a `core subcomplex'
\[
\mathcal D^\infty_{{\max}} \Omega^*(M,g) \subset L^2 \Omega^*(M,g)
\]
consisting of all elements $\omega$ which are in the domain of $\Delta_{M,m}^\ell$
for every $\ell \geq 0$.
\begin{proposition}[\cite{BL} Theorem 2.12]
The cohomology $H^*_{{\max}}(M,g)$ is equal to the cohomology of the
complex $(\mathcal D^\infty \Omega^*_{{\max}}(M,g), d_{{\max}})$.
\label{pr:core}
\end{proposition}

This implies that  $H^*(M)$ can be computed using the complex
of forms which are smooth on the interior of $M$.

\section{Proof of Theorem \ref{th:hodge}}

The two complexes $(D(d_{max}), d_{max})$ and $(D(d_{min}), d_{min})$ form Hilbert complexes as
defined in \cite{BL}, so we can use the theory developed in that paper to prove Theorem \ref{th:hodge}.
First we note that it suffices to prove the theorem for $(D(d_{max}), d_{max})$.  This is because
by Corollary 2.6 of \cite{BL}, the cohomology of this complex is finite dimensional if and only
if the cohomology of the dual complex $(D(\delta_{min}), \delta_{min})$ is.  The Hodge star operator
gives a isomorphism of complexes between this dual complex and the complex
$(D(d_{min}), d_{min})$, so the maximal complex has finite dimensional cohomology if and only if
the minimal complex does.  These two are Poincar\'{e} dual spaces, so if $H^*_{max} \cong
I\!H^*_{\mathfrak{p}}(X, B)$, then we must also have $H^*_{min} \cong I\!H^*_{\mathfrak{q}}(X, B)$, where
$\mathfrak{p}$ and $\mathfrak{q}$ are dual perversities.

Further, if these cohomologies are finite dimensional, for instance, if they are isomorphic
to intersection cohomologies, then by Corollary 2.5 in \cite{BL}, we immediately get the isomorphisms
$$
H^*_{min}(M, g_c) \cong  \mathcal{H}^*_{m,M}(M, g_c)
$$
and
$$
H^*_{max}(M, g_c) \cong  \mathcal{H}^*_{M, m}(M, g_c).
$$

We can create a complex of fine sheaves of $L^2$ forms on $X$ in the same manner as in
the proofs of Proposition 2 in \cite{HHM} and Proposition 1.18 in \cite{Z2}.  By Proposition \ref{pr:shchar},
we need only then prove a local Poincar\'{e} lemma to establish
the isomorphism between $H^*_{max}(M, g_c)$ and intersection cohomology of some perversity.
The local Poincar\'{e} lemma for points on the stratum $M \subset X$ is identical to the standard
Poincar\'{e} lemma for compact manifolds.  Near a point on the singular stratum, we need to establish a
Poincar\'{e} lemma for neighborhoods of the form $U = \Delta \times C(F)$, where $\Delta$ is a disk in
the singular stratum.  By the Kunneth Theorem, Corollary 2.15,  from \cite{BL}, this calculation
reduces:
$$
H^*_{max}(U, g_c) \cong H^*_{max}(\Delta) \otimes H^*_{max}(C(F), g_c) \cong H^*_{max}(C(F), g_c),
$$
\noindent
so the necessary Poincar\'{e} lemma is one for the cone on $F$ with respect to the metric
$g_c = dr^2 + r^{2c}ds_F^2$.

\subsection{Preliminary propositions}

The proof of the necessary Poincar\'{e} lemma is based on the techniques in \cite{Ch}.
Before proceeding to the proof, we'll lay out a few facts we will use.
First, by Theorem 2.12 of \cite{BL}, we know that the natural inclusion of smooth $L^2$ forms into
the space of $L^2$ forms induces an isomorphism on cohomology for both the maximal and
minimal complexes, so we may always assume a max cohomology class has a smooth
representative.  This is true for any smooth manifold.  Next we prove a few propositions specific to
the metrics $g_c$ on cones.

\begin{proposition}
Let $\phi$ be an $i$-form on $F$.  Then the pullback of $\phi$ to $C(F)$ is in
$L^2\Omega^i(C(F),g_c)$ if and only if $i < \frac{f}{2} + \frac{1}{2c}$, and in this case,
the pullback map is bounded.
\label{prop:pullbacks}
\end{proposition}
\begin{proof}
If $\phi$ is an $i$-form on $F$, then
$$
\int_{C(F)} ||\phi||^2_c dvol_c = ||\phi||^2_{L^2(F)} \int_0^1 r^{c(f-2i)} dr <\infty
$$
if and only if $c(f-2i)> -1$, which is if and only if $i < \frac{f}{2} + \frac{1}{2c}$.
Since the integral on the right is independent of $\phi$, the pullback map is bounded.
\end{proof}

\begin{proposition}
There exists a $K>0$ such that for all
$\alpha = \phi + dr \wedge \omega \in L^2 \Omega^i(C(F), g_c)$ and for any null set $S \subset (1/2, 1)$
there is an $a \in (1/2, 1) - S$ such that
$$
||\phi(a)||^2_{L^2(F)} \leq K ||\phi||^2_{L^2(C(F), g_c)} \leq K ||\alpha||^2_{L^2(C(F), g_c)}.
$$
\label{prop:bound}
\end{proposition}
\begin{proof}
Suppose not.  Then for any $K>0$, there is some family of forms on $F$,
$\phi(r)  \in L^2 \Omega^i(C(F), g_c)$ such that
$$
||\phi||^2_{L^2(C(F), g_c)} \geq \int_{1/2}^1 \int_F ||\phi||^2_F r^{-2ic} dvol_F r^{cf} dr
$$
$$
= \int_{1/2}^1 ||\phi(r)||^2_{L^2(F)} r^{c(f-2i)} dr > K ||\phi||^2_{L^2(C(F), g_c)}
 \int_{(1/2, 1) - S} r^{c(f-2i)} dr
 $$
 $$
 =K ||\phi||^2_{L^2(C(F), g_c)} \int_{1/2}^1 r^{c(f-2i)} dr.
 $$
So choose $K > (\int_{1/2}^1 r^{c(f-2i)} dr)^{-1}$ and we have a contradiction.
\end{proof}

\begin{proposition} If $i < \frac{f}{2} + \frac{1}{2c} +1$ and $\alpha = \phi + dr \wedge \omega \in L^2
\Omega^i(C(F), g_c)$, then for any $a\in (1/2, 1)$,
$$
K_a(\alpha) = \int_a^r \omega(s) ds \in L^2 \Omega^{i-1}(C(F), g_c).$$
\label{prop:K}
\end{proposition}
\begin{proof}
This proposition follows essentially from Schwartz's inequality.
\begin{equation}
||K_a||^2_{L^2(C(F),g_c)} = \int_0^1 \int_F ||\int_a^r \omega(s) \, ds||^2_{F} r^{c(f-2i+2)} \, dvol_F \,dr
\label{eq:Ka}
\end{equation}
since $\omega$ is a family of $i-1$ forms on $F$.  Consider just the inside of this:
$$
||\int_a^r \omega(s) \, ds||^2_{F} \leq ( \int_a^r ||\omega||_F)^2
$$
and for any $j$, by the Schwartz inequality,
$$
=(\int_a^r s^{-2j} || s^j \omega(s)||_F \, ds)^2
\leq \int_a^r s^{-2j} \, ds \int_a^r ||s^j \omega(s)||^2_F \, ds
$$
$$
= \left\{
\begin{array}{ll}
\frac{r^{1-2j} - a^{1-2j}}{1-2j} \int_a^r||s^j \omega(s)||^2_F \, ds & j \neq 1/2 \\
(\ln(r) - \ln(a)) \int_a^r||s^j \omega(s)||^2_F \, ds & j = 1/2
\end{array}
\right. .
$$
Choose $2j = c(f-2i + 2)$ and replace this expression in equation \ref{eq:Ka} to get
$$
||K_a||^2_{L^2(C(F),g_c)} \leq
\left\{
\begin{array}{ll}
\int_0^1 \frac{r - a(r/a)^{c(f-2i + 2)}}{1-c(f-2i+2)} \int_a^r|| \omega(s)||^2_F s^{c(f-2i+2)}\, ds \, dvol_F \, dr
& i \neq  \frac{f}{2} + 1 - \frac{1}{2c}\\
\int_0^1 r(\ln(r) - \ln(a)) \int_a^r||\omega(s)||^2_F s^{c(f-2i+2)} \, ds \, dvol_F \, dr&  i =  \frac{f}{2} + 1 -
\frac{1}{2c}
\end{array}
\right.
$$
$$
\leq
\left\{
\begin{array}{ll}
\int_0^1 \frac{r - a(r/a)^{c(f-2i + 2)}}{1-c(f-2i+2)} \, dr || \omega||^2_{L^2(C(F), g_c)}
& i \neq  \frac{f}{2} + 1 - \frac{1}{2c}\\
\int_0^1 r(\ln(r) - \ln(a)) \, dr || \omega||^2_{L^2(C(F), g_c)}&  i =  \frac{f}{2} + 1 - \frac{1}{2c}
\end{array}
\right. .
$$
Since $a \in (1/2, 1)$, the first integral is uniformly bounded in $a$ for $i \leq \frac{f}{2} + \frac{1}{2c}+1$
and the second integral is also bounded uniformly in $a$.  Thus $K_a$ is a bounded operator.
\end{proof}

\begin{proposition} Let $\rho >0$, and endow $(\rho,1) \times F$ with the metric $g_c$ restricted from
$C(F)$.    Let $\alpha = \phi + dr \wedge \omega  \in L^2\Omega^i(C(F), g_c)$
If $i \geq  \frac{f}{2} + \frac{1}{2c}$, then there exists a sequence $\epsilon_s \rightarrow 0$
such that
$$\lim_{\epsilon_s \rightarrow 0} ||\phi(\epsilon_s)||^2_{L^2((\rho,1) \times F, g_c)} = 0.
$$
\label{prop:epsilons}
\end{proposition}
\begin{proof}
Since $\alpha \in L^2\Omega^i(C(F), g_c)$, so is $\phi$, so we know that
$$
\int_0^1\int_F ||\phi||^2_F \,dvol_F r^{c(f-2i)} \, dr < \infty.
$$
That is,
$$
\int_F ||\phi(r)||^2_F \,dvol_F r^{c(f-2i)} \in L^1[(0,1)].
$$
Thus by Lemma 1.2 in \cite{Ch}, there is a sequence $\epsilon_s \rightarrow 0$ for which
$$
|\int_F ||\phi(\epsilon_s)||^2_F \,dvol_F \epsilon_s^{c(f-2i)} | < \frac{C}{\epsilon_s | \ln(\epsilon_s)|}
$$
for some constant $C>0$.  So we have
$$
|\int_F ||\phi(\epsilon_s)||^2_F \,dvol_F| < \frac{C \epsilon_s^{c(2i-f)-1}}{|\ln(\epsilon_s)|}.
$$
Since $i \geq  \frac{f}{2} + \frac{1}{2c}$, the right hand side tends to zero as $\epsilon_s \rightarrow 0$.
Thus
$$
||\phi(\epsilon_s)||^2_{L^2((\rho, 1)\times F, g_c)} = \int_{\rho}^1 \int_F
||\phi(\epsilon_s)||^2_F r^{c(f-2i)} \, dvol_F \, dr
$$
$$
=||\phi(\epsilon_s)||^2_{L^2(F)} \int_{\rho}^1 r^{c(f-2i)} \, dr \longrightarrow 0
$$
also as $\epsilon_s \rightarrow 0$.

\end{proof}

\subsection{Poincar\'{e} lemma}

The Poincar\'{e} lemma we need to prove for Theorem \ref{th:hodge}, is the following :
\begin{lemma}
Let $(F,g)$ be an $f$-dimensional compact manifold and consider the family of metrics on the cone
over $F$, $C(F)$, given by $ds_c^2 = dr^2 + r^{2c}ds_F^2$ for $0 < c < 1$.  Then
$$
H^i_{max}(C(F), g_c) \cong
\left\{
\begin{array}{ll}
H^i(N) & i < \frac{f}{2} + \frac{1}{2c} \\
0          & i \geq \frac{f}{2} + \frac{1}{2c}
\end{array} \right.
.
$$
\label{lem:poincare}
\end{lemma}

\begin{proof}
Consider first the case where $ i < \frac{f}{2} + \frac{1}{2c}$.  We want to create a bijective bounded
linear map
$$R: H^i_{max}(C(F), g_c) \longrightarrow H^i(F).$$
\noindent
 If $\alpha = \phi + dr \wedge \omega \in L^2 \Omega^i(C(F), g_c)$, then for all but a null set of points in
$(1/2,1)$, we get
$\phi \in L^2(F)$.  So let $[\alpha] \in H^i_{max}(C(F), g_c)$ and for any such value, $a$, define
$R([\alpha]) = [\phi(a)]$.  We need to check that this map makes sense, that it is independent of our
choice of cohomology representative and of a, that it is bijective, and that it is bounded with respect
to the natural norm on cohomology:
$$
|| [\alpha] ||_{L^2H} = \mbox{min}_{\gamma \in [\alpha]} ||\gamma||_{L^2}.
$$

First, the map clearly makes sense, since if $\alpha$ is closed, then
$d \alpha = d_F \phi + dr \wedge (\phi' - d_N \omega)$, thus $d_F \phi = 0$ for all a.  To show
that the map is well defined, suppose that $\alpha= \phi + dr \wedge \omega$ is a
smooth representative of $[\alpha]$ and
$\beta = \psi + dr \wedge \nu$ is any other representative.  Then
$\phi(a) \in L^2\Omega^i(F)$ for any $a$, so choose any $a$ such that $\psi(a) \in L^2\Omega^i(F)$.
Then there is some form $\eta = \rho + dr \wedge \theta \in L^2\Omega^{i-1}(C(F), g_c)$ such
that $\beta = \alpha + d_{max} \eta$, so $\psi = \phi + d_F \rho$ for all $a$.  Since $L^2$ cohomology
on a compact manifold is the same as absolute cohomology, this means that even if $\rho(a)$ is not
in $L^2(F)$, there must be some $L^2$ form $\tilde {\rho}_a$ which is and for which
$\psi(a) = \phi(a) + d_F \tilde{\rho}_a$.  Thus $[\psi(a)] = [\phi(a)]$, so the map $R$ is independent
of the choice of cohomology representatives.  To show that it is independent of the choice of $a$,
it suffices therefore to consider smooth representatives.    For $\alpha$ a representative as before,
since $\phi(a) \in L^2\Omega^i(F)$ for any $a$, we need to show that $[\phi(a)] = [\phi(b)]$ for
any $a,b \in (1/2, 1)$.  Since $\alpha$ is smooth,
$
\int_a^b \omega(s) \, ds \in L^2\Omega^{i-1}(F)
$
and
$$
d\int_a^b \omega(s) \, ds  = \int_a^b d_F \omega(s) \, ds = \int_a^b \phi'(s) \, ds
$$
$$
= \phi(b) - \phi(a).
$$
\noindent
Thus $[\phi(a)] = [\phi(b)]$, so the map $R$ is well-defined.

To show the map is linear, let $a$ be a value in $(1/2,1)$ such that $\alpha(a)$, $\beta(a)$ and
$(\alpha+\beta)(a)$ are all in $L^2\Omega^i(F)$.  Then $R([\alpha]) + R([\beta]) = [\alpha(a)]
+[\beta(a)] = [(\alpha + \beta)(a)] = R([\alpha+\beta])$.
The map is bounded by proposition \ref{prop:bound} and surjective by proposition \ref{prop:pullbacks}.
So we have only left to show that
it is injective.  So suppose that $R([\alpha])=[0]$ and let $\alpha= \phi + dr \wedge \omega$ be a smooth
representative.
Then $\phi(a) = d_F \eta$ for some $\eta \in L^2\Omega^{i-1}(F)$.  By proposition \ref{prop:pullbacks},
we can consider
$\eta$ as a form in $L^2\Omega^{i-1}(C(F), g_c)$, and by proposition \ref{prop:K},
$
\int_a^r \omega(s) \, ds \in L^2\Omega^{i-1}(C(F), g_c)
$.
Then
$$
d(\eta + \int_a^r \omega(s) \, ds) = d_F \eta + dr \wedge \omega(r) + \int_a^r d_F\omega(s)\, ds
$$
$$
= \phi(a) + dr \wedge \omega(r) + \int_a^r \phi'(s)\, ds = \alpha.
$$
Thus $\alpha$ is exact and $[\alpha] = [0]$.

Now consider the case where $i \geq \frac{f}{2} + \frac{1}{2c}$.  Since $c<1$, this implies that
$c(f-2i+2) < 1$.  We want to show that any class $[\alpha] \in H^i_{max}(C(F),g_c)$ is trivial.
First we know again that any such class can be represented by a smooth form.
Let $\alpha = \phi + dr \wedge \omega \in L^2 \Omega^i(C(F), g_c)$ be a smooth representative
of $[\alpha]$.  We need
to show that $\alpha = d_{max} \eta$ for some $\eta \in L^2 \Omega^{i-1}(C(F), g_c)$.
Consider $K_0 (\alpha) = \int_0^r \omega(s) \, ds$.  Then
$$
||K_0 \alpha||^2_{L^2(C(F),g_c)} = \int_0^1 \int_F || \int_0^r \omega(s)\, ds||_F^2 r^{c(f-21 + 2)} \, dvol_F \,
dr.
$$
By the same steps as in the proof of proposition \ref{prop:K}, we get that this is
$$
\leq \int_0^1 \int_F r^{c(f-2i+2)} [ \int_0^r s^{-2j} \int_0^r ||s^j \omega(s)||_F^2 \,ds]\, dvol_F \, dr
$$
for all $j$.  For $2j < 1$, this is
$$
= \int_0^1 \int_F r^{c(f-21+2)} r^{-2j+1} [ \int_0^r ||s^j \omega(s)||_F^2 \, ds] \, dvol_F \, dr.
$$
Let $2j = c(f-2i + 2)$.  Then this becomes
$$
\int_0^1 \int_F r \int_0^r ||\omega(s)||^2_F s^{c(f-2i+2)} \, ds \, dvol_F \, dr
\leq
\int_0^1r  ||\omega(s)||^2_{L^2(C(F), g_c) }\, dr \leq \frac{1}{2} ||\alpha||^2_{L^2(C(F), g_c) }.
$$
Thus $K_0: L^2 \Omega^i(C(F),g_c) \longrightarrow L^2 \Omega^{i-1}(C(F),g_c)$ is a bounded map.
We want to show that if $\alpha$ is closed, then $d_{max}(K_0(\alpha)) = \alpha$.
This means that we need to show for any $\beta \in C^{\infty}_0 \Omega^{i-1}(C(F))$, we have
\begin{equation}
\langle K_0(\alpha), \delta \beta \rangle_{L^2(C(F), g_c)} = \langle \alpha, \beta \rangle _{L^2(C(F), g_c)}.
\label{eq:K0dmax}
\end{equation}
So let $\beta \in C^{\infty}_0 \Omega^{i-1}(C(F))$.  Then for some $\rho > 0$, $\beta$ is supported in
$(\rho, 1) \times F$.  So equation \ref{eq:K0dmax} becomes
\begin{equation}
\langle K_0(\alpha), \delta \beta \rangle_{L^2((\rho,1) \times F, g_c)} = \langle \alpha, \beta \rangle
_{L^2((\rho, 1) \times F, g_c)}.
\label{eq:dmax2}
\end{equation}
We'd like to use the fundamental theorem of calculus as in the proof for $i< \frac{f}{2} + \frac{1}{2c}$,
but since we don't know that $\alpha$ is continuous at 0, we can't do this directly.  So to get around
this we use the following proposition:
\begin{proposition} Define
$$K_{\epsilon} (\alpha) = \int_{\epsilon}^r \omega(s)\, ds.$$
Then on $(\rho , 1) \times F$ with the restricted $g_c$ metric,
$$
K_{\epsilon}(\alpha) \stackrel{L^2((\delta,1) \times F, g_c)}{\longrightarrow} K_0(\alpha).
$$
\label{prop:Kepsilon}
\end{proposition}
\begin{proof}
We have
$$
||K_{\epsilon}(\alpha) - K_0(\alpha)||^2_{L^2((\delta,1) \times F, g_c)}
= \int_{\delta}^1 \int_F ||\int_0^{\epsilon} \omega(s)\, ds||_F r^{c(f-2i + 2)} \, dvol_F \, dr.
$$
As in the proof of Proposition \ref{prop:K}, this is
$$
 \leq \epsilon^{-c(f-2i + 2) +1} \int_{\delta}^1  r^{c(f-2i + 2)} \, dr ||\int_0^{\epsilon} \omega(s) \,
ds||^2_{L^2(C(F), g_c)}.
$$
Since $c(f-2i + 2) < 1$, the first integral in this product converges, and the whole expression
tends to 0 as $\epsilon \rightarrow 0$.
\end{proof}

So now we can consider for a closed smooth $\alpha$,
$$
dK_{\epsilon} (\alpha) = d \int_{\epsilon}^r \omega(s) \, ds = dr \wedge \omega(r) +
\int_{\epsilon}^{r} d_F \omega(s) \, ds
$$
$$
= dr \wedge \omega(r) + \int_{\epsilon}^r \phi'(s) \, ds
= dr \wedge\omega(r) + \phi(r) - \phi(\epsilon) = \alpha - \phi(\epsilon).
$$
By proposition \ref{prop:epsilons}, there is some sequence $\epsilon_s \rightarrow 0$ such that
$$\lim_{\epsilon_s \rightarrow 0} ||\phi(\epsilon_s)||^2_{L^2((\rho,1) \times F, g_c)} = 0.
$$
So our equation \ref{eq:dmax2} can be proved as follows:
$$
\langle K_0(\alpha), \delta \beta \rangle_{L^2((\rho,1) \times F, g_c)} =
\lim_{\epsilon \rightarrow 0}
 \langle K_{\epsilon}(\alpha), \delta \beta \rangle_{L^2((\rho,1) \times F, g_c)}
 = \lim_{\epsilon \rightarrow 0}
 \langle dK_{\epsilon}(\alpha), \beta \rangle_{L^2((\rho,1) \times f, g_c)}
$$
$$
= \langle \alpha, \beta \rangle_{L^2((\rho,1) \times F, g_c)} -
 \lim_{\epsilon \rightarrow 0} \langle \phi(\epsilon),  \beta \rangle_{L^2((\rho,1) \times F, g_c)} .
$$
Thus the limit
$$
 \lim_{\epsilon \rightarrow 0} \langle \phi(\epsilon),  \beta \rangle_{L^2((\rho,1) \times F, g_c)}
 $$
 must exist.  This means we can calculate it from any subsequence, such as the subsequence
 $\epsilon_s$ given in proposition \ref{prop:epsilons}.  So we get
 $$
  \lim_{\epsilon \rightarrow 0} \langle \phi(\epsilon),  \beta \rangle_{L^2((\rho,1) \times F, g_c)}
  =
   \lim_{\epsilon_s \rightarrow 0} \langle \phi(\epsilon_s),  \beta \rangle_{L^2((\rho,1) \times F, g_c)}
   $$
   $$
  \leq  \lim_{\epsilon_s \rightarrow 0} ||\phi(\epsilon_s)||_{L^2((\rho,1) \times F, g_c)}
  ||\beta||_{L^2((\rho,1) \times F, g_c)} = 0.
  $$
  So
$$
\langle K_0(\alpha), \delta \beta \rangle_{L^2((\rho,1) \times F, g_c)}
=  \langle \alpha, \beta \rangle_{L^2((\rho,1) \times F, g_c)}
$$
and we have that $d_{max} K_0(\alpha) = \alpha$ as required.
\end{proof}

\section{Intersection pairings and Novikov additivity}

In order to prove Theorem \ref{th:signature}, we first need to check that it makes sense to talk about
a signature on Im$(I\!H^{n/2}_p (X) \rightarrow I\!H^{n/2}_q (X) )$, where $p>q$ are dual
perversities for $X$.

\begin{lemma} Let $p>q$ be dual perversities for intersection cohomology on the
compactification $X$ of a manifold with fibration boundary $M$ described in
the introduction.  There is a well defined (degenerate) bilinear pairing
 $$I\!H^{n/2}_p (X) \otimes I\!H^{n/2}_p (X) \rightarrow \mathbb{R}$$
 given by the map
 $$I\!H^{n/2}_p (X) \rightarrow I\!H^{n/2}_q (X) )$$
 and the nondegenerate bilinear pairing
 $$I\!H^{n/2}_p (X) \otimes I\!H^{n/2}_q (X) ) \rightarrow \mathbb{R}.$$
 It descends to a well-defined nondegenerate bilinear pairing $B_X$
 $$ \mbox{Im}(I\!H^{n/2}_p (X) \rightarrow I\!H^{n/2}_q (X) ) \otimes
 \mbox{Im}(I\!H^{n/2}_p (X) \rightarrow I\!H^{n/2}_q (X) )  \rightarrow \mathbb{R}.
 $$
 \label{lem:intpair}
\end{lemma}
\begin{proof}
By Theorem \ref{th:hodge}, $I\!H^*_p(X)$ and $I\!H^*_q(X)$ can be identified with
$H^*_{min}(M,g_c)$ and $H^*_{max}(M,g_c)$ for some $0 \leq c \leq 1$.
To show that the pairing is well defined and descends to one which is well defined on
$\mbox{Im}(I\!H^{n/2}_p (X) \rightarrow I\!H^{n/2}_q (X))$,
let $\phi, \psi, \theta \in \mathcal{D}(d_{min})$ be closed
and assume that $[\phi]=[\psi] \in H^*_{max}(M,g_c)$.  Then $\delta_{min} * \theta =
\pm * d_{min} \theta = 0$ and
$\phi = \psi + d_{max}\eta$ for some $\eta \in \mathcal{D}(d_{max})$.  So
$$
\int_M \phi \wedge \theta = \int_M (\psi + d_{max} \eta ) \wedge \theta
= \int_M \psi \wedge \theta + \int_M d_{max}\eta \wedge \theta
$$
$$
= \int_M \psi \wedge \theta \pm \int_M d_{max} \wedge *^2 \theta
= \int_M \psi \wedge \theta \pm \langle d_{max}\eta , *\theta \rangle
$$
$$
= \int_M \psi \wedge \theta \pm \langle \eta , \delta_{min} *\theta \rangle
= \int_M \psi \wedge \theta.
$$
Thus the pairing is well defined and descends to one which is also well defined.
To show it descends to a nondegenerate pairing, we need to show that the subspace
$$
W = \{[\phi] \in H_{min}^{n/2}(M,g_c) | \int_M \phi \wedge \psi = 0 \mbox{ for all }
[\psi] \in H_{min}^{n/2}(M,g_c) \}
$$
is the same as the kernel of $i: H^{n/2}_{min}(M,g_c) \rightarrow H^{n/2}_{max}(M,g_c)$.
Let $[\phi] \in \mbox{Ker}(i)$.  Then $\phi = d_{max} \eta$ for some
$\eta \in \mathcal{D}(d_{max})$, and for $[\psi]$ in $\mathcal{D}(d_{min})$, we have
$$
\int_M \phi \wedge \psi = \int_M d_{max} \eta \wedge \psi =\pm  \int_M d_{max}\eta  \wedge *^2 \psi
$$
$$
=\pm  \langle d_{max} \eta , * \psi \rangle = \pm \langle \eta , \delta_{min} * \psi \rangle = 0.
$$
Thus $[\phi] \in W$.  Now suppose that $[\phi]$ is not in $ \mbox{Ker}(i)$.  Then
$[\phi] = i([\phi]) \neq [0] \in H^{n/2}_{max}(M,g_c)$.  Thus by Poincar\'{e} duality,
there is some $[\psi] \in H^{n/2}_{min}(M,g_c)$ with
$\int_M \phi \wedge \psi \neq 0$.  Thus
$[\phi]$ is not in $W$.
So $W$ is exactly Ker$(i)$, and we are done.
\end{proof}

We also need to define a signature on the cone-bundle neighborhood of
the singular stratum of $X$.  Let $Z$ be a pseudomanifold with boundary,
$Y$, which does not intersect the singular part of $Z$.
Then define $I\!H^{k}_{p}(Z,Y)$ to be the cohomology of $Z$ with $p$-perversity intersection
cohomology conditions near the singular set of $Z$ and with relative cohomology conditions
at the boundary, $Y$.  Define
$I\!H^{k}_{p}(Z)$ to be the cohomology of $Z$ with $p$-perversity intersection cohomology conditions
near the singular set of $Z$ and with absolute cohomology conditions at the boundary, $Y$.

Define a (degenerate) pairing on
$$
I\!H^{k}_{p}(Z,Y) \otimes I\!H^{n-k}_{q}(Z,Y)
$$
\noindent via the map
$$
I\!H^{k}_{p}(Z,Y) \longrightarrow I\!H^{k}_{q}(Z,Y) \longrightarrow I\!H^{k}_{q}(Z)
$$
\noindent
and the nondegenerate pairing $B_{ZY}$
$$
I\!H^{k}_{p}(Z,Y) \otimes I\!H^{n-k}_{q}(Z) \longrightarrow \mathbb{R}.
$$
\noindent
Call its signature $\hat{\sigma}_p(Z)$.  In the special case that $Z$ is just a manifold with
boundary, we suppress the $p$, since the signature will be the same for any perversity,
and will simply be the signature of the intersection pairing on
$Im(H^{n/2}(Z, \ensuremath{\partial} Z) \rightarrow H^{n/2}(Z))$.
The proof that this pairing is well defined is similar to the proof of Lemma \ref{lem:intpair}.

Next we need a theorem which allows us to study the signature of a neighborhood
of the boundary separately from the interior of $M$.  We use a version of Novikov Additivity for this.

\begin{theorem}
If $X$ is a pseudomanifold with a single compact smooth singular stratum and if $Y \subset X$ is a compact codimension 1 submanifold such that $X = Z \bigcup_Y Z^{\prime}$ where
$Z \subset \subset X^{\mbox{reg}}$, then
$$
\sigma_p(X) = \hat{\sigma}(Z) + \hat{\sigma}_p(Z^{\prime}).
$$
\label{th:Novikov}
\end{theorem}
\begin{proof}
The proof is a modification of the original Novikov additivity theorem in \cite{AS3}.
Assume that $X$ is  $n$ dimensional, and
let $\hat{H}^{n/2}(Z) = Im(H^{n/2}(Z,Y) \rightarrow H^{n/2}(Z)$ and
$\hat{I\!H}_p^{n/2}(Z^{\prime}) = Im(I\!H_p^{n/2}(Z^{\prime},Y) \rightarrow I\!H_q^{n/2}(Z^{\prime})$

We have exact relative cohomology sequences for $I\!H^*_p(X) $ and $I\!H^*_q(X) $:
$$
\xymatrix{
\cdots \ar[r] & I\!H^{n/2}_p(X,Z)  \ar[r]^{\alpha_p^{\prime}} & I\!H^{n/2}_p(X)  \ar[r]^{\beta_p}
	& I\!H^{n/2}_p(Z)  \ar[r] & \cdots \\
\cdots \ar[r] & I\!H^{n/2}_q(X,Z)  \ar[r]^{\alpha_q^{\prime}} & I\!H^{n/2}_q(X)  \ar[r]^{\beta_q}
	& I\!H^{n/2}_q(Z)  \ar[r] & \cdots \\
\cdots & I\!H^{n/2}_q(Z^{\prime})  \ar[l] & I\!H^{n/2}_q(X)  \ar[l]_{\beta_q^{\prime}}
	& I\!H^{n/2}_q(X,Z^{\prime})  \ar[l]_{\alpha_q} & \cdots \ar[l]\\
\cdots & I\!H^{n/2}_p(Z^{\prime})  \ar[l] & I\!H^{n/2}_p(X)  \ar[l]_{\beta_p^{\prime}}
	& I\!H^{n/2}_p(X,Z^{\prime})  \ar[l]_{\alpha_p} & \cdots \ar[l]\\
}
$$
\noindent
Because $Z$ is smooth, these sequences are the same as:
$$
\xymatrix{
\cdots \ar[r] & I\!H^{n/2}_p(Z^{\prime},Y)  \ar[r]^{\alpha_p^{\prime}} \ar[d]_{i_{Z^{\prime}Y}}
	& I\!H^{n/2}_p(X)  \ar[r]^{\beta_p} \ar[d]_{i_X}
	& H^{n/2}(Z)  \ar[r] \ar[d]_{=}
	& \cdots \\
\cdots \ar[r] & I\!H^{n/2}_q(Z^{\prime},Y)  \ar[r]^{\alpha_q^{\prime}} & I\!H^{n/2}_q(X)  \ar[r]^{\beta_q}
	& H^{n/2}(Z)  \ar[r] & \cdots \\
\cdots & I\!H^{n/2}_q(Z^{\prime})  \ar[l] & I\!H^{n/2}_q(X)  \ar[l]_{\beta_q^{\prime}}
	& H^{n/2}(Z,Y)  \ar[l]_{\alpha_q} & \cdots \ar[l]\\
\cdots & I\!H^{n/2}_p(Z^{\prime})  \ar[l] \ar[u]^{i_{Z^{\prime}}}
	& I\!H^{n/2}_p(X)  \ar[l]_{\beta_p^{\prime}} \ar[u]^{i_X}
	& H^{n/2}_p(Z,Y)  \ar[l]_{\alpha_p} \ar[u]^{=}& \cdots \ar[l] \\
}
$$
\noindent
The maps $i_{Z^{\prime}Y}$, $i_{Z^{\prime}}$ and $i_X$ are induced from the natural inclusion maps on the complexes of forms, and this entire diagram commutes.

Define
$$
A^{\prime} = Im(i_X \circ \alpha_p^{\prime}) \subset Ker(\beta)
$$
$$
A = Im (\alpha_q) = Ker(\beta_q^{\prime} )= Im(i_x \circ \alpha_p).
$$
\begin{proposition}
The spaces $A$ and $A^{\prime}$ are mutual annihilators in
$Im(I\!H^{n/2}_p(X)  \rightarrow I\!H^{n/2}_q(X))$ under the intersection pairing $B_x$.
\end{proposition}
\begin{proof}
Suppose that $i_X \circ \alpha_p^{\prime}[a] \in A^{\prime}$ and $\alpha_q[b] \in A$.
Then
$$
B_X(i_X \circ \alpha_p^{\prime}[a] , \alpha_q[b]) = B_{Z^{\prime}Y}([a], bq^{\prime} \circ \alpha_q[b])
=0.
$$
\noindent
So $A$ and $A^{\prime}$ are contained in each other's annihilators.

Now suppose that $[e] \in Im(I\!H_p^{n/2}(X) \rightarrow I\!H_q^{n/2}(X))$ annihilates $A^{\prime}$
under $B_X$.  Then for every $[c^{\prime}] \in I\!H_p^{n/2}(Z^{\prime},Y)$,
$$
0 = B_X(i_X \circ \alpha_p^{\prime}[c^{\prime}],[e]) = B_{Z^{\prime}Y}([c^{\prime},\beta_q^{\prime}[e]).
$$
\noindent
Since the pairing $B_{Z^{\prime}Y}$ is nondegenerate, this can only happen if
$\beta_q^{\prime}[e]=0$.  But then by exactness of the $I\!H_q$ sequence we have
$[e] \in Im(\alpha_q) = A$.  So $A$ is the annihilator of $A^{\prime}$ under $B_Z$.

Finally, suppose that $i_X[e^{\prime}] \in Im(I\!H_p^{n/2}(X) \rightarrow I\!H_q^{n/2}(X))$ annihilates
$A$ under $B_X$.  Then for all $[c] \in H^{n/2}(Z,Y)$,
$$
0 = B_X( \alpha_q[c], i_X[e^{\prime}]) = B_{ZY}([c], \beta_q\circ i_X[e^{\prime}])
= B_{ZY}([c], \beta_p[e^{\prime}]).
$$
\noindent
This implies that $\beta_p[e^{\prime}]=0$, so by exactness of the $I\!H_p$ sequence,
$[e^{\prime}] \in Im (\alpha_p^{\prime})$.  So $i_X[e^{\prime}] \in A^{\prime}$,
and we get that $A^{\prime}$ is also the annihilator of $A$.
\end{proof}

Since the pairing $B_Z$ is nondegenerate, we get as in the original Novikov additivity proof
that
$$
(A \cap A^{\prime})^* \cong \frac{Im(I\!H_p^{n/2}(X) \rightarrow I\!H_q^{n/2}(X))}{A+A^{\prime}}.
$$
\noindent
Also,
$$
\frac{A+A^{\prime}}{A \cap A^{\prime}} = \frac{A}{A \cap A^{\prime}} \oplus \frac{A^{\prime}}{A \cap A^{\prime}}
\cong
\frac{Im(\alpha_q)}{Im(\alpha_q) \cap Im(i_X \circ \alpha_p^{\prime})} \oplus
\frac{Im(i_X \circ \alpha_p^{\prime})}{Im(\alpha_q) \cap Im(i_X \circ \alpha_p^{\prime})}.
$$
\noindent
From the exact sequences, we know that
$$
\frac{Im(i_X \circ \alpha_p^{\prime})}{Im(\alpha_q) \cap Im(i_X \circ \alpha_p^{\prime})} \cong
\frac{Im(i_X \circ \alpha_p^{\prime})}{ker(\beta_q^{\prime}) \cap Im(i_X \circ \alpha_p^{\prime})}
\cong Im(\beta_q^{\prime} \circ i_z \circ \alpha_p^{\prime}) \cong
\hat{I\!H}^{n/2}_p(Z^{\prime}).
$$
\noindent
And also that
$$
\frac{Im(\alpha_q)}{Im(\alpha_q) \cap Im(i_X \circ \alpha_p^{\prime})}
\cong
\frac{Im(i_X \circ \alpha_p)}{Im(i_X \circ \alpha_p)  \cap i_X(Ker(\beta_q))}
\cong \frac{Im(i_X \circ \alpha_p)}{Im(i_X \circ \alpha_p)  \cap Im(i_X) \cap (Ker(\beta_q)}.
$$
\noindent
Since $Im(i_X \circ \alpha_p) \subset Im(i_X)$, we can eliminate the $Im(i_X)$ term in
the denominator to get
$$
\cong \frac{Im(i_X \circ \alpha_p)}{Im(i_X \circ \alpha_p)
\cap Ker(\beta_q)}
\cong
\frac{Im(\alpha_q)}{Im(\alpha_q) \cap Ker(\beta_q)} \cong \hat{H}^{n/2}(Z).
$$
\noindent
So altogether, we have
$$
\frac{A+A^{\prime}}{A \cap A^{\prime}} \cong \hat{H}^{n/2}(Z) \oplus \hat{I\!H}^{n/2}_p(Z^{\prime}).
$$

By splitting the maps
$$
A \cap A^{\prime} \rightarrow A + A^{\prime} \rightarrow Im(I\!H_p^{n/2}(X) \rightarrow I\!H_q^{n/2}(X))
$$
\noindent
we get that
$$
Im(I\!H_p^{n/2}(X) \rightarrow I\!H_q^{n/2}(X)) \cong  \hat{H}^{n/2}(Z) \oplus \hat{I\!H}^{n/2}_p(Z^{\prime})
\oplus [(A \cap A^{\prime}) \oplus (A \cap A^{\prime})^*].
$$
As in the proof of the original Novikov additivity, by choosing a good splitting, we can arrange
for the form $B_X$ to be given with respect to this splitting by the block matrix
$$
\left(
\begin{array}{lll}
B_Z & 0 & 0 \\
0 & B_{Z^{\prime}} & 0 \\
0 & 0 & C
\end{array}
\right),
$$
\noindent
where $C$ is the natural form on $[(A \cap A^{\prime}) \oplus (A \cap A^{\prime})^*]$.  The
signature of such a form is always 0 (p. 580 \cite{AS3}), so we obtain
$$
\sigma_p(X) = \hat{\sigma}_p(Z) + \hat{\sigma}_p(Z^{\prime})
$$
\noindent
as we wanted.

\end{proof}

\section{Signature of the end}

Now return to our original situation, where $\overline{M}$ is an $n$ dimensional manifold
with boundary $Y\stackrel{\phi}{\rightarrow} B$
with fibre $F$ and $X$ is the pseudomanifold formed by coning off the fibres of $Y$.  As before,
let $f$ be the dimension of F and $b$ be the dimension of $B$ so that $f+b+1=n$.
Decompose $X$ as $X^{reg} \cup_Y C_{\phi}Y$, where $C_{\phi}Y$ is the pseudomanifold with
boundary formed by coning off the fibres of $Y$.  Assume that $p = \underline{\mathfrak{m}} + k$ and
$q = \overline{\mathfrak{m}} -k$.
By Theorem \ref{th:Novikov}, to prove Theorem \ref{th:signature}, it suffices to prove the
\begin{lemma}
The signature on $C_{\phi}Y$ is
$$
\hat{\sigma}_p(C_{\phi}Y^{\prime}) = - \sum_{i= 2 + 2k} ^{\infty}\tau_i.
$$
\label{lem:signend}
\end{lemma}
\noindent
Once we have this, reversing orientation to glue, we get
$$
\sigma_p(X) = \hat{\sigma} (M) + \hat{\sigma}_p(C_{\phi}Y) =
sgn \, Im(H^{n/2}(M, \ensuremath{\partial} M) \rightarrow H^{n/2}(M)) + \sum_{i= 2 + 2k} ^{\infty}\tau_i
$$
\noindent as required.

The proof of Lemma \ref{lem:signend} generalizes the techniques of \cite{CD}.
\begin{proof}
The first step is to identify $I\!H_q^*(C_{\phi}Y)$ and $Im(I\!H_p^*(C_{\phi}Y, Y) \rightarrow I\!H_q^*(C_{\phi}Y))$ in terms of the Leray-Serre spectral sequence for the fibration on $Y$.
We will prove the
\begin{proposition} In terms of the $E_\infty^{i,j}(Y)$ terms of the Leray-Serre spectral sequence and
its differentials $d_r^{i,j}: E_r^{i,j}(Y) \rightarrow E_r^{i+r, j-r+1}(Y)$, we get the isomorphisms:
$$
I\!H_q^{n/2}(C_{\phi}Y) \cong \left\{
\begin{array}{ll}
\bigoplus_{\stackrel{i+j=n/2}{j\leq f/2 + k}} (E_{\infty}^{i,j}(Y) \oplus \sum_{s=1}^{\infty} Im(d_s^{i-s, j+s-1})
& b/2 - k >0 \\
\bigoplus_{\stackrel{i+j=n/2}{ j\leq f/2 + k}} E_{\infty}^{i,j}(Y) & b/2 - k \leq 0
\end{array} \right.
$$
$$
Im(I\!H_p^{n/2}(C_{\phi}Y, Y) \rightarrow I\!H_q^{n/2}(C_{\phi}Y)) \cong  \left\{
\begin{array}{ll}
\bigoplus_{\stackrel{i+j=n/2}{ j \geq (f+3)/2 + k}}  \sum_{t \geq j-(f-1)/2 +k} Im(d_t^{i, j-1})
& b/2 - k >0 \\
0 & b/2 - k \leq 0
\end{array} \right.
$$
\end{proposition}
\begin{proof}
Recall that by Theorem \ref{th:hodge}, $I\!H_q^*(C_{\phi}Y) \cong H_{max}^*(C_{\phi}Y, g_c)$
for some metric $g_c$, $0\leq c \leq 1$
and $I\!H_p^*(C_{\phi}Y) \cong H_{min}^*(C_{\phi}Y, g_c)$, where $H_{min}^*(C_{\phi}Y, g_c)$ means
cohomology which is  minimal with respect to the metric $g_c$ near the singular stratum of
$C_{\phi}Y$ and absolute near the boundary $Y$.  Further, as for regular cohomology,
$I\!H^*_p(C_{\phi}Y,Y)$ is the cohomology of the complex
$(\Omega^{*-1}(Y) \oplus \Omega_{min}^*(C_{\phi}Y, g_c), d_{rel})$ where $\Omega_{min}^*(C_{\phi}Y)$
is the space of smooth forms on $C_{\phi}Y$ which satisfy minimal boundary conditions
for $g_c$ near the singular stratum and absolute boundary conditions
near the boundary $Y$ and where $d_{rel}(\theta, \alpha) = (d_Y \theta - \alpha(1), d_{c} \alpha)$.
So the short exact sequence
$$
0 \rightarrow \Omega^{*-1}(Y) \stackrel{\gamma}{\rightarrow} \Omega^{*-1}(Y) \oplus \Omega^*_{min}(C_{\phi}Y)
\stackrel{\beta}{\rightarrow} \Omega^*_{min}(C_{\phi}Y) \rightarrow 0
$$
\noindent
induces a long exact sequence on cohomologies.  Similarly, we get a long exact sequence on
cohomologies corresponding to the perversity $q$:
$$
0 \rightarrow \Omega^{*-1}(Y) \stackrel{\gamma}{\rightarrow} \Omega^{*-1}(Y) \oplus \Omega^*_{max}(C_{\phi}Y)
\stackrel{\beta}{\rightarrow} \Omega^*_{max}(C_{\phi}Y) \rightarrow 0
$$
Using the metric $g_c$, each of these spaces of forms can be filtered by
(conic) fibre and base bi-degree as in section ?? of \cite{HHM}.
For any any base degree, {i}, we get the following commutative diagram
relating $q$-perversity complexes:
\begin{equation}
\begin{xy}
\xymatrix{
&\vdots &  \vdots  & \vdots & \\
0 \ar[r] &
\Omega^{i,1}(Y) \ar[u]^{d_F} \ar[r]^{\gamma} &
\Omega^{i,1}(Y) \oplus \Omega^{i,2}_{max}(C_{\phi}Y) \ar[u]^{d_{rel}} \ar[r]^{\beta} &
 \Omega^{i,2}_{max}(C_{\phi}Y) \ar[u]^{d_c} \ar[r] & 0\\
0 \ar[r] &
\Omega^{i,0}(Y) \ar[u]^{d_F} \ar[r]^{\gamma} &
\Omega^{i,0}(Y) \oplus \Omega^{i,1}_{max}(C_{\phi}Y) \ar[u]^{d_{rel}} \ar[r]^{\beta}  &
 \Omega^{i,1}_{max}(C_{\phi}Y) \ar[u]^{d_c} \ar[r]  & 0\\
&
0 \ar[u] \ar[r] &
0 \oplus \Omega^{i,0}_{max}(C_{\phi}Y) \ar[u]^{d_{rel}} \ar[r]^{\beta} &
 \Omega^{i,0}_{max}(C_{\phi}Y) \ar[u]^{d_c} \ar[r] & 0
}
\end{xy}
\label{eq:fibreexactsequences}
\end{equation}
The $E_1$ terms of the Leray-Serre spectral sequence for each base degree $i$ fit into the exact sequence:
$$
\cdots \rightarrow E_1^{i, j-1}(C_{\phi}Y,q) \rightarrow
E_1^{i,j-1}(Y) \rightarrow E_1^{i,j}(C_{\phi}Y, Y,q) \rightarrow
E_1^{i, j}(C_{\phi}Y,q) \rightarrow E_1^{i,j}(Y) \rightarrow\cdots .
$$
\noindent
By the Poincar\'{e} lemma for the maximal complex, Lemma \ref{lem:poincare} we get  that for $j \leq \frac{f-1}{2} +k$ if $f$ is odd and $j \leq \frac{f}{2} +k$ if $f$ is even, this is
$$
\rightarrow \Omega^i(B, H^{j-1}(F))
\rightarrow \Omega^i(B,H^{j-1}(F)) \rightarrow E_1^{i,j}(C_{\phi}Y, Y,q) \rightarrow
\Omega^i(B, H^j(F)) \rightarrow  \Omega^i(B,H^{j}(F)) \rightarrow .
$$
\noindent
So for $j \leq \frac{f-1}{2} +k$ if $f$ is odd and $j \leq \frac{f}{2} +k$ if $f$ is even, we find
$E_1^{i,j}(C_{\phi}Y, Y,q)  =0$.
For $j \geq \frac{f-1}{2} +k +2$ if $f$ is odd and $j \geq \frac{f}{2} +k+2$ if $f$ is even, we have
$$
\cdots \rightarrow 0
\rightarrow \Omega^i(B,H^{j-1}(F)) \rightarrow E_1^{i,j}(C_{\phi}Y, Y,q) \rightarrow
0 \rightarrow \Omega^i(B,H^{j}(F)) \rightarrow\cdots .
$$
\noindent
So for $j \geq \frac{f-1}{2} +k +2$  if $f$ is odd and $j \leq \frac{f}{2} +k+2$ if $f$ is even, we find $E_1^{i,j}(C_{\phi}Y, Y,q) \cong \Omega^i(B,H^{j-1}(F))$.
Thus for the boundary degree $j= \frac{f-1}{2} +k+1$ if $f$ is odd and $j = \frac{f}{2} +k+1$ if $f$ is even, we have (dropping the last term and adding one previous term):
$$
\rightarrow 0 \rightarrow \Omega^i(B, H^{(f-1)/2 + k}(F))
\rightarrow \Omega^i(B,H^{(f-1)/2 +k}(F)) \rightarrow E_1^{i,(f-1)/2+k+1}(C_{\phi}Y, Y,q) \rightarrow
0 \rightarrow ,
$$
\noindent
so $E_1^{i,(f-1)/2+k+1}(C_{\phi}Y, Y,q)=0$ if $f$ is odd and $E_1^{i,f/2+k+1}(C_{\phi}Y, Y,q)=0$ if $f$ is even.

The differential $d_1$ is horizontal, so it does not
see the truncations.  Thus the $E_2$ levels of the relative and absolute $p$-perversity spectral sequences are for $f$ odd:

Relative $q$ sequence:
\begin{equation}
\begin{array}{llllll}
j=f+1 & &H^0(B, H^{f}(F)) & H^1(B, H^{f}(F)) & H^1(B, H^{f}(F)) 	&\cdots \\
\vdots && \vdots                 & \vdots                  & \vdots                    & \cdots \\
j = \frac{f+3}{2} +k & & H^0(B, H^{(f+1)/2 +k}(F)) & H^1(B, H^{(f+1)/2 +k}(F)) & H^2(B, H^{(f+1)/2 +k}(F))
	&\cdots \\
j = \frac{f+1}{2} +k & & 0&0&0&\cdots \\
\vdots && \vdots & \vdots & \vdots & \cdots \\
j =0 & & 0&0&0&\cdots \\
E_2^{i,j}(C_{\phi}Y, Y, q) && i=0&i=1&i=2& \cdots
\end{array}
\label{eq:relq}
\end{equation}

Absolute $q$ sequence:
\begin{equation}
\begin{array}{l l l l l l}
                     \vdots & & \vdots  				& \vdots & \vdots & \\
j = \frac{f+1}{2} +k & & 0        				 &           0&          0&\cdots \\
j = \frac{f-1}{2} +k  & & H^0(B, H^{(f-1)/2 +k}(F)) & H^1(B, H^{(f-1)/2 +k}(F)) & H^2(B, H^{(f-1)/2 +k}(F))
	&\cdots \\
\vdots && \vdots & \vdots & \vdots & \cdots \\
j =0 & & H^0(B, H^{0}(F)) & H^1(B, H^{0}(F)) & H^2(B, H^{0}(F)) &\cdots \\
E_2^{i,j}(C_{\phi}Y, q) && i=0&i=1&i=2& \cdots
\end{array}
\label{eq:absq}
\end{equation}

\noindent And using an analogous argument for the minimal complex, we get that the $E_2$ term of the
relative $p$-perversity spectral sequence is:

\begin{equation}
\begin{array}{llllll}
j=f+1 & &H^0(B, H^{f}(F)) & H^1(B, H^{f}(F)) & H^1(B, H^{f}(F)) 	&\cdots \\
\vdots && \vdots                 & \vdots                  & \vdots                    & \cdots \\
j = \frac{f+3}{2} -k & & H^0(B, H^{(f+1)/2 -k}(F)) & H^1(B, H^{(f+1)/2 -k}(F)) & H^2(B, H^{(f+1)/2 -k}(F))
	&\cdots \\
j = \frac{f+1}{2} -k & & 0&0&0&\cdots \\
\vdots && \vdots & \vdots & \vdots & \cdots \\
j =0 & & 0&0&0&\cdots \\
E_2^{i,j}(C_{\phi}Y, Y, p) && i=0&i=1&i=2& \cdots
\end{array}
\label{eq:relp}
\end{equation}

In the case that $f$ is even, these diagrams are similar, but with zeros
below level $f/2 + k$ for the relative $q$ spectral sequence, and shifted up by one as in the odd case,
zeros below $f/2-1- k$ for the relative $p$ spectral sequence, and again shifted up by one,
and zeros above level $f/2 + k$ for the absolute $q$ spectral sequence.

These are all truncated (and in the relative case, shifted) copies of the Leray-Serre spectral sequence for
the fibration on $Y$.  So the higher levels of these spectral sequences will be the same as the
higher levels of the spectral sequence for $Y$, but with term added because of the truncation.
By definition, $d_s \circ d_{s+1} = 0$ for any sequential differentials in the spectral sequence on $Y$.  Thus in the case of $E_{\infty}(I\!H_q^*(C_{\phi}Y))$ and $E_{\infty}(I\!H_p^*(C_{\phi}Y,Y)$, the extra terms will always have the form of $Im(d_s^{i,j})$ and will come from cases where there is no longer any image to quotient by because of the truncation above.  In the
case of $E_{\infty}(I\!H_q^*(C_{\phi}Y,Y)$, the extra terms will have the form of a preimage $Im(d_s^{i,j})^*$ and will come from the
cases where because of the truncation below, the kernel of the differential in the truncated sequence contains what would have been the image of the differential in the spectral sequence for $Y$.

It is easiest to understand this in an example.  Take the case when the base dimension $b=6$, the fibre dimension $f=5$ and the perversities are $p=\underline{\mathfrak{m}} + 1$ and $q = \overline{\mathfrak{m}} -1$.
Refer to the figures at the end of this paper for this example.
The the $E_{\infty}$ term of the spectral sequence for $I\!H_q^*(C_{\phi}Y)$ looks like figure \ref{fig:qabs},
the $E_{\infty}$ term of the spectral sequence for $I\!H_q^*(C_{\phi}Y, Y)$ looks like figure \ref{fig:qrel}
and the $E_{\infty}$ term of the spectral sequence for $I\!H_p^*(C_{\phi}Y, Y)$ looks like figure \ref{fig:prel}.
So in this example, we get for instance:
$$
E^{2,5}_\infty(I\!H_{\overline{\mathfrak{m}} -1}^*(C_{\phi}Y,Y)) = E_{\infty}^{2,4}(Y) + Im(d_2^{2,4})^* + Im(d_3^{2,4})^*+
Im(d_4^{2,4})^*
$$
$$
E^{3,3}_\infty(I\!H_{\overline{\mathfrak{m}} -1}^*(C_{\phi}Y)) = E_{\infty}^{3,3}(Y) + Im(d_2^{1,4}) + Im(d_3^{0,5})
$$
\noindent and
$$
E^{2,5}_\infty(I\!H_{\underline{\mathfrak{m}} + 1}^*(C_{\phi}Y,Y)) = E_{\infty}^{2,4} (Y)+ Im(d_4^{2,4})^*
$$
In general we get that:
$$
E^{i,j}_\infty(I\!H_q^*(C_{\phi}Y)) \cong
\left\{
\begin{array}{ll}
0 & j> f/2 +k \\
E_{\infty}^{ij}(Y) + \sum_{s \geq 1} Im(d_s^{i-s,j+s-1}) & j \leq f/2 + k, \\
&i+j > (f+1)/2 +k \\
E_{\infty}^{ij}(Y) & \mbox{ otherwise, }
\end{array} \right.
$$
$$
E^{i,j}_\infty(I\!H_q^*(C_{\phi}Y,Y)) \cong
\left\{
\begin{array}{ll}
0 & j<(f+3)/2 +k \\
E_{\infty}^{i,j-1}(Y) + \sum_{s \geq j - f/2 - k} Im(d_s^{i,j-1})^* & j \geq (f+3)/2 + k,\\
& i+j < (f+1)/2+b +k \\
E_{\infty}^{i,j-1}(Y) & \mbox{ otherwise, }
\end{array} \right.
$$
\noindent and
$$
E^{i,j}_\infty(I\!H_p^*(C_{\phi}Y,Y)) \cong
\left\{
\begin{array}{ll}
0 & j<f/2 +1-k \\
E_{\infty}^{i,j-1}(Y) + \sum_{s \geq j - (f-1)/2 + k} Im(d_s^{i,j-1})^* & j \geq f/2 + 1-k, \\
&i+j < (f+1)/2+b -k \\
E_{\infty}^{i,j-1}(Y) & \mbox{ otherwise.}
\end{array} \right.
$$
When we map $I\!H_p^*(C_{\phi}Y,Y)$ to $I\!H_q^*(C_{\phi}Y,Y)$ we simply truncate the bottom of
$E^{i,j}_\infty(I\!H_p^*(C_{\phi}Y,Y)) $ by $2k$ levels, if $f$ is odd, and by $2k+1$ levels if $f$
is even.  Modfying slightly the argument from \cite{CD} we have the
\begin{proposition}
In terms of the spectral sequence, the map $I\!H_q^*(C_{\phi}Y,Y) \rightarrow I\!H_q^*(C_{\phi}Y)$
is zero on the terms of the form $E_{\infty}^{i,j}(Y)$ and is given by applying the appropriate
$d_r$'s to the other factors.
\label{prop:incmap}
\end{proposition}
\begin{proof}
First consider a form $[\theta]$ in $Im(d_s^{i,j-1})^*$.  Then $d_s \theta$ will be in
$E_s^{i+s, j-s}$ where $j-s$ is below the level of truncation for the spectral sequence.
That is, $j-s \leq (f-1)/2 + k$ if $f$ is odd and $f \leq f/2 + k$ if $f$ is even.  So $d \theta$ will have
fibre degree $\leq f/2 + k$.  From Theorem \ref{th:hodge}, we know that in terms of
maximal cohomology on $(C_{\phi}Y, g_c)$, $k = [[1+1/2c]]$ if $f$ is even and $k = [[1/2 + 1/2c]]$
if $f$ is odd.  So in either case, the fibre degree of $d\theta < f/2 + 1/2c$.  By Proposition 2,
this guarantees that $d\theta$ extends to an $L^2$ form on $C_{\phi}Y$.
In terms of the complex for relative maximal cohomology, the class
$[\theta, 0] \in \Omega^{*-1}(Y) \oplus \Omega^*(C_{\phi}Y, g_c)$ is equivalent to
$[\theta, \alpha]$ in $E_s$ for any $\alpha in  \Omega^*(C_{\phi}Y, g_c)$, so choose
$\alpha = d\theta$ to get a representative $[\theta, d\theta]$ which is in $I\!H_q^*(C_{\phi}Y,Y)$.
Then under the map to absolute cohomology, this goes to $[d\theta] \in I\!H_q^*(C_{\phi}Y)$.
The map from $\Omega^*(C_{\phi}Y,g_c)$ to $\Omega^*(Y)$ which induced the identification
of $I\!H_q^*(C_{\phi}Y)$ with spectral sequence terms for $Y$ is given by restriction.
Since $d\theta$ is constant in the $r$ direction, this is just $d_Y\theta$, which by
identification with spectral sequence terms is just $d_s \theta$.

Now suppose that $[\theta] \in E_{\infty}^{i,j-1}(Y)$.  Then $d_Y \theta = 0$, so by the same
argument as above, we get that $[\theta]$ goes to $[0]$ under the map
$I\!H_q^*(C_{\phi}Y,Y) \rightarrow I\!H_q^*(C_{\phi}Y)$.

\end{proof}

So we get that
$$
Im(E_{\infty}^{ij}(C_{\phi}Y, Y, p) \rightarrow E_{\infty}^{ij}(C_{\phi}Y, q)) \cong
\left\{
\begin{array}{ll}
\sum_{s \geq j - (f-1)/2+k} Im(d_s^{i,j-1}) & i+j < (f+1)/2 +b-k ,\\
&j \geq (f+3)/2 +k\\
0 & \mbox{ otherwise}
\end{array}
\right. .
$$
\noindent
We obtain the result of the lemma by summing over $i+j=n/2=(b+f+1)/2$.
\end{proof}

Now we have to understand the signature pairing on
$Im(I\!H_p^{n/2}(C_{\phi}Y, Y) \rightarrow I\!H_q^{n/2}(C_{\phi}Y))$
in terms of this decomposition.  To do this, first return to our example where $f=5$, $b=6$ and $k=1$.
We have
$$
I\!H_{\overline{\mathfrak{m}}-1}^6(C_{\phi}Y)) \cong
 E_{\infty}^{6,0}(I\!H^*_q(C_{\phi}Y))+ E_{\infty}^{5,1}(I\!H^*_q(C_{\phi}Y))+ E_{\infty}^{4,2}(I\!H^*_q(C_{\phi}Y))
+ E_{\infty}^{3,3}(I\!H^*_q(C_{\phi}Y))
$$
and
$$
Im(I\!H_{\underline{\mathfrak{m}}+1}^6(C_{\phi}Y, Y) \rightarrow I\!H_{\overline{\mathfrak{m}}-1}^6(C_{\phi}Y,Y))
$$
$$
\cong (Im(E_{\infty}^{0,6}(I\!H^*_p(C_{\phi}Y, Y)) \rightarrow E_{\infty}^{0,6}(I\!H^*_q(C_{\phi}Y,Y)))
+( Im(E_{\infty}^{1,5}(I\!H^*_p(C_{\phi}Y, Y)) \rightarrow E_{\infty}^{1,5}(I\!H^*_q(C_{\phi}Y,Y)) )
$$
Only forms with complementary bi-degrees for the the cone bundle $C_{\phi}Y$ can have nontrivial pairing.  By the filtration on Leray Serre spectral sequences, (see \cite{CD}), we can
always represent a class in $E_s^{i,j}(C_{\phi}Y, q)$ or  $E_s^{i,j}(C_{\phi}Y, Y, p)$ by a form $\theta$ which
is a sum of forms of bidegree $(i-a, j+a)$ for $a >0$ and for which $d \theta$ is a sum of forms whose
base degree is at least $j+s$.   Thus terms in the decomposition \ref{eq:decomp} whose $i$ indices add
to more than the dimension of the base will have trivial intersection.
Therefore we can represent the intersection form by a 4 by 2 block lower-triangular matrix:
$$
\left(
\begin{array}{llll}
A & 0 & 0 & 0\\
* & B & 0 & 0
\end{array}
\right),
$$
\noindent
where $A$ is the pairing between
$Im(E_{\infty}^{0,6}(I\!H^*_p(C_{\phi}Y, Y)) \rightarrow E_{\infty}^{0,6}(I\!H^*_q(C_{\phi}Y,Y))$ and
$E_{\infty}^{6,0}(I\!H^*_q(C_{\phi}Y))$ and $B$ is the pairing between
$Im(E_{\infty}^{1,5}(I\!H^*_p(C_{\phi}Y, Y)) \rightarrow E_{\infty}^{1,5}(I\!H^*_q(C_{\phi}Y,Y))$ and
$E_{\infty}^{5,1}(I\!H^*_q(C_{\phi}Y))$.

Now we have to understand these pieces.  First consider $A$.
We can further decompose $A$ into a 3 by 4 block matrix corresponding to the
decompositions:
$$
E_{\infty}^{6,0}(I\!H^*_q(C_{\phi}Y)) \cong E_{\infty}^{6,0}(Y) + Im(d_6^{0,5})+ Im(d_5^{1,4}) .
$$
$$
(Im(E_{\infty}^{0,6}(I\!H^*_p(C_{\phi}Y, Y)) \rightarrow E_{\infty}^{0,6}(I\!H^*_q(C_{\phi}Y,Y)))
\cong
E_{\infty}^{0,5}(Y) + Im(d_6^{0,5})^*+ Im(d_5^{0,5})^* + Im(d_4^{0,5})^*
$$
We know that the entire intersection pairing descends to a nondegenerate pairing on
$$
Im(I\!H_p^{n/2}(C_{\phi}Y, Y) \rightarrow I\!H_q^{n/2}(C_{\phi}Y))
$$
\noindent
which contains no terms of the form $E_{\infty}^{i,j}(Y)$ by Proposition \ref{prop:incmap}.
Thus such terms must pair trivially with everything.  So the first row and the first
column of block matrices in $A$ are trivial.

From the exact sequences in \ref{eq:fibreexactsequences}, we know that under the identifications
above, an element $[\theta] \in E_r^{i,j}(Y)$ lifts to the element
$[d\theta, \theta] \in I\!H_{\underline{\mathfrak{m}}+1}^6(C_{\phi}Y, Y)$, which is constant in the $r$-direction
on $C_{\phi}Y$, and an element $[\alpha] \in E_\infty^{i,j}(M) + \sum Im(d_r)$ lifts to
$[\alpha] \in I\!H_q^{n/2}(C_{\phi}Y)$.
Then for the lifts $[d\theta, \theta] \in I\!H_p^{n/2}(C_{\phi}Y, Y)$ and $\alpha \in I\!H_q^{n/2}(C_{\phi}Y)$,
the pairing is given by
$$
\langle [\theta],[\alpha] \rangle = \int_{C_{\phi}Y} d\theta \wedge \alpha - \int_M \theta \wedge \alpha.
$$
Neither $d\theta$ nor $\alpha$ contains a $dr$ term, so the first integral always vanishes.
Thus we are left with just the second term.

If the level of $[\theta]$ is greater than the level of $[\alpha]$ in the spectral sequence for $Y$, e.g. $[\theta] \in Im(d_6^{0,5})^*$ and $[\alpha] \in Im(d_5^{1,4})$,
then we can choose $\theta$ to be a sum of forms with base degree greater than or equal to 0 and such that $d\theta$ has base degree greater than or equal to $i+r=6$ and fibre degree less than or equal to $j-r+1=0$. Similarly, we can choose $\alpha$ to have base degree greater than or equal to 6 and to equal $d_5\eta$ for a form of base degree greater than or equal to $1$ and fibre degree less than or equal to $4$.  So we have
$$
\langle [\theta],[\alpha] \rangle = -\int_Y \theta \wedge \alpha = -\int_Y \theta \wedge d_5 \eta
$$
\noindent Only the lowest base degree
parts of these forms will pair nondegenerately on $Y$, so we can add the other terms of $d \eta$
(which will all have higher base degree) to $d_5 \eta$ without changing the integral.  So we have
$$
= - \int_Y \theta \wedge d\eta = \int_Y d\theta \wedge \eta
$$
and similarly, we can eliminate all but $d_6 \theta$ without changing the integral to get:
$$
= \int_Y d_6 \theta \wedge \eta.
$$

Now $d_6 \theta$ has base degree greater than or equal to 6
and $\eta$ has base degree greater than or equal to 1.
So none of their component forms have complementary bidegree since the base degrees add to
more than 6.  So this integral vanishes.
Thus the block matrix for $A$ is also of lower triangular form:
$$
A = \left(
\begin{array}{lll}
0 & 0 & 0  \\
0 & A_1& 0 \\
0 & * & A_2 \\
0 & * & *
\end{array}
\right),
$$
\noindent
where $A_1$ is the intersection matrix for $Im(d_6^{0,5})^* \otimes Im(d_6^{0,5})$
and $A_2$ is the intersection matrix for $Im(d_5^{0,5})^* \otimes Im(d_5^{1,4})$ .
The pairing $A_1$ is given by
$$
Im(d_6^{0,5})^* \otimes Im(d_6^{0,5})^* \rightarrow \mathbb{R}
$$
$$
\omega  , \alpha \rightarrow -\langle \omega \cdot d_6 \alpha, \zeta_6 \rangle
$$
where $\zeta_6$ is the volume form in $E_6$.
The signature of $A_1$ is therefore exactly $-\tau_6$.

By a similar decomposition, $B$ has the form:
$$
B = \left(
\begin{array}{lll}
0 & 0& 0 \\
0 & B_1 & 0 \\
0 & * & B_2
\end{array}
\right),
$$
\noindent where $B_1$ is the intersection matrix for $Im(d_5^{1,4})^* \otimes  Im(d_5^{0,5})$
and $B_2$ is the intersection matrix for $Im(d_4^{1,4})^* \otimes  Im(d_4^{1,4})$.
The signature of $B_2$ is exactly $-\tau_4$, and the matrix $B_1 = - A_2$ since for
any $\omega \in Im(d_5^{1,4})^*$ and $\alpha \in Im(d_5^{0,5})$, we have
$$
\langle [\omega],[\alpha] \rangle = - \int_Y \omega \wedge \alpha
= - \int_Y \omega \wedge d_5 \eta
$$
$$
= - \int_Y \omega \wedge d \eta = \int_Y d\omega \wedge \eta = \int_Y d_5 \omega \wedge \eta
$$
$$
= - \langle [\gamma], [\eta] \rangle
$$
for some classes $[\gamma] \in Im(d_5^{1,4})$ and $\eta \in Im(d_5^{0,5})^*$.
So the signatures of $B_1$ and $A_2$ cancel.  We can observe from this argument also that
in this case $\tau_5 = 0$, and so we are left with
$$
\hat{\sigma}_p(C_{\phi}Y) = -\tau_4 - \tau_6 = \sum_{s=2+2k} \tau_s.
$$

The general argument is similar.  That is, since only complementary bidegrees will pair, we can always represent the intersection pairing on
$Im(I\!H_p^{n/2}(C_{\phi}Y, Y) \rightarrow I\!H_q^{n/2}(C_{\phi}Y))$
by a block lower triangular matrix.  By integration by parts and the degeneracy of terms of the form
$E_{\infty}^{i,j}$ in the pairing, each block will further decompose into a block lower triangular matrix.  One of these sublocks will have signature $\tau_s$ for some $s$
and the rest will have signature which cancels the signature of one of the sub-blocks of
another block in the large matrix.  The signature of the large matrix will thus be the
sum of $\tau_s$'s.  If $f$ is odd, only even $\tau_s$ will contribute and if $f$ is even, only
odd $\tau_s$ will contribute.  The first $s$ to appear comes from the lowest appropriate parity
$d_s$ which appears in
$Im(E_{\infty}^{i,j} (I\!H_p^*(C_{\phi}Y, Y)) \rightarrow E_{\infty}^{i,j} (I\!H_q^*(C_{\phi}Y, Y))$,
which will be $s = 2 + 2k$ if $f$ is odd and $s = 3 + 2k$ if $f$ is even.  Since the opposite parity
$\tau_s$ vanish, we can simply write
$$
\hat{\sigma}_p(C_{\phi}Y) = - \sum_{s=2+ 2k} \tau_s
$$
and we are done.

\end{proof}

\section{Further work}
It seems likely that the theorems in this paper should generalize.  For instance, by relying more on strictly topological methods and using the basic definitions and results about intersection cohomology, it should be possible to define a signature pairing for any perversity intersection cohomology on any pseudomanifold and to prove a more general version of Novikov additivity:
\begin{conjecture}
If $X^n$ is a pseudomanifold and if $Y \subset X$ is a compact codimension 1 submanifold such that
$X = Z \bigcup_Y Z^{\prime}$ where
$Y \subset \subset X^{\mbox{reg}}$, then for any perversity function, $\mathfrak{p}$, the signature
of the intersection pairing defined on $IH^{n/2}_{\mathfrak{p}}(X)$ satisfies:
$$
\sigma_{\mathfrak{p}}(X) = \hat{\sigma}_{\mathfrak{p}}(Z) + \hat{\sigma}_{\mathfrak{p}}(Z^{\prime}),
$$
\noindent
where as before, $\hat{\sigma}_{\mathfrak{p}}(Z)$ is the signature for the cohomology that satisfies
perversity $\mathfrak{p}$ type conditions away from $Y$ and relative boundary conditions at $Y$.
\end{conjecture}

Together with Daniel Grieser the author is currently proving the following conjecture which also relates the topological signatures considered in this paper to $L^2$ signatures for a family of complete metrics on $M$ that interpolates between fibred cusp and cylindrical metrics:

\begin{conjecture} If $M$ is a manifold with boundary fibration
$\ensuremath{\partial} M \stackrel{\phi}{\rightarrow} B$
and the metric on
$M$ is quasi-isometric near the boundary to one of the form
$$
ds_c^2 = R^{2c}(dR^2 + k) + \phi^*ds_B^2,
$$
where $R \in [1, \infty)$, $-1 \leq c<0$ and $k$ is a symmetric two-tensor on $\ensuremath{\partial} M$ which restricts
to a metric on each fiber, then for $m$ even,
$$
\mathcal{H}^i_{(2)}(M,g_c) \cong
 \mbox{Im} \, \big(I\!H^k_{ \underline{\mathfrak{m}} + [[-1/2c]]}(X,B)
 \longrightarrow I\!H^k_{ \overline{\mathfrak{m}}  - [[-1/2c]]}(X,B) \big)
 $$
 and for $m$ odd,
 $$
\mathcal{H}^i_{(2)}(M,g_c) \cong
\mbox{Im} \, \big(I\!H^k_{ \underline{\mathfrak{m}}  +[[(-1/2c)-(1/2)]]}(X,B)
\longrightarrow I\!H^k_{ \overline{\mathfrak{m}}  -[[(-1/2c)-(1/2)]]}(X,B) \big),
$$
\noindent where $[[x]]$ denotes the smallest integer strictly greater than $x$.
Thus we obtain the signature theorem:
$$
L^2-\sigma(M)= \mbox{sgn}\,
\mbox{Im}\,\big( H^*(M, \ensuremath{\partial} M) \longrightarrow H^*(M))\big) + \sum_{i=2+2k}^{\infty} \tau_i,
$$
\noindent
where $k= [-1/2c]$ when $m$ is even and $[(-1/2c)-(1/2)]$ when $m$ is odd.

\end{conjecture}

It seems likely that $IH^*_{\mathfrak{p}}(X)$ should be isomorphic to the space of $L^2$-harmonic forms on $X^{reg}$ for an incomplete metric on $X^{reg}$ with degeneration conditions near the singular strata related to those for the family of metrics considered in this paper.  It would be interesting to explore
which metrics these might be, and also to consider if it might be possible to define a general
$\tau$-invariant for singular sets of pseudomanifolds with more than one singular stratum.  Such a
generalization would make it possible to derive signature theorems from the Hodge theorems
for noncompact manifolds with more complicated singularity structures than we have so far considered.

redpajama_set_name: RedPajamaArXiv


text:
Marina Spine Center »
Discectomy
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Great Surgical Care at Marina Del Rey Hospital
Guide to Advanced Surgery for The Spine
Why Choose Marina Spine Center for Discectomy?
Following a discectomy procedure performed at the highly regarded Marina Spine Center, our well experienced and highly skilled medical professionals can immediately release the pressure you are feeling off the inflicted area.
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There are three major portions of the spine. Nerve roots and the spinal cord transmit information and commands throughout the body. Surrounding them are the vertebrae, bones that encase and protect the nerves. Because the vertebrae are arranged as a column of smaller bones, rather than as one long bone, the spine can move, bend, twist, and straighten—but for this to work, there has to be some way to make sure bones don't grind against each other. So between each pair of vertebrae are the vertebral discs, which are made of softer tissue and act as cushions.
Many cases of back pain are caused by a disc slipping from its place in between two vertebrae. When a slipped disc occurs, the disc can press against the nerve roots or spinal cord, and that causes pain, numbness, weakness, and other symptoms. When a vertebral disc slips and the nerve pressure causes these symptoms in the legs, it is called sciatica.
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Will numbness in my legs go away after microdiscectomy?
In some cases it may take weeks or months, perhaps even a year for post-surgical numbness to disappear. If you still experience this numbness after a year, it is probably permanent. This is a subject you may want to discuss further with your physician.
See Full Answer
What is a slipped disc condition, and what can be done to correct it?
A slipped disc occurs when the spinal disc or a soft cushion-like substance found in between two vertebrae is pushed out of its normal position.
What type of physical activity is recommended after discectomy?
Following discectomy, there are certain activities that should be avoided and others that are recommended. Your doctor will tell you everything you need to know about them before you are discharged.
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text:
Home/News/BMW makes the all-new BMW M2 debut in the region at the Hankook 24 Hours in Dubai
BMW makes the all-new BMW M2 debut in the region at the Hankook 24 Hours in Dubai
The Mechanical Center of the Arabian Gulf, the exclusive agent and authorized distributor, announced The BMW Group in Dubai, Sharjah and the Northern Emirates, announced the launch of the all-new BMW M2, the flagship compact car in the group, for the first time in the region, during the Hankook Race 24 an hour in Dubai, which runs from 13 to 100 January at the Dubai Autodrome.
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text:
Power Mac G4 – linia komputerów osobistych produkowanych przez firmę Apple. Zastosowano w nich procesory serii PowerPC G4 (PPC74xx).

Pierwsze modele z procesorami taktowanymi zegarem o częstotliwości 400, 450 i 500 MHz weszły do sprzedaży 31 sierpnia 1999. Seria Power Mac G4 została wycofana z produkcji 9 czerwca 2004 roku, a ostatni model miał dwa procesory PowerPC 7455 taktowane zegarem 1,42 GHz.

PowerMac G4

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text:
Q: python error: TypeError: list indices must be integers or slices, not list I am writing this python code and I'm getting an error. I've searched through the internet and StackOverflow but nothing works for me!
The error:
    TypeError: list indices must be integers or slices, not list (line 7)
My code:
products = [
    [
        'name',
        'id',
        'number in stock',
        'price ($)'
    ],
    ['iPhone', '1', '100', '999']
]

def instore(item):
    for x in products:
        if products[x][0] == item:
            print(products[x][0] + ': \n id: ' + products[x][1] + ' In Stock: ' + products[x][2])

A: x inside the for loop is already the relevant item from products. It is not a counter and there is no need to say products[x].
for x in products:
   if x[0] == item:
       ...

(Also, you probably want to use a dictionary rather than a list here; then you can find products[item] directly rather than iterating through the list each time.)

A: The variable x is assigned with each item in the products list with the for statement, and is not an index, so you should access it as an item in the products list:
def instore(item):
    for x in products:
        if x[0] == item:
            print(x[0] + ': \n id: ' + x[1] + ' In Stock: ' + x[2])

A: As the other's have already told you about x being an element of the list, instead of being a counter variable. If you still want to use a counter, you can do something like,
for x in range(len(products)):
   if products[x][0] == item:
       ...

The range(n) function creates a range (list) from 0 to n-1. You can read more about range, from here.

A: Because you are not iterating the index in the list but what you are doing is iterating the element present in the products list in the form of the variable "x" . i think the point will be cleared to you . and checkout the previous answers for the correct code

redpajama_set_name: RedPajamaStackExchange


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Today's picks: Warriors-Grizzlies best bet, plus more college basketball and NFL picks for Tuesday
SportsLine's proven experts share their favorite bets on the Early Edge podcast for Tuesday, Jan. 11.
Jonathan Coachman is joined by top sports betting experts Mike Barner, Justin Perri, and Larry Hartstein on the Early Edge podcast, powered by SportsLine. Get highlighted plays for the NBA, college basketball and the NFL for Tuesday, Jan. 11, and you can stream the show live every morning on the SportsLine YouTube page.
Mike takes a look at the Detroit Pistons, who upended Utah not long after knocking off Milwaukee. Will the Pistons upset Chicago next? Larry, the Coach and Justin examine the controversy surrounding the Draymond Green prop bet from Monday.
Mike's featured NBA play is the Grizzlies +2.5 over the Warriors at 8 p.m. ET. "The Grizzlies have won nine straight games, and in four of them they were underdogs," he says. "Yes, the Warriors have Klay Thompson back, but Draymond Green is out. The Warriors are 18-3 at home, but they are not as great on the road, 12-6."
Justin delivers a pair of college basketball picks, including Akron -7.5 over Ball State at 7 p.m. ET. The Zips have shown they can cover this number in recent meetings with the Cardinals. He is also on Stetson +6.5 over Jacksonville at 7 p.m. ET. The Dolphins may win, but the Hatters will keep the score close.
Larry looks ahead to Steelers-Chiefs in an AFC Wild Card round matchup on Saturday at 8:15 p.m. ET. He is taking Kansas City -12.5. Yes, the number is massive, but Larry provides the numbers to back the selection up and reminds us that the last meeting between these teams was a blowout.
Follow our crew on Twitter: @TheCoachRules, @rotomikebarner, @JustinPerri8, and @LarryHartstein. You can listen to Early Edge on your smart speakers! Simply say "Alexa, play the latest episode of the Early Edge podcast" or "Hey Google, play the latest episode of the Early Edge podcast." For more betting picks and analysis, visit SportsLine.com. To hear more from the CBS Sports Podcast Network, visit CBSSports.com/podcasts.


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Orientation Programs for the Local Leaders on the Services of UPHCSDP Held
The Urban Primary Health Care Services Delivery Project (UPHCSDP) aims to improve health status of the urban population, especially the poor, including women and children through improved access to and utilization of efficient, effective and sustainable Primary Health Care (PHC) services. Bangladesh Center for Communication Programs (BCCP) is responsible for implementing the Behavior Change Communication and Marketing (BCCM) Component of UPHCSDP. As part of its designed activities, BCCP organized 25 orientation programs in all the 25 PA Units of UPHCSDP in which local women leaders, women ward councilors, members of Ward Urban Health Care Coordination Committee, members of User Forum, youth leaders, faith-based leaders and other local leaders participated. In these programs, the participants were oriented on the services being provided by the Rainbow marked health centers of UPHCSDP particularly focusing on issues like Maternal and Child Health, Violence Against Women (VAW), Family Planning etc. Another objectives of the program was to discuss about how the local leaders can contribute better in improving the services of UPHCSDP. Following a structured plan, the first orientation program was held at PA-04, Pallabi, Dhaka North City Corporation on May 12, 2016. The last programs were held in two PA Units at KgM PA-01 under Kishorgonj Municipality and at SCC PA-01 under Sylhet City Corporation on June 07, 2016. The Project Director, UPHCSDP was present in a couple of orientation programs. Mr. Md. Abu Bakr Siddique, Project Director, UPHCSDP graced the Orientation Program at Sirajgonj Municipality on May 26, 2016 as the chief guest. Dr. A.K.M. Farhad Hossain, Program Officer, PIU, SM & Mr. Md. Abdul Alim Mondol, Ward Councilor, Ward No. 12, Sirajgonj Municipality were present as special guests in the program. Mr. Jamini Kumar Roy, Project Coordinator, ESDO delivered the welcome speech. Ms. Kanta Devi, Campaign Specialist, BCCM Component then briefly spoke on behalf of BCCP. Mr. Mahitur Rahman Khan, Technical Officer, BCCM Component facilitated the introduction session and updated the activities and services of UPHCSDP by a short power-point presentation. Mr. Mahbubur Rahman, Technical Officer, BCCM component facilitated the session on VAW followed by brainstorming with the participants on selective issues like VAW, services of UPHCSDP. The Orientation Program ended with a concluding remarks by Mr. Md. Abdul Alim Mondol.
Inauguration of SBCC Digital Archive
Communication and Community Mobilization training for EMCRP partners and stakeholders
Inter-unit coordination plays a vital role in implementation of SBCC activities
Validation Workshop on Draft Communication & Knowledge Management Strategy & Outreach Plan for SREDA
Training Program on IPC/BCC, Community Mobilization & Outreach Activities Continues
Request for Applications : Tobacco Control Policy Research Grant Program 2022


Doug coauthored lex & yacc, another O'Reilly & Associates Nutshell Handbook. For a pure reference such as the Python Essential Reference or Python in a Nutshell. This wasn't the first book I picked up to learn C++. Nutshell Handbook, the Nutshell Handbook logo, and the O'Reilly logo are registered .. An in-depth, tutorial introduction to Python core language fundamentals, .. This review is from: C++ the Core Language (Nutshell Handbooks) (Paperback). Take a look at our companion book, Practical C++ Programming. This book covers C#, the CLR, and the core Framework assemblies. Python is a very powerful high level object oriented language and has easy bindings with C,C++,Java etc. C++: The Core Language Nutshell Handbook - Satir, Gregory. C++: The Core Language is for C programmers transitioning to C++.


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Megapixel cameras can provide a radically higher resolution image than traditional analogue systems. Whilst not the only factor in providing excellent CCTV image quality, recording at a greater resolution means more information is captured in a single frame. When compared to a traditional analogue D1 image (720 x 576), a 1.3 megapixel image (1280 x 960) captures 3x the pixels, whereas a 2.0 megapixel image (1920 x 1080) captures over 5x the pixels.
This increase in information results in better resolved shapes, sharper object edges and preservation of fine details. It is this information that can make the difference between usable and unusable footage. View each footage sample below to see the difference a megapixel system can make for you.
*Be sure to watch each video in its maximum resolution.
The VIP Vision 3.0 megapixel vandal resistant mini dome ip camera recording 3.0MP at 10fps (2048 x 1536). Observe fine details in the separation in the tree leaves and definition of the number plate.
The Watchguard 1.3 megapixel weatherproof mini bullet camera is capable of recording 1.3MP at 25fps (1280 x 960). Here it is recording at 720p (1280 x 720) at 25fps. While not as sharp as the 3.0MP, plenty of detail is still visible.
A typical analogue CCTV surveillance system recording at CIF resolution (352 x 288). It is difficult to distinguish between small objects and fine details, such as the number of the truck, are lost completely. Even in ideal conditions, only large shapes are clear enough to identified accurately.
Wagner Online carries Australia's largest range of IP Cameras, Analogue Cameras, DVR and NVR Systems as well as all Security Camera Mounting Brackets, Installation Hardware and Accessories.
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