text:
\section{Introduction.}

\subsection{Summary of the contents}
Let $M$ and $N$ be manifolds with some additional structure, e.g holomorphic, symplectic, real algebraic etc. The relation between the topology of the space of continuous maps preserving this structure and that of the space of all continuous maps has long been an object of study in several areas of topology and geometry. Early examples  were provided by Gromov's h-principle for holomorphic maps \cite{Grom}. In these cases the manifolds are complex, the structure preserving maps are the holomorphic ones, and the spaces of holomorphic and continuous maps turn out to be homotopy equivalent. However,  in many other cases, the space of structure preserving maps approximates, in some sense, the space of all continuous ones and becomes homotopy equivalent to it only after some kind of stabilization.  A paradigmatic example of this type was given in a seminal paper of Segal \cite{Se}, where the space of rational (or holomorphic) maps of a fixed degree from the Riemann sphere to a complex projective space was shown to approximate the space of all continuous maps in homotopy, with the approximation becoming better as the degree increases. Segal's result was extended to a variety of other target spaces by various authors (e.g. \cite{BHM}). Although it has been sometimes stated that these phenomena are inherently related to complex or at least symplectic structures, real analogs of Segal's result were given in \cite{Se}, \cite{Mo1}, \cite{GKY2}, \cite{Va}. In fact, Segal  formulated the complex and real approximation theorems which he had proved  as a single statement involving equivariant equivalence, with respect to complex conjugation (see the remark after Proposition 1.4 of \cite{Se}). A similar idea was used in \cite{GKY2}, Theorem 3.7.  This theorem amounts to two equivariant ones, one of which is equivalent to (a stable version of)  Segal's equivariant one, while the other one is related to the \lq real version\rq\  of Segal's theorem proved in \cite{Mo1}.

All the results mentioned above (except the ones involving Gromov's h-principle), assume that the domain of the mappings is one dimensional (complex or real). It is natural to try to generalize them to the situation where the domain is higher dimensional. That such generalizations might be possible was first suggested by Segal (see the remark under Proposition 1.3 of \cite{Se}).
A large step in this direction appeared to have been made when Mostovoy
 \cite{Mo2}  showed that the homotopy types of spaces of holomorphic maps
from $\Bbb C {\rm P}^m$ to $\Bbb C {\rm P}^n$
 (for $m\le n$) approximate the homotopy types of the spaces of continuous maps, with the approximation becoming better as the degree increases. Unfortunately Mostovoy's published argument contains several gaps.  A new version of the paper, currently only available from the author, appears to correct all the mistakes, with the main results remaining essentially unchanged. There are two major changes in the proofs. One is that the space $\mbox{{\rm Rat}}_f(p,q)$ of $(p,q)$ maps from $\Bbb C {\rm P}^m$ to $\Bbb C {\rm P}^n$  that restrict to a fixed map $f$ on a fixed hyperplane, used in section 2 of the published article is replaced by the space $\overline {\mbox{{\rm Rat}}}_f(p,q)$ of pairs of $n+1$-tuples of polynomials in $m$ variables that produce these maps. In the published version of the article it is assumed that these two spaces are homotopy equivalent, which is clearly not the case. However, they are homotopy equivalent after stabilisation, both being equivalent to $\Omega^{2m}\Bbb C {\rm P}^n$ - the space of continuous maps that restrict to $f$ on a fixed hyperplane.
 The second important change is the introduction of a new filtration on the simplicial resolution $X^\Delta \subset \Bbb R^N\times Y $ of a map $h: X \to Y$ and an embedding $i: X \to \Bbb R^N = \Bbb C^{N/2}$. This filtration is defined by means of complex skeleta  (where the complex $k$-skeleton of a simplex in a complex affine space is the union of all its faces that are contained in complex affine subspaces of dimension at most $k$) and replaces the analogous \lq\lq real\rq\rq\ filtration in the arguments of section 4.

 In \cite{AKY1}  a variant of Mostovoy's idea  was applied to the case of algebraic maps from $\Bbb R\mbox{{\rm P}}^m$ to $\Bbb R\mbox{{\rm P}}^n$.  This leads naturally to the question whether one can generalize the $\Bbb Z/2$-equivariant Theorem 3.7 of \cite{GKY2} to an analogous equivariant equivalence of the spaces of algebraic maps and
continuous maps between projective spaces, in which the domain is either a real or a complex projective space of dimension $m>1$, the range is a  complex projective space of dimension greater or equal to that of the domain.
(Here, the $\Bbb Z/2$-action is induced by complex conjugation.)

Note that Theorem 3.7 of \cite{GKY2} has two parts, in the first the domain being complex and in the second real. Clearly, to prove the  first part we need a  Mostovoy's complex theorem. We plan to consider this problem in a future paper.
Here we concentrate on the second part, concerning equivariant algebraic maps from real projective spaces to complex projective ones. Our main theorem is new, but it uses the main result of   \cite{AKY1} and where  our arguments are very similar to those used in that paper we omit their details and refer the reader to  \cite{AKY1}.
\par\vspace{2mm}\par
In the remainder of this section we introduce our notation  and state the main definitions and theorems.
\subsection{Notation and Main Results}

We first introduce notation which is analogous to the one used in \cite{AKY1}, the presence of $\Bbb C$ indicating that the complex case is being considered (i.e. maps take values in $\Bbb C {\rm P}^n$ or polynomials have coefficients in $\Bbb C$).
\par
Let $m$ and $n$ be positive integers such that
$1\leq m <2\cdot (n+1)-1$.
We choose  ${\bf e}_m=[1:0:\cdots :0]\in \Bbb R\mbox{{\rm P}}^m$ and
${\bf e}_n^{\prime}=[1:0:\cdots :0]\in\Bbb \Bbb C {\rm P}^n$
as the base points of $\Bbb R\mbox{{\rm P}}^m$ and  $\Bbb  \Bbb C {\rm P}^n$, respectively.
Let $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb  \Bbb C {\rm P}^n)$ denote the space
consisting of all based maps
$f:(\Bbb R\mbox{{\rm P}}^m,{\bf e}_{m})\to (\Bbb \Bbb C {\rm P}^n,{\bf e}^{\prime}_n)$.
When $m\not= 1)$,
we denote
by $\mbox{{\rm Map}}_{\epsilon}^* (\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
the corresponding path component
of $\mbox{{\rm Map}}^* (\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
for each
$\epsilon \in \Bbb Z/2=\{0,1\}=
\pi_0(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb  \Bbb C {\rm P}^n))$
(\cite{CS}).
Similarly, let $\mbox{{\rm Map}} (\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ denote the space
of all free maps $f:\Bbb R\mbox{{\rm P}}^m\to\Bbb C {\rm P}^n$ and
$\mbox{{\rm Map}}_{\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ the corresponding path component
of $\mbox{{\rm Map}} (\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$.
\par
We shall use the symbols $z_i$ when we refer to complex valued coordinates or  variables or when we refer to complex and real valued ones at the same time while the notation $x_i$ will be restricted to the purely real case.
\par\vspace{2mm}\par
A  map $f:\Bbb R\mbox{{\rm P}}^m\to \Bbb C {\rm P}^n$ is called a
{\it algebraic map of the degree }$d$ if it can be represented as
a rational map of the form
$f=[f_0:\cdots :f_n]$ such that
 $f_0,\cdots ,f_n\in\Bbb C [z_0,\cdots ,z_m]$ are homogeneous polynomials of the same degree $d$
with no common {\it real } roots except ${\bf 0}_{m+1}=(0,\cdots ,0)\in\Bbb R^{m+1}$.

 We  denote by $\mbox{{\rm Alg}}_d(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ (resp. $\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$) the space
 consisting of all (resp. based) algebraic maps $f:\Bbb R\mbox{{\rm P}}^m\to \Bbb C {\rm P}^n$
 of degree $d$.
 It is easy to see that there are inclusions $\mbox{{\rm Alg}}_d(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\subset \mbox{{\rm Map}}_{[d]_2}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
 and
 $\mbox{{\rm Alg}}^*_d(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\subset \mbox{{\rm Map}}^*_{[d]_2}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$,
 where $[d]_2\in\Bbb Z/2=\{0,1\}$ denotes the integer $d$ mod $2$.
\par
Let
$ A_{d}(m,n)(\Bbb C)$ denote  the space consisting of all $(n+1)$-tuples
$(f_0,\cdots ,f_n)\in \Bbb C[z_0,\cdots ,z_m]^{n+1}$
of  homogeneous polynomials of degree $d$  with coefficients in $\Bbb C$  and without non-trivial common real roots
(but possibly with non-trivial common {\it non-real} ones).
\par
Let $ A_{d}^{\Bbb C}(m,n)\subset A_d(m,n)(\Bbb C)$ be the subspace consisting of $(n+1)$-tuples
$(f_0,\cdots ,f_n)\in A_d(m,n)(\Bbb C)$
such that the coefficient of $z_0^d$ in $f_0$ is 1 and $0$ in the other $f_k$'s
$(k\not= 0$).
Then there is a natural surjective projection map
\begin{equation}\label{Psi}
\Psi_d^{\Bbb C}:A_d^{\Bbb C}(m,n) \to \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n).
\end{equation}
If $d=2d^*\equiv 0$ $\mbox{{\rm (mod $2$)}}$ is an even positive integer,
we also have a natural projection map
\begin{equation}\label{}
j_d^{\Bbb C}:A_d^{\Bbb C}(m,n) \to \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C^{n+1}\setminus\{{\bf 0}\})
\simeq
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})
\end{equation}
defined by
$$
j_d^{\Bbb C}(f)([x_0:\cdots :x_m])=
\Big(\frac{f_0(x_0,\cdots ,x_m)}{(\sum_{k=0}^mx_k^2)^{d^*}},\cdots
,\frac{f_n(x_0,\cdots ,x_m)}{(\sum_{k=0}^mx_k^2)^{d^*}}\Big)
$$
for $f=(f_0,\cdots ,f_n)\in A_d^{\Bbb C}(m,n)$.
Note that the map
$j_d^{\Bbb C}$ is well defined only if $d\geq 2$ is an even integer.
\par\vspace{2mm}\par
For $m\geq 2$ and $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ a fixed algebraic map,
we denote by
$\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)$ and $F_d^{\Bbb C}(m,n;g)$  the spaces defined by
$$
\begin{cases}
\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g) &=\ \{f\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n):f\vert \Bbb R\mbox{{\rm P}}^{m-1}=g\},
\\
F_d^{\Bbb C}(m,n;g) & =\ \{f\in \mbox{{\rm Map}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n):f\vert \Bbb R\mbox{{\rm P}}^{m-1}=g\}.
\end{cases}
$$
\par
It is well-known that there is a homotopy equivalence
$F_d^{\Bbb C}(m,n;g)\simeq  \Omega^m\Bbb C {\rm P}^n$ (\cite{Sasao}).
Let $A_d^{\Bbb C}(m,n;g)\subset A_d^{\Bbb C}(m,n)$ denote the subspace
given
by
$$A_d^{\Bbb C}(m,n;g)=(\Psi_d^{\Bbb C})^{-1}(\mbox{{\rm Alg}}^{\Bbb C}_d(m,n;g)).
$$
Observe that if an algebraic map $f\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ can be represented as
$f=[f_0:\cdots :f_n]$ for some
$(f_0,\cdots ,f_n)\in A_d^{\Bbb C}(m,n)$ then  the same map can also be represented as
$f=[\tilde{g}_mf_0:\cdots :\tilde{g}_{m}f_n]$, where
$\tilde{g}_{m}=\sum_{k=0}^mz_k^2$.
So there is an inclusion
$\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\subset \mbox{{\rm Alg}}_{d+2}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
and we can define
{\it the stabilization map}
$s_d:A_d^{\Bbb C}(m,n)\stackrel{}{\to} A_{d+2}^{\Bbb C}(m,n)$
by $s_d(f_0,\cdots ,f_n)=(\tilde{g}_mf_0,\cdots ,\tilde{g}_mf_n).$
It is easy to see that there is a commutative diagram
$$
\begin{CD}
A_d^{\Bbb C}(m,n) @>s_d>> A_{d+2}^{\Bbb C}(m,n)
\\
@V{\Psi_d^{\Bbb C}}VV @V{\Psi_{d+2}^{\Bbb C}}VV
\\
\mbox{{\rm Alg}}^{*}_d(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n) @>\subset>> \mbox{{\rm Alg}}_{d+2}^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{CD}
$$
A map $f\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ is  called an algebraic map of
{\it minimal degree} $d$ if
$f\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\setminus \mbox{{\rm Alg}}_{d-2}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$.
It is easy to see that
if $g\in\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ is an algebraic map of minimal degree $d$, then
the restriction
\begin{equation}\label{restriction}
\Psi_d^{\Bbb C}\vert A_d^{\Bbb C}(m,n;g):A_d^{\Bbb C}(m,n;g)\stackrel{\cong}{\rightarrow}
\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)
\end{equation}
is a homeomorphism.
Let
\begin{equation}
\begin{cases}
i_{d,\Bbb C}:\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\stackrel{\subset}{\rightarrow} \mbox{{\rm Map}}_{[d]_2}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\\
i_{d,\Bbb C}^{\prime}:\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)\stackrel{\subset}{\rightarrow} F(m,n;g)\simeq \Omega^m\Bbb C {\rm P}^n
\end{cases}
\end{equation}
denote the inclusions and let
\begin{equation}
i_d^{\Bbb C}=i_{d,\Bbb C}\circ \Psi_d^{\Bbb C}:A_d^{\Bbb C}(m,n)\to \mbox{{\rm Map}}_{[d]_2}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n).
\end{equation}
be the natural projection.
For a connected space $X$,
let $F(X,r)$ denote the configuration space  of distinct $r$ points in $X$.
The symmetric group $S_r$ of $r$ letters acts on $F(X,r)$ freely by permuting
coordinates. Let $C_r(X)$ be the configuration space of unordered
$r$-distinct  points in $X$ given by
$C_r(X)=F(X,r)/S_r$.
Note that there is a stable homotopy equivalence
$\Omega^mS^{m+l}\simeq_s \bigvee_{r=1}^{\infty}F(\Bbb R^m,r)_+\wedge_{S_r}\big(\bigwedge^rS^l\big)$
(\cite{Sn}), and it is known that
there is an isomorphism
$H_k(F(\Bbb R^m,r)_+\wedge_{S_r}\big(\bigwedge^rS^l\big),\Bbb Z)
\cong H_{k-rl}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes r})$ for $k,l\geq 1$
(\cite{CLM},  \cite{Va}),
where
$\bigwedge^rX=X\wedge \cdots \wedge X$
($r$ times).
\par
Let $G^d_{m,N}$ denote the abelian group $G^d_{m,N}=\bigoplus_{r=1}^{\lfloor \frac{d+1}{2}\rfloor}
H_{k-(N-m)r}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (N-m)} )$, where
the meaning of  $(\pm \Bbb Z)^{\otimes (N-m)}$  is the same as in \cite{Va}.

Let   $D_{\Bbb K}(d;m,n)$
be the positive integer defined by

$$D_{\Bbb K}(d;m,n)=
\begin{cases}
(n-m)\big(\lfloor \frac{d+1}{2}\rfloor  +1\big) -1
& \mbox{if } \Bbb K =\Bbb R,
\\
(2n-m+1)\big(\lfloor \frac{d+1}{2}\rfloor  +1\big) -1
& \mbox{if } \Bbb K =\Bbb C,
\end{cases}
$$
, and
$\lfloor x\rfloor$ is the integer part of a real number $x$.
Note that $D_{\Bbb C}(d;m,n)=D_{\Bbb R}(d;m,2n+1)$.
\par\vspace{2mm}\par
First, recall
the following 3 results. Here we use the notation of \cite{AKY1}.

\begin{thm}[\cite{KY1}, \cite{Y5}]\label{thm: A1}
If $n\geq 2$ and then
the natural projection
$i_{d}^{\Bbb R}:A_d^{\Bbb R}(1,n)\to  \mbox{{\rm Map}}^*_{[d]_2}(\Bbb R\mbox{{\rm P}}^1,\Bbb R\mbox{{\rm P}}^n)\simeq \Omega S^{n}$
is a homotopy equivalence
up to dimension $D_1(d,n)=(d+1)(n-1)-1$.
\end{thm}
\begin{thm}[\cite{AKY1}]
\label{thm: AKY1-I}
Let $2\leq m <n$ be integers and
let $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb R\mbox{{\rm P}}^n)$ be an algebraic map of minimal degree $d$.
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
The inclusion
$i_{d,\Bbb R}^{\prime}:\mbox{{\rm Alg}}_d^{\Bbb R}(m,n;g)\to F_d(m,n;g)\simeq \Omega^mS^n$
is a homotopy equivalence through dimension $D_{\Bbb R}(d;m,n)$ if $m+2\leq n$
and a homology equivalence through dimension $D_{\Bbb R}(d;m,n)$ if $m+1=n$.
\item[$\mbox{{\rm (ii)}}$]
For any $k\geq 1$,
$H_k(\mbox{{\rm Alg}}_d^{\Bbb R}(m,n;g),\Bbb Z)$ contains the subgroup
$G^d_{m,n}$ as a direct summand.
\end{enumerate}
\end{thm}
\begin{thm}[\cite{AKY1}]
\label{thm: AKY1-II}
If $2\leq m <n$ and $d\equiv 0$ $\mbox{{\rm (mod $2$)}}$ are positive integers,
$$\begin{cases}
j_d^{\Bbb R}:A_d^{\Bbb R}(m,n)\to \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^n)
\\
i_d^{\Bbb R}:A_d^{\Bbb R}(m,n)\to \mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^n)
\end{cases}
$$
are homotopy equivalence through dimension $D_{\Bbb R}(d;m,n)$ if
$m+2\leq n$ and homology equivalences through dimension
$D_{\Bbb R}(d;m,n)$ if $m+1=n$.
\end{thm}
\par
{\it Remark. }
A map $f:X\to Y$ is called {\it a homotopy} (resp. {\it a homology}) {\it equivalence up to dimension} $D$ if
$f_*:\pi_k(X)\to \pi_k(Y)$ (resp.$ f_*:H_k(X,\Bbb Z)\to H_k(Y,\Bbb Z)$) is an isomorphism for any
$k<D$ and an epimorphism for $k=D$.
Similarly, it is called {\it a homotopy} (resp. {\it a homology}) {\it equivalence through dimension} $D$ if
$f_*:\pi_k(X)\to \pi_k(Y)$ (resp.$ f_*:H_k(X,\Bbb Z)\to H_k(Y,\Bbb Z)$) is an isomorphism for any $k\leq D$.
\par\vspace{3mm}\par
In this paper, from now on let $m,n\geq 2$ be positive integers. Our main results are as follows.
\begin{thm}\label{thm: I}
Let  $2\leq m\leq 2n$, and
let $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ be an algebraic map of minimal degree $d$.
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
The inclusion
$i_{d,\Bbb C}^{\prime}:\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)\to F_d(m,n;g)\simeq \Omega^mS^{2n+1}$
is a homotopy equivalence through dimension $D_{\Bbb C}(d;m,n)$ if
$m<2n$ and a homology equivalence through dimension $D_{\Bbb C}(d;m,n)$
if $m=2n$.
\item[$\mbox{{\rm (ii)}}$]
For any $k\geq 1$,
$H_k(\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g),\Bbb Z)$ contains the subgroup
$G^d_{m,2n+1}$ as a direct summand.
\end{enumerate}
\end{thm}

\begin{thm}\label{thm: II}
If $2\leq m\leq 2n$ and $d\equiv 0$ $\mbox{{\rm (mod $2$)}}$ are positive integers,
$$
\begin{cases}
j_d^{\Bbb C}:A_d^{\Bbb C}(m,n)\to \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})
\\
i_d^{\Bbb C}:A_d^{\Bbb C}(m,n)\to \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{cases}
$$
are homotopy equivalences through dimension $D_{\Bbb C}(d;m,n)$
if
$m<2n$ and homology equivalences through dimension $D_{\Bbb C}(d;m,n)$
if $m=2n$.
\end{thm}

\begin{cor}\label{cor: III}
If $2\leq m\leq 2n$ and $d\equiv 0$ $\mbox{{\rm (mod $2$)}}$ are positive integers,
the stabilization map
$s_d:A_d^{\Bbb C}(m,n)\to A_{d+2}^{\Bbb C}(m,n)$
is a homotopy equivalence through dimension
$D_{\Bbb C}(d;m,n)$ if
$m<2n$ and a homology equivalence through dimension $D_{\Bbb C}(d;m,n)$
if $m=2n$.
\end{cor}

Note that the
complex conjugation on $\Bbb C$ naturally induces $\Bbb Z/2$-actions on
the spaces $\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)$ and $A_d^{\Bbb C}(m,n)$.
In the same way it also induces a $\Bbb Z/2$-action on $\Bbb C {\rm P}^n$ and this action extends to actions on the spaces
$\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})$ and $\mbox{{\rm Map}}^*_{\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$,
 where we identify
$S^{2n+1}=\{(w_0,\cdots ,w_n)\in\Bbb C^{n+1}:\sum_{k=0}^n\vert w_k\vert^2 =1\}$
and regard $\Bbb R\mbox{{\rm P}}^m$ as a $\Bbb Z/2$-space with the trivial $\Bbb Z/2$-action.
Since the maps $i^{\prime}_{d,\Bbb C}$, $j_d^{\Bbb C}$, $i_d^{\Bbb C}$ are
$\Bbb Z/2$-equivariant and
$(i^{\prime}_{d,\Bbb C})^{\Bbb Z/2}=i^{\prime}_{d,\Bbb R}$, $(j_d^{\Bbb C})^{\Bbb Z/2}=j_d^{\Bbb R}$,
$(i_d^{\Bbb C})^{\Bbb Z/2}=i_d^{\Bbb R}$, we easily obtain the following result.

\begin{cor}\label{cor: IV}
Let $2\leq m\leq 2n$ and $d\geq 1$ be  positive integers.
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
The inclusion map
$i_{d,\Bbb C}^{\prime}:\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)\to F_d(m,n;g)\simeq \Omega^mS^{2n+1}$
is a $\Bbb Z/2$-equivariant homotopy equivalence through dimension $D_{\Bbb R}(d;m,n)$ if
$m<2n$ and a $\Bbb Z/2$-equivariant homology equivalence through dimension $D_{\Bbb R}(d;m,n)$
if $m=2n$.
\item[$\mbox{{\rm (ii)}}$]
If $d\equiv 0$ $\mbox{{\rm (mod $2$)}}$, then
$$
\begin{cases}
j_d^{\Bbb C}:A_d^{\Bbb C}(m,n)\to \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})
\\
i_d^{\Bbb C}:A_d^{\Bbb C}(m,n)\to \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{cases}
$$
are $\Bbb Z/2$-equivariant homotopy equivalences through dimension $D_{\Bbb R}(d;m,n)$
if
$m<2n$ and $\Bbb Z/2$-equivariant homology equivalences through dimension $D_{\Bbb R}(d;m,n)$
if $m=2n$.
\end{enumerate}
\end{cor}
\par
{\it Remark. }
Let $G$ be a finite group and let
$f:X\to Y$ be a $G$-equivariant map.
Then a map $f:X\to Y$ is called
{\it a $G$-equivariant homotopy }(resp. {\it homology})
{\it equivalence through dimension }$D$ if the induced
homomorphism
$f_*^{H}:\pi_k(X^{H})\stackrel{\cong}{\rightarrow}\pi_k(Y^{H})$
(resp.
$f_*^{H}:H_k(X^{H},\Bbb Z)\stackrel{\cong}{\rightarrow}H_k(Y^{H},\Bbb Z)$)
are isomorphisms for any $k\leq D$ and any subgroup $H\subset G$.
\par\vspace{2mm}\par

Of course we would also like to understand the cases $d\equiv 1$ $\mbox{{\rm (mod $2$)}}$.
The homotopy type of
$\mbox{{\rm Alg}}_d^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ appears hard to investigate in general.
However, for $d=1$,
$\Psi_1^{\Bbb K}:A^{\Bbb C}_1(m,n)\stackrel{\cong}{\rightarrow}\mbox{{\rm Alg}}_1^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is a homeomorphism and we can prove the following results.
\begin{thm}\label{thm: V}
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
If $2\leq m< 2n$,
the inclusion
$$
i_{1,\Bbb C}:\mbox{{\rm Alg}}_1^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\to
\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
$$
is a homotopy equivalence up to dimension
$D_{\Bbb C}(1;m,n)=4n-2m+1$.
\item[$\mbox{{\rm (ii)}}$]
If $m=2n\geq 4$, the inclusion $i_{1,\Bbb C}$ induces an isomorphism
$$
(i_{1,\Bbb C})_*:\pi_1(\mbox{{\rm Alg}}_1^{*}(\Bbb R\mbox{{\rm P}}^{2n},\Bbb C {\rm P}^n))
\stackrel{\cong}{\longrightarrow}
\pi_1(\mbox{{\rm Map}}^*_1(\Bbb R\mbox{{\rm P}}^{2n},\Bbb C {\rm P}^n))\cong\Bbb Z/2 .
$$
\end{enumerate}
\end{thm}
\begin{cor}\label{cor: V}
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
If $2\leq m< 2n$ and $d\equiv 0$ $\mbox{{\rm (mod $2$)}}$ are positive integers,
the space $\mbox{{\rm Alg}}_d^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is $(2n-m)$-connected and
$$
\pi_{2n-m+1}(\mbox{{\rm Alg}}_{d}^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n))
\cong
\begin{cases}
\Bbb Z & \mbox{if }\ m\equiv 1\ \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z/2 & \mbox{if }\ m\equiv 0\ \mbox{{\rm (mod $2$)}} .
\end{cases}
$$
\item[$\mbox{{\rm (ii)}}$]
If $2\leq m\leq 2n$ and $\epsilon\in\{0,1\}$,  the
two spaces $\mbox{{\rm Alg}}_1^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ and
$\mbox{{\rm Map}}_{\epsilon}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ are $(2n-m)$-connected, and
\begin{eqnarray*}
\pi_{2n-m+1}(\mbox{{\rm Alg}}_{1}^{*}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n))
&\cong&
\pi_{2n-m+1}(\mbox{{\rm Map}}_{\epsilon}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n))
\\
&\cong&
\begin{cases}
\Bbb Z & \mbox{if }\ m\equiv 1\ \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z/2 & \mbox{if }\ m\equiv 0\ \mbox{{\rm (mod $2$)}} .
\end{cases}
\end{eqnarray*}
\end{enumerate}
\end{cor}
\par
{\it Remark. }
We conjecture that $\pi_1(\mbox{{\rm Alg}}^{*}_{d}(\Bbb R\mbox{{\rm P}}^{m},\Bbb C {\rm P}^n))=\Bbb Z/2$
if $m=2n\geq 4$ and $d\equiv 0$ $\mbox{{\rm (mod $2$)}}$, but at this time
we cannot prove this.
\par\vspace{2mm}\par
This paper is organized as follows.
In section 2, we consider the space of algebraic maps
$\Bbb R\mbox{{\rm P}}^m\to \Bbb C {\rm P}^n$ and recall the stable Theorem obtained in
\cite{AKY1}.
In section 3 we study simplicial resolutions and
the spectral sequences induced from them and prove Theorem \ref{thm: I}.
In section 4 we prove Theorem \ref{thm: II} and
Corollary \ref{cor: III}
by using Theorem \ref{thm: AKY1-II}, and in section 5 we study the homotopy type of the stabilized space
$A^{\Bbb C}_{\infty +\epsilon}(m,n)$ for $\epsilon =0$.
Finally in section 6, we investigate
 $A_d^{\Bbb C}(m,n)$ for the case $d=1$ and prove Theorem \ref{thm: V}
and Corollary \ref{cor: V}.

\section{Spaces of algebraic maps.}

An algebraic  map $f:\Bbb R\mbox{{\rm P}}^m\to\Bbb C {\rm P}^n$ can always be
 represented as
$f=[f_0:f_1:\cdots :f_n]$, where
$f_0,\cdots ,f_n\in \Bbb C [z_0,z_1,\cdots ,z_m]$
are homogeneous polynomials of the same degree $d$
with no common {\it real} root other than
${\bf 0}_{m+1}=(0,\cdots ,0)\in\Bbb R^{m+1}$
(but possibly with common {\it non-real} roots).
\par
Clearly, an element of $\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
can always be represented in the form $f=[f_0:f_1:\cdots :f_n]$,
such that the coefficient of $z_0^d$ in $f_0$
is $1$
and in the other polynomials $f_i$ $(i\not= 0)$ $0$.
In general, such a representation is also not unique.
For example, if we multiply all polynomials $f_i$ by two different homogeneous polynomials in $z_0,z_1,\cdots ,z_m$, which contain a power of $z_0$ with coefficient   $1$
and are always positive on $\Bbb R\mbox{{\rm P}}^m$, we will obtain two distinct representations of the same algebraic map.
So the map
$\Psi_d^{\Bbb C}:A_d^{\Bbb C}(m,n)\to \mbox{{\rm Alg}}_{d}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ is a surjective projection map.
It is easy to see that any fiber of $\Psi_d^{\Bbb C}$ is homeomorphic to the space
consisting of non-negative positive homogeneous polynomial functions of some
fixed even degree. So it is convex and contractible.
Hence, it seems plausible to expect the following may be true.
\begin{conj}\label{conj: Psi}
The map
$\Psi_d^{\Bbb C}:A_d^{\Bbb C}(m,n) \to \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is a homotopy equivalence.
\end{conj}
Although  we cannot prove this conjecture, we
show in section 6 that it is true if  $d\to \infty$ through even integers .
\par\vspace{2mm}\par
We always have $\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n) \subset \mbox{{\rm Alg}}_{d+2}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
and $\mbox{{\rm Alg}}_d(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n) \subset \mbox{{\rm Alg}}_{d+2}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$,
because
$[f_0:f_1:\cdots :f_n]=
[\tilde{g}_{m} f_0:\tilde{g}_mf_1:\cdots :\tilde{g}_{m} f_n]$.
\par\vspace{2mm}\par
\begin{defi}
{\rm
For $\epsilon \in\{0,1\}$,  define  subspaces
$\mbox{{\rm Alg}}_{\epsilon}^*(m,n)\subset\mbox{{\rm Map}}_{\epsilon}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
and
$\mbox{{\rm Alg}}_{\epsilon}(m,n)\subset\mbox{{\rm Map}}_{\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
by
$$
\begin{cases}
\mbox{{\rm Alg}}^*_{\epsilon}(m,n)   &=\ \bigcup_{k=1}^{\infty}
\mbox{{\rm Alg}}^*_{\epsilon+2k}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n),
\\
\mbox{{\rm Alg}}_{\epsilon}(m,n)   &=\ \bigcup_{k=1}^{\infty}
\mbox{{\rm Alg}}_{\epsilon+2k}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n).
\end{cases}
$$
}
\end{defi}
\begin{thm}\label{thm: stable}
If $1\leq m\leq 2n$ and $\epsilon=0$ or $1$,
the inclusion maps
$$
\begin{cases}
i:\mbox{{\rm Alg}}_{\epsilon}^*(m,n)\stackrel{\simeq}{\rightarrow}
 \mbox{{\rm Map}}_{\epsilon}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\\
j:\mbox{{\rm Alg}}_{\epsilon}(m,n)\stackrel{\simeq}{\rightarrow}
 \mbox{{\rm Map}}_{\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{cases}
$$
are homotopy equivalences.
\end{thm}
\begin{proof}
It follows from  [\cite{AKY1}, Theorem 2.3] that $j$ is a homotopy equivalence.
The statement and the proof in it
 are valid also for spaces of
based maps, and we can show
 that $i$ is also a homotopy equivalence.
\end{proof}

\section{Spectral sequences of the Vassiliev type.}\label{section 3}
From now on, we assume $2\leq m\leq 2n$ and
let $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ be a fixed algebraic map of
minimal degree $d$, such that
$g=[g_0:\cdots :g_n]$ with
$(g_0,\cdots ,g_n)\in A_d^{\Bbb C}(m-1,n)$.
Note that $(g_0,\cdots ,g_n)$ is uniquely determined by $g$ (because of the minimal degree condition).
\par\vspace{2mm}\par

Let $\mathcal{H}_d
\subset \Bbb C [z_0,\cdots ,z_m]$
denote the subspace consisting of all
homogeneous polynomials of degree $d$.
For $\epsilon \in \{0,1\}$,
let $H_d^{\epsilon}\subset \mathcal{H}_d$ be the subspace
consisting of all homogeneous polynomials $f\in \mathcal{H}_d$
such that the coefficient of $(z_0)^d$ of $f$ is
$\epsilon$.

Since $A_d^{\Bbb C}(m,n)$ is the space
consisting of all $(n+1)$-tuples
$(f_0,\cdots ,f_n)\in A_d(m,n)(\Bbb C)$
such that the coefficient of $z_0^d$ in $f_0$ is $1$ and
those of other $f_k$'s are all zero,
$A_d^{\Bbb C}(m,n)\subset \mathcal{H}_d^0\times (\mathcal{H}_d^1)^n$.
Note that
$\mathcal{H}_d^0\times (\mathcal{H}_d^1)^n$ is an affine space of real dimension of
$N_d=2(n+1)\Big(\binom{m+d}{m}-1\Big)$.

\par\vspace{2mm}\par
Next, we set $B_k=\{g_k+z_mh:h\in \mathcal{H}_{d-1}\}$ $(k=0,1,\cdots ,n)$ and
define the subspace $A_d^*\subset \mathcal{H}_d^0\times (\mathcal{H}_d^1)^n$ by
$A_d^*=B_0\times B_1\times \cdots \times B_n$.
Note that
$A_d^*$ is an affine space of real dimension $N_d^*=2(n+1)\binom{m+d-1}{m}.$
\par\vspace{2mm}\par
\begin{defi}
{\rm
Let $A_d^{\Bbb C}(m,n;g)\subset A_d^*$ be the subspace
$A_d^{\Bbb C}(m,n;g)=A_d^*\cap A_d^{\Bbb C}(m,n).$
Let
 $\Sigma_d\subset A_d$
denote the discriminant of $A_d^{\Bbb C}(m,n;g)$ in $A_d^*$
defined by
}
$\Sigma_d=A_d^*\setminus A_d^{\Bbb C}(m,n;g).$
\end{defi}

Since $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb R\mbox{{\rm P}}^n)$ has minimal degree $d$,
clearly  the restriction
\begin{equation}\label{Psi1}
\Psi_d^{\Bbb C}\vert_{A_d^{\Bbb C}(m,n;g)}:A_d^{\Bbb C}(m,n;g)\stackrel{\cong}{\rightarrow}
\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)
\end{equation}
 is a homeomorphism.
\par\vspace{2mm}\par

\begin{lemma}\label{lemma: number}
$\mbox{{\rm (i)}}$
If $(f_0,\cdots ,f_n)\in \Sigma_d$ and
${\bf x}=(x_0,\cdots ,x_m)\in \Bbb R^{m+1}$ is a non-trivial common root
of $f_0,\cdots ,f_n$, then $x_m\not= 0$.
\par
$\mbox{{\rm (ii)}}$
 $A_d^{\Bbb C}(m,n;g)$ and
$A_d^{\Bbb C}(m,n)$ are simply connected if $m<2n$.
\end{lemma}
\begin{proof}
The proof is completely analogous to that of [\cite{AKY1}, Lemma 4.1].
\end{proof}

\begin{defi}
{\rm
(i) For a finite set ${\bf x} =\{x_1,\cdots ,x_l\}\subset \Bbb R^N$,
let $\sigma ({\bf x})$ denote the convex hull spanned by ${\bf x}.$
Note that   $\sigma ({\bf x})$
is an $(l-1)$-dimensional simplex if and only if vectors
$\{x_k-x_1\}_{k=2}^l$ are linearly independent.
In particular, it is in general position if $x_1,\cdots ,x_l$ are linearly independent over $\Bbb R$.
\par
(ii) Let $h:X\to Y$ be a surjective map such that
$h^{-1}(y)$ is a finite set for any $y\in Y$, and let
$i:X\to \Bbb R^n$ be an embedding.
Let  $\mathcal{X}^{\Delta}$  and $h^{\Delta}:{\mathcal{X}}^{\Delta}\to Y$
denote the space and the map
defined by
$$
\mathcal{X}^{\Delta}=
\big\{(y,w)\in Y\times \Bbb R^N:
w\in \sigma (i(h^{-1}(y)))
\big\}\subset Y\times \Bbb R^N,
\ h^{\Delta}(y,w)=y.
$$
The pair $(\mathcal{X}^{\Delta},h^{\Delta})$ is called
{\it a simplicial resolution of }$(h,i)$.
In particular, $(\mathcal{X}^{\Delta},h^{\Delta})$
is called {\it a non-degenerate simplicial resolution} if for each $y\in Y$
and any $k$ points of $i(h^{-1}(y))$ span $(k-1)$-dimensional simplex of $\Bbb R^N$.
\par
(iii)
For each $k\geq 0$, let $\mathcal{X}^{\Delta}_k\subset \mathcal{X}^{\Delta}$ be the subspace
given by
$$
\mathcal{X}_k^{\Delta}=\big\{(y,\omega)\in \mathcal{X}^{\Delta}:
\omega\in\sigma ({\bf u}),
{\bf u}=\{u_1,\cdots ,u_l\}\subset i(h^{-1}(y)),l\leq k\big\}.
$$
We make identification $X=\mathcal{X}^{\Delta}_1$ by identifying the point $x\in X$ with the pair
$(h(x),i(x))\in \mathcal{X}^{\Delta}_1$,
and we note that  there is an increasing filtration
$$
\emptyset =
\mathcal{X}^{\Delta}_0\subset X=\mathcal{X}^{\Delta}_1\subset \mathcal{X}^{\Delta}_2\subset
\cdots \subset \mathcal{X}^{\Delta}_k\subset \mathcal{X}^{\Delta}_{k+1}\subset
\cdots \subset \bigcup_{k= 0}^{\infty}\mathcal{X}^{\Delta}_k=\mathcal{X}^{\Delta}.
$$
}
\end{defi}

\begin{lemma}[\cite{Mo2}, \cite{Va}]\label{lemma: simp}
Let $h:X\to Y$ be a surjective map such that
$h^{-1}(y)$ is a finite set for any $y\in Y$, and let
$i:X\to \Bbb R^N$ be an embedding.
\par
$\mbox{{\rm (i)}}$
If $X$ and $Y$ are closed semi-algebraic spaces and the
two maps $h$, $i$ are polynomial maps, then
$h^{\Delta}:\mathcal{X}^{\Delta}\stackrel{\simeq}{\rightarrow}Y$
is a homotopy equivalence.
\par
$\mbox{{\rm (ii)}}$
There is an embedding $j:X\to \Bbb R^M$ such that the associated simplicial resolution
$(\tilde{\mathcal{X}}^{\Delta},\tilde{h}^{\Delta})$
of $(h,j)$ is non-degenerate, and
the space $\tilde{\mathcal{X}}^{\Delta}$
is uniquely determined up to homeomorphism.
Moreover,
there is a filtration preserving homotopy equivalence
$q^{\Delta}:\tilde{\mathcal{X}}^{\Delta}\stackrel{\simeq}{\rightarrow}{\mathcal{X}}^{\Delta}$ such that $q^{\Delta}\vert X=\mbox{id}_X$.
\qed
\end{lemma}

\par
{\it Remark. }
Even when $h$ is not finite to one,  it is still possible to define its simplicial resolution and associated non-degenerate one.
We omit the details of this construction and refer the reader to  \cite{Mo2}.

\par
\begin{defi}
{\rm
Let  $Z_d\subset \Sigma_d\times \Bbb R^m$
denote
the tautological normalization of $\Sigma_d$
consisting of all pairs
$({\bf f},{\bf x})=((f_0,\cdots ,f_n),
(x_0,\cdots ,x_{m-1}))\in \Sigma_d\times\Bbb R^m$
such that the polynomials $f_0,\cdots ,f_n$ have a non-trivial common real root
$({\bf x},1)=(x_0,\cdots ,x_{m-1},1)$.
Projection on the first factor  gives a surjective map
$\pi_d^{\prime} :Z_d\to\Sigma_d.$
\par
Let $\phi_d:A_d^*\stackrel{\cong}{\rightarrow}\Bbb R^{N_d^*}$
be any fixed homeomorphism, and
let ${\rm H}_d$ be the set consisting of all
monomials $\varphi_I=z^I=z_0^{i_0}z_1^{i_1}\cdots z_m^{i_m}$
of degree $d$
($I=(i_0,i_1,\cdots ,i_m)\in\Bbb Z^{m+1}_{\geq 0}$,
$\vert I\vert =\sum_{k=0}^mi_k = d$).
Next, we define the  Veronese embedding, which will play a key role in our argument.
Let
$\psi_d^* :\Bbb R^m\to \Bbb R^{M_d}$
be the map given by
$\displaystyle
\psi_d^* (x_0,\cdots ,x_{m-1})=
\Big(\varphi_I(x_0,\cdots ,x_{m-1},1)
\Big)_{\varphi_I\in {\rm H}_d},$
where $M_d:=
\binom{d+m}{m}.$
Now define the embedding
$\Phi_d^* :Z_d^* \to \Bbb R^{N_d^*+M_d}$ by
$$
\Phi_d^*  ((f_0,\cdots ,f_n),{\bf x})=
(\phi_d^* (f_0,\cdots ,f_n),\psi_d^* ({\bf x})).
$$
}
\end{defi}

\begin{defi}
{\rm
Let
$({\mathcal{Z}}^{\Delta}(d),\ ^{\prime}{\pi_d^{\Delta}}:{\mathcal{Z}}^{\Delta}(d)\to\Sigma_d)$
and
$(\tilde{{\mathcal{Z}}^{\Delta}}(d),\ ^{\prime}{\tilde{\pi}_d^{\Delta}}:\tilde{{\mathcal{Z}}^{\Delta}}(d)\to\Sigma_d)$
denote the simplicial resolution of
$(\pi_d^{\prime},\Phi_d^*)$
and the corresponding non-degenerate
simplicial resolution
with the natural increasing filtrations
$$
\begin{cases}
\displaystyle
{\mathcal{Z}}^{\Delta}(d)_0=\emptyset
\subset {\mathcal{Z}}^{\Delta}(d)_1\subset
{\mathcal{Z}}^{\Delta}(d)_2\subset \cdots
\subset
{\mathcal{Z}}^{\Delta}(d)=\bigcup_{k= 0}^{\infty}{\mathcal{Z}}^{\Delta} (d)_k,
\\
\displaystyle
\tilde{\mathcal{Z}}^{\Delta}(d)_0
=\emptyset
\subset \tilde{\mathcal{Z}}^{\Delta}(d)_1
\subset
\tilde{\mathcal{Z}}^{\Delta}(d)_2\subset \cdots
\subset \tilde{\mathcal{Z}}^{\Delta}(d)=\bigcup_{k=0}^{\infty}
\tilde{\mathcal{Z}}^{\Delta}(d)_k.
\end{cases}
$$
}
\end{defi}
By Lemma \ref{lemma: simp}, the map
$^{\prime}{\pi_d^{\Delta}}:
{\mathcal{Z}}^{\Delta}(d)\stackrel{\simeq}{\rightarrow}\Sigma_d$ is a homotopy equivalence. It is easy to see that it
extends to a homotopy equivalence
$^{\prime}{\pi_{d+}^{\Delta}}:{\mathcal{Z}}^{\Delta}(d)_+\stackrel{\simeq}{\rightarrow}{\Sigma_{d+}},$
where $X_+$ denotes the one-point compactification of a
locally compact space $X$.
\par
Since
${{\mathcal{Z}}^{\Delta} (d)_r}_+/{{\mathcal{Z}}^{\Delta} (d)_{r-1}}_+
\cong ({\mathcal{Z}}^{\Delta} (d)_r\setminus {\mathcal{Z}}^{\Delta} (d)_{r-1})_+$,
we have the  Vassiliev type spectral sequence
$$
\big\{E_t^{r,s}(d),
d_t:E_t^{r,s}(d)\to E_t^{r+t,s+1-t}(d)
\big\}
\Rightarrow
H^{r+s}_c(\Sigma_d,\Bbb Z),
$$
where
$H_c^k(X,\Bbb Z)$ denotes the cohomology group with compact supports given by
$H_c^k(X,\Bbb Z):= H^k(X_+,\Bbb Z)$ and
$E_1^{r,s}(d):=H^{r+s}_c({\mathcal{Z}}^{\Delta}(d)_r\setminus{\mathcal{Z}}^{\Delta}(d)_{r-1},\Bbb Z)$.
\par

It follows from the Alexander duality that there is a natural
isomorphism
\begin{equation}\label{Al}
H_k(A_d^{\Bbb C}(m,n;g),\Bbb Z)\cong
H_c^{N_d^*-k-1}(\Sigma_d,\Bbb Z)
\quad
\mbox{for }1\leq k\leq N_d^*-2.
\end{equation}
Using (\ref{Al}) and
reindexing we obtain a
spectral sequence
\begin{eqnarray}\label{SS}
&&\big\{\tilde{E}^t_{r,s}(d), \tilde{d}^t:\tilde{E}^t_{r,s}(d)\to \tilde{E}^t_{r+t,s+t-1}(d)\big\}
\Rightarrow H_{s-r}(A_d^{\Bbb C}(m,n;g),\Bbb Z)
\end{eqnarray}
if $s-r\leq N_d^*-2$,
where
$\tilde{E}^1_{r,s}(d)=
H^{N_d^*+r-s-1}_c({\mathcal{Z}}^{\Delta}(d)_r\setminus{\mathcal{Z}}^{\Delta}(d)_{r-1},\Bbb Z).$

\begin{lemma}\label{lemma: simplex*}
$\mbox{{\rm (i)}}$
If
$\{y_1,\cdots ,y_{r}\}\in C_r(\Bbb R^{m})$ is  any set of $r$ distinct points
in $\Bbb R^m$ and $r\leq d+1$, then the $r$ vectors $\{\psi_d^*(y_k):1\leq k\leq r\}$
are linearly independent
over $\Bbb R$ and
span an $(r-1)$-dimensional simplex in $\Bbb R^{M_d}$.
\par
$\mbox{{\rm (ii)}}$
If $1\leq r\leq d+1$,
there is a homeomorphism
$$
{\mathcal{Z}}^{\Delta} (d)_r\setminus {\mathcal{Z}}^{\Delta} (d)_{r-1}\cong
\tilde{{\mathcal{Z}}^{\Delta}} (d)_r\setminus \tilde{{\mathcal{Z}}^{\Delta}} (d)_{r-1}
$$
\end{lemma}
\begin{proof}
The proof is completely analogous to that of
[\cite{AKY1}, Lemma 4.3].
\end{proof}

\begin{lemma}\label{lemma: vector bundle*}
If $1\leq r\leq \lfloor\frac{d+1}{2}\rfloor$,
${\mathcal{Z}}^{\Delta}(d)_r\setminus{\mathcal{Z}}^{\Delta}(d)_{r-1}$
is homeomorphic to the total space of a real
vector bundle $\xi_{d,r}$ over $C_r(\Bbb R^m)$ with rank
$l_{d,r}^*=N_d^*-(2n+1)r-1$.
\end{lemma}
\begin{proof}
The proof is completely analogous to that of [\cite{AKY1}, Lemma 4.4].
\end{proof}

\begin{lemma}\label{lemma: range**}
All non-zero entries of $\tilde{E}^1_{r,s}(d)$ are
situated in the range
$s\geq r\big(2n+2-m\big)$ if $1\leq r\leq d^m$,
and $\tilde{E}^1_{r,*}(d)=0$ if $r>d^m$.
\end{lemma}
\begin{proof}
The proof is completely analogous to that of
[\cite{AKY1}, Lemma 4.5].
\end{proof}
\begin{lemma}\label{lemma: range*}
If $1\leq r\leq \lfloor \frac{d+1}{2}\rfloor$, there is a natural isomorphism
$$
\tilde{E}^1_{r,s}(d)\cong
H_{s-(2n-m+2)r}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)}).
$$
\end{lemma}
\begin{proof}
By using the Thom isomorphism and Poincar\'e duality,
we obtain the desired isomorphism.
\end{proof}
\par\vspace{3mm}\par

Now we recall the spectral sequence constructed by V. Vassiliev
\cite{Va}.
From now on, we will assume that $m\leq 2n$, and
we identify $\mbox{{\rm Map}} (S^m,S^{2n+1})$ with the space
$\mbox{{\rm Map}} (S^m,\Bbb R^{2n+2}\setminus \{{\bf 0}_{2n+2}\})$.
We also choose  a map
$\varphi :S^m\to \Bbb R^{2n+2}\setminus\{{\bf 0}_{2n+2}\}$
and fix it.
Observe that $\mbox{{\rm Map}} (S^m,\Bbb R^{2n+2})$ is a linear space
and consider
the complements
${\frak A}_m^n=\mbox{{\rm Map}} (S^m,\Bbb R^{2n+2})\setminus \mbox{{\rm Map}} (S^m,S^{2n+1})$
and
$\tilde{\frak A}_m^n=\mbox{{\rm Map}}^* (S^m,\Bbb R^{2n+2})\setminus \mbox{{\rm Map}}^*(S^m,S^{2n+1}).$
\par
Note that  ${\frak A}_m^n$ consists of all continuous maps
$f:S^m\to \Bbb R^{2n+2}$ passing through ${\bf 0}_{2n+2}.$
We will denote by $\Theta^k_{\varphi}\subset \mbox{{\rm Map}} (S^m,\Bbb R^{2n+2})$ the subspace
consisting of all maps $f$ of the forms $f=\varphi +p$,
where $p$ is the restriction to $S^m$ of a polynomial map
$\Bbb R^{m+1}\to \Bbb R^{2n+2}$ of degree $\leq k$.
Let
$\Theta^k\subset \Theta^k_{\varphi}$ denote the subspace
consisting of all $f\in \Theta_{\varphi}^k$
passing through ${\bf 0}_{n+1}.$
In [\cite{Va}, page 111-112] Vassiliev uses the space
$\Theta^k$ as a finite dimensional approximation of ${\frak A}^n_m$
\par
Let $\tilde{\Theta}^k$ denote the subspace of $\Theta^k$ consisting of all maps
$f\in \Theta^k$ which preserve
the base points.
By a variation of the preceding argument, Vassiliev also shows that
$\tilde{\Theta}^k$ can be used as a finite dimensional approximation of
$\tilde{\frak A}_m^n$ [\cite{Va}, page 112].
\par
Let $\mathcal{X}_k\subset \tilde{\Theta}^k\times \Bbb R^{m+1}$ denote the
subspace consisting of all pairs
$(f,\alpha)\in\tilde{\Theta}^k \times \Bbb R^{m+1}$
such that $f(\alpha )={\bf 0}_{2n+2}$, and
let $p_k:\mathcal{X}_k\to \tilde{\Theta}^k$ be the projection onto the first factor.
Then,  by making use of simplicial resolutions of the surjective maps
$\{p_k:k\geq 1\}$,
one can construct an associated geometric resolution
$\{\tilde{\frak A}_m^n\}$ of $\tilde{\frak A}_m^n,$
whose cohomology approximates the homology of $\mbox{{\rm Map}}^*(S^m,S^{2n+1})=\Omega^mS^{2n+1}$ to any desired dimension.
From the natural filtration on the approximating space
$\displaystyle F_1\subset F_2\subset F_3\subset
\cdots \subset \bigcup_{k=1}^{\infty}F_k=\{\tilde{\frak A}_m^n\},$
we obtain the associated spectral sequence:
\begin{equation}\label{SSS}
\{E^t_{r,s},d^t:
E^t_{r,s}\to
E^t_{r+t,s+t-1}\}
\Rightarrow
H_{s-r}(\Omega^mS^{2n+1},\Bbb Z).
\end{equation}
The following result follows easily from
[\cite{Va}, Theorem 2 (page 112) and (32) (page 114)].

\begin{lemma}[\cite{Va}]
\label{lemma: Va}
Let $2\leq m\leq 2n$ be integers and let
$X$ be a finite $m$-dimensional simplicial complex
with a fixed base point $x_0\in X$.
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
$E^1_{r,s}=
H_{s-(2n-m+2)r}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)})$ if $r\geq 1$,
and $E^1_{r,s}=0$ if $r<0$ or $s<0$ or $s<(2n-m+2)r$.
\item[$\mbox{{\rm (ii)}}$]
For any $t\geq 1$,
$d^t=0:
E^t_{r,s}\to
E^t_{r+t,s+t-1}$ for all $(r,s)$,
and $E^1_{r,s}=E^{\infty}_{r,s}$.
Moreover,
for any $k\geq 1$, the extension problem for
the graded group
$Gr (H_k(\Omega^mS^{2n+1},\Bbb Z))=
\bigoplus_{r=1}^{\infty}E^{\infty}_{r,r+k}
=\bigoplus_{r=1}^{\infty}E^{1}_{r,r+k}$
is trivial and
there is an isomorphism
$$
H_{k}(\Omega^mS^{2n+1},\Bbb Z)
\cong
\bigoplus_{r=1}^{\infty}
H_{k-(2n-m+1)r}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)}).
\qed
$$
\end{enumerate}
\end{lemma}

\begin{defi}
{\rm
We identify $\Omega^mS^{2n+1}=\Omega^m(\Bbb C^{n+1}\setminus \{{\bf 0}\})$ and
define the map $j_d^{\prime}:A_d^{\Bbb C}(m,n;g)\to \Omega^mS^{2n+1}$ by
$$
j_d^{\prime}(f_0,\cdots ,f_n)(x_0,\cdots ,x_m)=
(f_0(x_0,\cdots ,x_m),\cdots\cdots ,f_n(x_0,\cdots ,x_m))
$$
for $((f_0,\cdots ,f_n),(x_0,\cdots ,x_m))\in A_d^{\Bbb C}(m,n;g)\times S^m$.
}
\end{defi}

Now, by applying the spectral sequence
(\ref{SSS}),  we prove the following result,  which plays a key role the proof of Theorem \ref{thm: I} .

\begin{thm}\label{thm: I*}
Let $m,n\geq 2$ be positive integers such that $2\leq m\leq 2n$, and
let $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ be an algebraic map of minimal degree $d$.
\begin{enumerate}
\item[$\mbox{{\rm (i)}}$]
The map
$j_{d}^{\prime}:A_d^{\Bbb C}(m,n;g)\to  \Omega^mS^{2n+1}$
is a homotopy equivalence through dimension $D_{\Bbb C}(d;m,n)$ if
$m<2n$ and a homology equivalence through dimension $D_{\Bbb C}(d;m,n)$
if $m=2n$.
\item[$\mbox{{\rm (ii)}}$]
For any $k\geq 1$,
$H_k(A_d^{\Bbb C}(m,n;g),\Bbb Z)$ contains the subgroup
$$
G^d_{m,2n+1}=\bigoplus_{r=1}^{\lfloor \frac{d+1}{2}\rfloor}
H_{k-(2n-m+1)r}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)} )
$$
as a direct summand.
\end{enumerate}
\end{thm}

\begin{proof}
Consider the spectral sequence (\ref{SSS}).
First, note that, by Lemma \ref{lemma: simp}, there is a filtration
preserving homotopy equivalence
$^{\prime}q^{\Delta}:\tilde{{\mathcal{Z}}^{\Delta}} (d)\stackrel{\simeq}{\rightarrow}{\mathcal{Z}}^{\Delta} (d).$
Note also that the image of the  map $j_d^{\prime}$ lies
in a space of polynomial mappings, which approximates  the space of continuous mappings
$S^m\to S^{2n+1}$.
Since $\tilde{\mathcal{Z}}^{\Delta}(d)$ is non-degenerate,
the map$j_d^{\prime}$ naturally extends to a filtration
preserving map
$^{\prime}{\tilde{\pi}}: \tilde{{\mathcal{Z}}^{\Delta}} (d)\to \{\tilde{\frak A}_m^n\}$
between  resolutions.
Thus the filtration preserving maps
$$
\begin{CD}
{\mathcal{Z}}^{\Delta} (d) @<^{\prime}q^{\Delta}<\simeq< \tilde{{\mathcal{Z}}^{\Delta}} (d)
@>^{\prime}\tilde{\pi}>> \{\tilde{\frak A}_m^n\}
\end{CD}
$$
induce a homomorphism of spectral sequences
$\{\tilde{\theta}^t_{r,s}:\tilde{E}^t_{r,s}(d)\to E^t_{r,s}\},$
where $\{E^t_{r,s},d^t\}\Rightarrow H_{s-r}(\Omega^mS^n,\Bbb Z).$
\par
Observe that,
by Lemma \ref{lemma: range*},
(ii) of Lemma \ref{lemma: Va} and
the naturality of Thom isomorphism,
for $r\leq \lfloor \frac{d+1}{2}\rfloor$
 there is a
commutative diagram
\begin{equation}\label{Thom2}
\begin{CD}
\tilde{E}^1_{r,s}(d)
@>T>\cong> H_{s-r(2n-m+2)}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)})
\\
@V\tilde{\theta}^1_{r,s}VV  \Vert @.
\\
E^1_{r,s}
@>T>\cong> H_{s-r(2n-m+2)}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)} )
\end{CD}
\end{equation}
Hence, if $r\leq \lfloor \frac{d+1}{2}\rfloor$,
$\tilde{\theta}^1_{r,s}:\tilde{E}^1_{r,s}(d)\stackrel{\cong}{\rightarrow}
E^1_{r,s}$ and thus so is
$\tilde{\theta}^{\infty}_{r,s}:\tilde{E}^{\infty}_{r,s}(d)\stackrel{\cong}{\rightarrow}
E^{\infty}_{r,s}$.
\par
Next, we will compute the number
$$
D_{min}=\min\{
N\vert \ N\geq s-r,\
s\geq (2n+2-m)r,\
1\leq r<\lfloor \frac{d+1}{2}\rfloor +1
\}.
$$
It is easy to see that  $D_{min}$ is the
largest integer $N$ which satisfies the inequality
$(n+1-m)r> r+N$
for $r=\lfloor\frac{d+1}{2}\rfloor +1$, hence
\begin{equation}\label{Dmin}
D_{min}=
(2n-m+1)(\lfloor\frac{d+1}{2}\rfloor +1)-1=D_{\Bbb C}(d;m,n).
\end{equation}
We note that, for dimensional reasons,
$\tilde{\theta}^{\infty}_{r,s}:
\tilde{E}^{\infty}_{r,s}(d)\stackrel{\cong}{\rightarrow}
E^{\infty}_{r,s}$ is always
an isomorphism if
$r\leq \lfloor \frac{d+1}{2}\rfloor$ and $s-r\leq D_{\Bbb C}(d;m,n)$.
\par
On the other hand,  from Lemma \ref{lemma: range**}
it easily follows that
 $\tilde{E}^1_{r,s}(d)=E^1_{r,s}=0$ if
$s-r\leq D_{\Bbb C}(d;m,n)$ and $r>\lfloor \frac{d+1}{2}\rfloor$.
Hence,
we have:
\begin{enumerate}
\item[(\ref{Dmin}.1)]
If  $s\leq r+D_{\Bbb C}(d;m,n)$, then
$\tilde{\theta}^{\infty}_{r,s}:
\tilde{E}^{\infty}_{r,s}(d)\stackrel{\cong}{\rightarrow}
E^{\infty}_{r,s}$ is
always an isomorphism.
\end{enumerate}
Hence, by using the comparison Theorem of spectral sequences,
we have that
$j_{d}^{\prime}$
is a homology equivalence through dimension $D_{\Bbb C}(d;m,n)$.
Since $A_d^{\Bbb C}(m,n;g)$ and $\Omega^mS^{2n+1}$ are simply connected
if $m<2n$,
$j_{d}^{\prime}$ is a homotopy equivalence through dimension $D_{\Bbb C}(d;m,n)$
if  $m<2n$.
Hence, (i) is proved.
\par
It remains to show (ii).
Since
$d^t=0$ for any $t\geq 1$, from
the equality
$d^t\circ \tilde{\theta}^t_{r,s}
=\tilde{\theta}^t_{r+t,s+t-1}\circ \tilde{d}^t$
and some diagram chasing,
we obtain $\tilde{E}^1_{r,s}(d)=\tilde{E}^{\infty}_{r,s}(d)$ for all $r\leq \lfloor \frac{d+1}{2}\rfloor$.
Moreover, since
the extension problem for the graded group
$$
Gr (H_k(\Omega^mS^{2n+1},\Bbb Z))=\bigoplus_{r=1}^{\infty}E^{\infty}_{r,k+r}
=\bigoplus_{r=1}^{\infty}E^{1}_{r,k+r}
$$
is trivial,  by using (\ref{Dmin}.1) we  can  prove
that the associated graded group
$$
Gr (H_k(A_d^{\Bbb C}(m,n;g),\Bbb Z))=\bigoplus_{r=1}^{\infty}\tilde{E}^{\infty}_{r,k+r}(d)
=\bigoplus_{r=1}^{\infty}\tilde{E}^{1}_{r,k+r}(d)
$$
is also trivial until the $\lfloor\frac{d+1}{2}\rfloor$-th term of the filtration.
Hence,  $H_k(A_d^{\Bbb C}(m,n;g),\Bbb Z)$ contains the subgroup
$$
\bigoplus_{r=1}^{\lfloor \frac{d+1}{2}\rfloor}\tilde{E}^1_{r,k+r}(d)
=
\bigoplus_{r=1}^{\lfloor \frac{d+1}{2}\rfloor}\tilde{E}^{\infty}_{r,k+r}(d)
\cong
\bigoplus_{r=1}^{\lfloor \frac{d+1}{2}\rfloor}
H_{k-r(2n-m+1)}(C_r(\Bbb R^m),(\pm \Bbb Z)^{\otimes (2n-m+1)})
$$
as a direct summand, which proves the assertion (ii).
\end{proof}

\begin{cor}\label{cor: I*}
Let $m,n\geq 2$ be positive integers such that $2\leq m\leq 2n$,
let $g\in \mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ be an algebraic map of minimal degree $d$,
and let $\Bbb F =\Bbb Z/p$ $($p: prime$)$ or $\Bbb F =\Bbb Q$.
Then the map $j_d^{\prime}:A_d^{\Bbb C}(m,n;g)\to \Omega^mS^{2n+1}$ induces an isomorphism on
the homology group $H_k(\ ,\Bbb F)$ for any $1\leq k\leq D_{\Bbb C}(d;m,n)$.
\end{cor}
\begin{proof}
In the  proof of Theorem \ref{thm: I*} replace the homology groups $H_k(\ ,\Bbb Z)$ and $H_k(\ ,(\pm \Bbb Z)^{\otimes k})$ by $H_k(\ ,\Bbb F)$
and $H_k(\ ,(\pm \Bbb F)^{\otimes k})$ and use the same argument.
\end{proof}

\par
Let $\gamma_m:S^{m}\to \Bbb R\mbox{{\rm P}}^m$ and $\gamma_n^{\Bbb C}:S^{2n+1}\to\Bbb C {\rm P}^n$
denote the usual double covering and the
Hopf fibration map, respectively.
Let
$\gamma_m^{\#}:\mbox{{\rm Map}}^*_{\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\to \Omega^m\Bbb C {\rm P}^n$ be given by
$\gamma_m^{\#}(h)=h\circ \gamma_m$.
It is easy to verify that the following  diagram
\begin{equation}\label{gamma}
\begin{CD}
A_d^{\Bbb C}(m,n;g) @>\Psi_d^{\prime}>\cong> \mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g) @>i_d^{\prime}>\subset> F_d(m,n;g)
\\
@V{j_d^{\prime}}VV @. @V{i^{\prime}}V{\cap}V
\\
\Omega^mS^{2n+1} @>\Omega^m\gamma_n^{\Bbb C}>\simeq>
\Omega^m\Bbb C {\rm P}^n @<\gamma_m^{\#}<< \mbox{{\rm Map}}^*_{[d]_2}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{CD}
\end{equation}
is commutative,
where $i^{\prime}:F_d(m,n;g)\stackrel{\subset}{\rightarrow} \mbox{{\rm Map}}_{[d]_2}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
and $\Psi_d^{\prime}$ denote the inclusion and the restriction
$\Psi_d^{\prime}=\Psi_d^{\Bbb C}\vert A_d^{\Bbb C}(m,n;g)$,
respectively.
\begin{lemma}\label{lemma: II}
If $2\leq m<2n$  and
$g\in\mbox{{\rm Alg}}_d^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb R\mbox{{\rm P}}^n)$ a fixed map of minimal degree $d$,
then the map
$\gamma_m^{\#}\circ i^{\prime}:F_d(m,n;g)\to\Omega^m\Bbb C {\rm P}^n$
is a homotopy equivalence through dimension $D_{\Bbb C}(d;m,n)$.
\end{lemma}
\begin{proof}
Since there is a homotopy equivalence $F_d(m,n;g)\simeq \Omega^m\Bbb C {\rm P}^n$,
the two spaces $F_d(m,n;g)$ and $\Omega^m\Bbb C {\rm P}^n$ are simple.
So it suffices to show that the map $\gamma_m^{\#}\circ i^{\prime}$ is a homology equivalence
through dimension $D_{\Bbb C}(d;m,n)$.
\par
Let $\Bbb F =\Bbb Z/p$ $(p$: prime) or $\Bbb F =\Bbb Q$, and consider the induced homomorphism
$(\gamma_m^{\#}\circ i^{\prime})_*=H_k(\gamma_m^{\#}\circ i^{\prime},\Bbb F)
:H_k(F_d(m,n;g),\Bbb F)\to H_k(\Omega^m\Bbb C {\rm P}^n,\Bbb F)$.
Since $\Omega^m\gamma_n^{\Bbb C}$ is a homotopy equivalence by
 Corollary \ref{cor: I*} and the commutativity of the diagram (\ref{gamma})
$(\gamma_m^{\#}\circ i^{\prime})_*$ is an epimorphism for any $1\leq k\leq D_{\Bbb C}(d;m,n)$.
\par
However, since there is a homotopy equivalence
$F_d(m,n;g)\simeq \Omega^m\Bbb C {\rm P}^n$, we have
$\dim_{\Bbb F}H_k(F_d(m,n;g),\Bbb F)=\dim_{\Bbb F}H_k(\Omega^m\Bbb C {\rm P}^n,\Bbb F)<\infty$
for any $k$.
Hence,
$H_k(\gamma_m^{\#}\circ i^{\prime},\Bbb F )$ is an isomorphism for any $1\leq k\leq D_{\Bbb C}(d;m,n)$.
By the Universal Coefficient Theorem $\gamma_m^{\#}\circ i^{\prime}$ induces an isomorphism on $H_k(\ ,\Bbb Z)$ for any $1\leq k\leq D_{\Bbb C}(d;m,n)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm: I}]
Since $\Psi_d^{\prime}:A_d^{\Bbb C}(m,n;g)\stackrel{\cong}{\rightarrow}\mbox{{\rm Alg}}_d^{\Bbb C}(m,n;g)$ is a
homeomorphism, the assertion (ii) follows from Theorem \ref{thm: I*}.
Because $\gamma_m^{\#}\circ i^{\prime}$ a homotopy equivalence through dimension $D(d;m,n)$,
by using the  diagram (\ref{gamma}) and Theorem \ref{thm: I*}
 we easily obtain the assertion (i).
\end{proof}

\section{The space $A_d^{\Bbb C}(m,n)$.}\label{section 3.3}

In this section we shall consider the unstable problem for the space
$A_d^{\Bbb C}(m,n)$, where $d = 2 d^*\geq 2$ is even.

\begin{defi}
{\rm
Define
$\psi_{m,n}:A_d^{\Bbb R}(m,2n+1)\to A_d^{\Bbb C}(m,n)$ by
$$
\psi_{m,n}(f_0,\cdots ,f_{2n+1})=
(f_0+\sqrt{-1}f_1,f_2+\sqrt{-1}f_3,\cdots ,
f_{2n}+\sqrt{-1}f_{2n+1}).
$$
}
\end{defi}
It is easy to see that
\begin{lemma}\label{lemma: varphi}
$\psi_{m,n}:A_d^{\Bbb R}(m,2n+1)
\stackrel{\cong}{\rightarrow}
A_d^{\Bbb C}(m,n)$ is a homeomorphism.
\qed
\end{lemma}

\begin{lemma}\label{lemma: simply}
$\mbox{{\rm (i)}}$
$\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})$ is $(2n-m)$-connected.
\par
$\mbox{{\rm (ii)}}$
$\pi_{2n-m+1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1}))\cong
\begin{cases}
\Bbb Z & \mbox{ if }\ m\equiv 1\ \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z/2 & \mbox{ if }\ m\equiv 0\  \mbox{{\rm (mod $2$)}} .
\end{cases}
$
\end{lemma}
\begin{proof}
(i)
We argue by induction on $m$.
For $m=1$ the result follows from the homotopy equivalence
$\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^1,S^{2n+1})\simeq
\Omega S^{2n+1}$ .
Suppose that the space
$\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m-1},S^{2n+1})$
is $(2n-m+1)$-connected for some $m\geq 2$.
Since $\Omega^mS^{2n+1}$ is $(2n-m)$-connected,
from the restriction fibration sequence
$\Omega^mS^{2n+1} \to \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1}) \stackrel{}{\rightarrow}
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m-1},S^{2n+1})$,
we deduce that $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})$ is $(2n-m)$-connected.
Hence, (i) has been proved.
\par
(ii) First, consider the case $m=1$. Since $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^1,S^{2n+1})\simeq
\Omega S^{2n+1}$, (ii) clearly holds for $m=1$.
Next, consider the case $m=2$.
If we consider the fibration sequence
$\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^2,S^{2n+1})\to \Omega S^{2n+1}\stackrel{2}{\rightarrow}\Omega S^{2n+1}$
induced from
the cofibration sequence $S^1\stackrel{2\iota_1}{\rightarrow}S^1 \to \Bbb R\mbox{{\rm P}}^2$,
an easy computation shows that $\pi_{2n-1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^2,S^{2n+1}))\cong \Bbb Z/2.$
Hence, (ii) holds for $m=2$, too.
\par
Now we assume that $m\geq 3$ and consider the fibration sequence
$$
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1})\to
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})\stackrel{res}{\longrightarrow}
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1})
$$
Since $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1})$ is $(2n-m+2)$-connected,
there is an isomorphism
$\pi_{2n-m+1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1}))\cong
\pi_{2n-m+1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1}))$.
Thus it remains to show the following:
\begin{equation}\label{picomp}
\pi_{2n-m+1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1}))
\cong
\begin{cases}
\Bbb Z & \mbox{ if }\ m\equiv 1\ \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z/2 & \mbox{ if }\ m\equiv 0\  \mbox{{\rm (mod $2$)}} .
\end{cases}
\end{equation}
If $m\equiv 1$ $\mbox{{\rm (mod $2$)}}$, $\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2}=S^{m-1}\vee S^m$ and
there is an isomorphism
$$
\pi_{2n-m+1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1}))\cong
\pi_{2n-m+1}(\Omega^{m-1}S^{2n+1}\times
\Omega^mS^{2n+1})\cong \Bbb Z .
$$
Hence (\ref{picomp}) holds for $m\equiv 1$ $\mbox{{\rm (mod $2$)}}$.
Finally suppose that $m\equiv 0$ $\mbox{{\rm (mod $2$)}}$.
Then, because
 $\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2}=S^{m-1}\cup_2e^m$,
there is a fibration sequence
$$
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1})\to \Omega^{m-1}S^{2n+1}
\stackrel{2}{\rightarrow}
\Omega^{m-1}S^{2n+1}.
$$
From  the homotopy exact sequence induced by the above sequence,
we deduce that
$\pi_{2n-m+1}(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m/\Bbb R\mbox{{\rm P}}^{m-2},S^{2n+1}))\cong \Bbb Z/2$.
\end{proof}

\begin{defi}
{\rm
(i)
Let $\gamma_n:S^n\to \Bbb R\mbox{{\rm P}}^n$ and
$\gamma_n^{\Bbb C}:S^{2n+1}\to \Bbb C {\rm P}^n$ denote the usual double covering
and the Hopf fibering as before.
Define the following two maps
$$
\begin{cases}
{\gamma_n}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^n)\to\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^n)
\\
{\gamma_n^{\Bbb C}}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})\to
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{cases}
$$
by ${\gamma_n}_{\#}(h)=\gamma_n\circ h$ and
${\gamma_n^{\Bbb C}}_{\#}(h^{\prime})=\gamma_n^{\Bbb C}\circ h^{\prime}$.
\par
Since $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^n)$ and $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})$) contain the subspace of constant maps,
the images of ${\gamma_n}_{\#}$ and that of ${\gamma_n^{\Bbb C}}_{\#}$ are contained in
$\mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^n)$ and
$\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$, respectively.
Thus we obtain two maps
$$
\begin{cases}
{\gamma_n}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^n)\to \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^n),
\\
{\gamma_n^{\Bbb C}}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1}) \to \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n).
\end{cases}
$$
\par
(ii) Let $\mu_n:\Bbb R\mbox{{\rm P}}^{2n+1}\to \Bbb C {\rm P}^n$ denote the usual projection given by
$$
\mu_n([x_0:x_1:\cdots :x_{2n+1}]=[x_0+\sqrt{-1}x_1:
\cdots :x_{2n}+\sqrt{-1}x_{2n+1}],
$$
and define the map
${\mu_n}_{\#}:\mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^{2n+1})\to\mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ by
${\mu_n}_{\#}(h)=\mu_n\circ h$.
}
\end{defi}

\begin{lemma}\label{lemma: Hopf}
$\mbox{{\rm (i)}}$
If $1\leq m<n$,
${\gamma_n}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^n)\stackrel{\simeq}{\rightarrow} \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^n)$
is a homotopy equivalence.
\par
$\mbox{{\rm (ii)}}$
If $2\leq m\leq 2n$,
${\gamma_n^{\Bbb C}}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})
\stackrel{\simeq}{\rightarrow}
\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is a homotopy equivalence.
\end{lemma}
\par
{\it Remark. }
Since $\pi_1(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^1,S^{2n+1}))=0$ and
$\pi_1(\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^1,\Bbb C {\rm P}^n))=\Bbb Z$, (ii) of Theorem \ref{lemma: Hopf}
does not hold for $m=1$.
\begin{proof}
Since (i) follows from [\cite{AKY1}, Lemma 4.17],
it remains to prove (ii).
We prove it by induction on $m$.
First, assume that $m=2$, and consider the following commutative diagram
of restriction fibration sequences
\begin{equation}\label{****}
\begin{CD}
\Omega^2S^{2n+1} @>>> \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^2,S^{2n+1}) @>r>> \Omega S^{2n+1}
\\
@V{\Omega^2\gamma_n^{\Bbb C}}V{\simeq}V @V{\gamma_n}_{\#}VV @V{\Omega \gamma_n^{\Bbb C}}VV
\\
\Omega^2\Bbb C {\rm P}^n @>>> \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n) @>r>> \Omega \Bbb C {\rm P}^n
\end{CD}
\end{equation}
Since $\Omega^2\gamma_n^{\Bbb C}$ is a homotopy equivalence
and
${\Omega \gamma_n^{\Bbb C}}_*:\pi_k(\Omega S^{2n+1})
\stackrel{\cong}{\rightarrow}\pi_k(\Omega \Bbb C {\rm P}^n)$
is an isomorphism for any $k\geq 2$,
by the Five Lemma
the induced homomorphism
${{\gamma_n^{\Bbb C}}_{\#}}_*:\pi_k(\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^2,S^{2n+1}))
\stackrel{\cong}{\rightarrow}
\pi_k(\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))$ is an isomorphism for any $k\geq 2$.
On the other hand, because
the homotopy exact sequence of the lower row of (\ref{****})
is
$$
0\to \pi_1(\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))\to
\Bbb Z =\pi_1(\Omega \Bbb C {\rm P}^n) \stackrel{\partial}{\rightarrow}
\pi_0(\Omega^2\Bbb C {\rm P}^n)=\Bbb Z \to 0,
$$
we see that
$\pi_1(\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))=0$.
Thus, since  $\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^2,S^{2n+1})$ is $(2n-2)$-connected
(by Lemma \ref{lemma: simply}),
the induced homomorphism
${\Omega \gamma_n^{\Bbb C}}_*:\pi_1(\Omega S^{2n+1})
\stackrel{\cong}{\rightarrow}\pi_1(\Omega \Bbb C {\rm P}^n)$
is  an isomorphism.
Hence, $\pi_k({\gamma_n^{\Bbb C}}_{\#})$ is an isomorphism for any $k\geq 1$ and
the assertion is true for $m=2$.
\par
Now suppose that
${\gamma_n}_{\#}^{\prime}:
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m-1},S^{2n+1})\stackrel{\simeq}{\rightarrow}
\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$
is a  homotopy equivalence for some $m\geq 3$, and
consider the following commutative diagram
of restriction fibration sequences
\begin{equation*}\label{***}
\begin{CD}
\Omega^mS^{2n+1} @>>> \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1}) @>r>> \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m-1},S^{2n+1})
\\
@V{\Omega^m\gamma_n^{\Bbb C}}V{\simeq}V @V{\gamma_n}_{\#}VV @V{\gamma_n}_{\#}^{\prime}V{\simeq}V
\\
\Omega^m\Bbb C {\rm P}^n @>>> \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n) @>r>> \mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)
\end{CD}
\end{equation*}
Since $\Omega^m\gamma_n^{\Bbb C}$ and ${\gamma^{\prime}_n}_{\#}$ are homotopy equivalences,
by  the Five Lemma,
we see that
${\gamma_n}_{\#}:\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^{m},S^{2n+1})\stackrel{\simeq}{\rightarrow}
\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^{m},\Bbb C {\rm P}^n)$
is also a homotopy equivalence.
\end{proof}

\begin{cor}\label{cor: M0-homotopy}
If $2\leq m\leq 2n$,
$\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ is $(2n-m)$-connected and
$$
\pi_{2n-m+1}(\mbox{{\rm Map}}^*_0(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n))\cong
\begin{cases}
\Bbb Z & \mbox{ if }\ m\equiv 1\ \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z/2 & \mbox{ if }\ m\equiv 0\  \mbox{{\rm (mod $2$)}} .
\end{cases}
$$
\end{cor}
\begin{proof}
This follows from Lemma \ref{lemma: simply} and
Lemma \ref{lemma: Hopf}.
\end{proof}

\begin{cor}\label{cor: mu}
If $2\leq m\leq 2n$, the map
${\mu_n}_{\#}:\mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^{2n+1})\stackrel{\simeq}{\rightarrow}
\mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is a homotopy equivalence.
\end{cor}
\begin{proof}
Consider the commutative diagram
$$
\begin{CD}
\mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1}) @>{\gamma_n^{\Bbb C}}_{\#}>\simeq> \mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\\
@V{{\gamma_{2n+1}}_{\#}}V{\simeq}V \Vert @.
\\
\mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^{2n+1}) @>{\mu_n}_{\#}>> \mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{CD}
$$
Since ${\gamma_{2n+1}}_{\#}$ and ${\gamma_n^{\Bbb C}}_{\#}$ are homotopy equivalences
by Lemma \ref{lemma: Hopf},
the assertion easily follows from the above commutative diagram.
\end{proof}

Now we can prove Theorem \ref{thm: II} and Corollary \ref{cor: III}.

\begin{proof}[Proof of Theorem \ref{thm: II}]
First, it is easy to see
the diagram
\begin{equation}\label{basic CD}
\begin{CD}
A_d^{\Bbb R}(m,2n+1) @>j_d^{\Bbb R}>> \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,S^{2n+1})
\\
@V\psi_{m,n}V{\cong}V \Vert @.
\\
A_d^{\Bbb C}(m,n) @>j_d^{\Bbb C}>> \mbox{{\rm Map}}^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\end{CD}
\end{equation}
is commutative.
Since $j_d^{\Bbb R}$ is a homotopy equivalence through dimension
$D_{\Bbb R}(d;m,2n+1)$ if $m<2n$ and a homology equivalence through
dimension $D_{\Bbb R}(d;m,2n+1)$ if $m=2n$,
by Theorem \ref{thm: AKY1-II},so is  the map
$j_d^{\Bbb C}$.
Because $D_{\Bbb C}(d;m,n)=D_{\Bbb R}(d;m,2n+1)$,
we see that the map $j_d^{\Bbb C}$ is a homotopy equivalence through
dimension $D_{\Bbb C}(d;m,n)$ if $m<2n$ and a homology equivalence
through dimension $D_{\Bbb C}(d;m,n)$ if $m=2n$.
It remains to show that the same holds for map $i_d^{\Bbb C}$.
However, this follows easily from the facts that
$i_d^{\Bbb C}={\gamma_n^{\Bbb C}}_{\#}\circ j_d^{\Bbb C}$ and
${\gamma_n^{\Bbb C}}_{\#}$ is a homotopy equivalence
(by Lemma \ref{lemma: Hopf}).
This completes the proof of Theorem \ref{thm: II}.
\end{proof}

\begin{proof}[Proof of Corollary \ref{cor: III}]
Since $j_{d+2}^{\Bbb C}\circ s_d=j_d^{\Bbb C}$, the assertion easily follows from
Theorem \ref{thm: II}.
\end{proof}

\section{The stabilized space $A_{\infty+\epsilon}^{\Bbb C}(m,n)$.  }
Although we cannot prove Conjecture \ref{conj: Psi}, we can prove the following
stabilized version.

\begin{defi}
{\rm
For $\epsilon =0$ or $1$, let $A^{\Bbb C}_{\infty +\epsilon}(m,n)$ denote the
stabilized space
$\displaystyle A^{\Bbb C}_{\infty +\epsilon}(m,n)=
\lim_{k\to\infty}A_{2k+\epsilon}^{\Bbb C}(m,n)$, where
the limit is taken over  the stabilization maps
$$
A_{\epsilon}^{\Bbb C}(m,n)\stackrel{s_{\epsilon}}{\longrightarrow}
A_{2+\epsilon}^{\Bbb C}(m,n)\stackrel{s_{2+\epsilon}}{\longrightarrow}
A_{4+\epsilon}^{\Bbb C}(m,n)\stackrel{s_{4+\epsilon}}{\longrightarrow}
A_{6+\epsilon}^{\Bbb C}(m,n)\stackrel{s_{6+\epsilon}}{\longrightarrow}
\cdots
$$
From the commutative diagram
$$
\begin{CD}
 @>>> A^{\Bbb C}_{2k+\epsilon}(m,n) @>s_{2k+\epsilon}>> A^{\Bbb C}_{2k+2+\epsilon}(m,n)
@>s_{2k+2+\epsilon}>> \cdots
\\
@. @V{\Psi_{2k+\epsilon}^{\Bbb C}}VV @V{\Psi^{\Bbb C}_{2k+2+\epsilon}}VV @.
\\
 @>>> \mbox{{\rm Alg}}^*_{2k+\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n) @>\subset>>
\mbox{{\rm Alg}}^*_{2k+2+\epsilon}(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
@>\subset>> \cdots
\end{CD}
$$
we obtain a stabilized map
$\displaystyle
\Psi^{\Bbb C}_{\infty +\epsilon}=
\lim_{k\to\infty}\Psi_{2k+\epsilon}^{\Bbb C}:
A^{\Bbb C}_{\infty +\epsilon}(m,n)\to \mbox{{\rm Alg}}^*_{\epsilon}(m,n).$
}
\end{defi}

\begin{prop}\label{prop: infty}
If  $2\leq m\leq 2n$ and $\epsilon =0$,
the map
$$
\Psi_{\infty +0}^{\Bbb C}:
A^{\Bbb C}_{\infty +0}(m,n)\stackrel{\simeq}{\rightarrow}
\mbox{{\rm Alg}}^*_{0}(m,n)
\simeq \mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
$$
is a homotopy equivalence if $m<2n$ and a homology equivalence
if $m=2n$.
\end{prop}
\par
{\it Proof. }
The assertion easily follows from
Theorem \ref{thm: stable}, Corollary  \ref{cor: IV} and the commutative diagram
$$
\begin{CD}
A^{0}_{\infty}(m,n) @>\lim_{k}i_{2k}^{\Bbb C}>\simeq> \mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\\
@V{\Psi_{\infty +0}^{\Bbb C}}VV \Vert @.
\\
\mbox{{\rm Alg}}^*_{0}(m,n) @>i>\simeq> \mbox{{\rm Map}}_0^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)
\qed
\end{CD}
$$

\section{The space $\mbox{{\rm Alg}}^{*}_1(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$.}

In this section we investigate the homotopy  of
$A_1^{\Bbb C}(m,n)\cong \mbox{{\rm Alg}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb R\mbox{{\rm P}}^n)$.

\begin{defi}
{\rm
For integers $1\leq m\leq 2n$, let
$V_{2n+1,m}$ denote the real Stiefel manifold  of all orthogonal $m$-frames
in $\Bbb R^{2n+1}$ and
$({\bf b}_1,\cdots ,{\bf b}_m)\in V_{2n+1,m}$ any element such that
${\bf b}_k=\ ^t(b_{k,1},\cdots ,b_{k,2n+1})\in\Bbb R^{2n+1}$ $(1\leq k\leq m)$.
Consider
the $(n+1)$-tuple of polynomial
defined by
$$
(f_0,\cdots ,f_n)=(z_0,\cdots ,z_m)
\left(\begin{array}{cccccc}
1 & 0 & 0 &\cdots & 0 & 0
\\
\sqrt{-1}{\bf b} & {\bf c}_1 & {\bf c}_2 & \cdots & {\bf c}_{n-1} & {\bf c}_n
\end{array}
\right),
$$
where
${\bf b}\in \Bbb R^m$ and ${\bf c}_k\in \Bbb C^m$ $(k=1,2,\cdots ,n)$ are given by
$$
\begin{cases}
{\bf b}=& ^{t}(b_{1,1},b_{1,2},b_{1,3},\cdots .b_{1,m}),
\\
{\bf c}_k=& ^t(b_{2k,1}+\sqrt{-1}b_{2k+1,1},b_{2k,2}+\sqrt{-1}b_{2k+1,2},\cdots ,b_{2k,m}+\sqrt{-1}b_{2k+1,m})
\end{cases}
$$
Since it is easy to see that $(f_0,\cdots ,f_n)\in A_1^{\Bbb C}(m,n)$, one can define the map
$\varphi_{m,n}:V_{2n+1,m}\to A_1^{\Bbb C}(m,n)$ by
$\varphi_{m,n}({\bf b}_1,\cdots ,{\bf b}_m)=(f_0,\cdots ,f_n)$.
}
\end{defi}

\begin{lemma}\label{lemma: V}
If $1\leq m\leq 2n$, the map
$\varphi_{m,n}:V_{2n+1,m}\stackrel{\simeq}{\rightarrow}
A_1^{\Bbb C}(m,n)$ is a homotopy equivalence.
\end{lemma}
\begin{proof}
Let us consider the element
$(f_0,\cdots ,f_n)\in\Bbb C [z_0,\cdots ,z_m]^{n+1}$ of the form
\begin{equation}\label{matrix}
(f_0,\cdots ,f_n)=
(z_0,\cdots ,z_m)
\left(
\begin{array}{cccc}
1 & 0 & \cdots & 0
\\
a_{1,0} & a_{1,1} & \cdots & a_{1,n}
\\
\vdots & \vdots & \vdots & \vdots
\\
a_{m,0} & a_{m,1} & \cdots & a_{m,n}
\end{array}
\right),
\end{equation}
where $a_{k,j}=b_{k,j}+\sqrt{-1}c_{k,j}$ $(b_{k,j},c_{k,j}\in\Bbb R)$
and we write
$$
\begin{cases}
{\bf a}=\ ^t(b_{1,0},b_{2,0},\cdots ,b_{m,0})\in \Bbb R^m
\\
B=\big(b_{k,j}\big)_{1\leq k\leq m,1\leq j\leq n},\
C=\big(c_{k,j}\big)_{1\leq k\leq m,0\leq j\leq n}.
\end{cases}
$$
It is easy to see that
the polynomials $f_0,\cdots ,f_n$ have no common
 {\it real} root beside zero if and only if
 the equation
$$
(z_0,z_1,\cdots ,z_m)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 &   \cdots & 0 & 0
\\
b_{1,0}&c_{1,0} & b_{1,1} & c_{1,1} &   \cdots & b_{1,n} & c_{1,n}
\\
\vdots &\vdots & \vdots  & \vdots  &    \vdots & \vdots  & \vdots
\\
b_{m,0}&c_{m,0} & b_{m,1} & c_{m,1} &   \cdots & b_{m,n} & c_{m,n}
\end{array}
\right)=
\left(
\begin{array}{c}
0
\\
0
\\
\vdots
\\
0
\end{array}
\right)
$$
has no  non-zero solution.
So we see that
$(f_0,\cdots ,f_n)\in A_1^{\Bbb C}(m,n)$ if and only if
the $\big(m\times (2n+1)\big)$-matrix $\big(B, C)$ has rank $m$.
Thus the map
$\Phi : V_{2n+1,m}\times \Bbb R^m \to A_1^{\Bbb C}(m,n)$
given by $( B,C,{\bf a})\mapsto (f_0,\cdots ,f_n)$
is clearly a homeomorphism, where $(f_0,\cdots ,f_n)$ is defined by
(\ref{matrix}).
Since $\varphi_{m,n}=\Phi \vert  V_{2n+1,m}\times \{{\bf 0}_m\},$
the map
$\varphi_{m,n}$ is a homotopy equivalence.
\end{proof}

\begin{lemma}\label{lemma: 1-connected}
$\mbox{{\rm (i)}}$
If $2\leq m \leq 2n$,  $\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ is $(2n-m)$-connected.
\par
$\mbox{{\rm (ii)}}$
If $1\leq m\leq 2n$, $V_{2n+1,m}$ is $(2n-m)$-connected.
\par
$\mbox{{\rm (iii)}}$
If $1\leq m \leq 2n$,
$\pi_{2n-m+1}(V_{2n+1,m})=
\begin{cases}
\Bbb Z & \mbox{if }\ m\equiv 1\  \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z /2 & \mbox{if }\ m\equiv 0\  \mbox{{\rm (mod $2$)}} .
\end{cases}$
\end{lemma}
\begin{proof}
(i)
Consider the restriction fibration sequence
\begin{equation}\label{res-fib}
\Omega^m\Bbb C {\rm P}^n \stackrel{\hat{j}}{\longrightarrow}
 \mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)\stackrel{r}{\longrightarrow}\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n).
\end{equation}
We prove the assertion (i) by induction on $m$.
First, consider the case $m=2$.
Since $\pi_1(\Omega^2\Bbb C {\rm P}^n)=0$ and $\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^1,\Bbb C {\rm P}^n)=\Omega\Bbb C {\rm P}^n$,
taking $m=2$  in (\ref{res-fib}) and using the induced exact sequence
$$
0\to
\pi_1(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))\stackrel{r_*}{\longrightarrow}
\pi_1(\Omega\Bbb C {\rm P}^n)=\Bbb Z\stackrel{\partial}{\rightarrow}
\Bbb Z=\pi_0(\Omega^2\Bbb C {\rm P}^n)\to 0,
$$
we see that $\pi_1(\mbox{{\rm Map}}^*_1(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))=0$.
Since $\pi_k(\Omega^2\Bbb C {\rm P}^n)\cong\pi_{k+2}(S^{2n+1})=0$
and $\pi_k(\Omega\Bbb C {\rm P}^n)\cong\pi_{k+1}(S^{2n+1})=0$ for $2\leq k\leq 2n-2$,
applying (\ref{res-fib}) with $m=2$
we see that $\pi_k(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))=0$
for $2\leq k\leq 2n-2$.
Hence, the case $m=2$ has been proved.
Next,  assume that $\mbox{{\rm Map}}^*_1(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)$ is
$(2n-m+1)$-connected for some $m\geq 3$.
Since $3\leq m <2n$, $\Omega^m\Bbb C {\rm P}^n=\Omega^mS^{2n+1}$ is
$(2n-m)$-connected.
Hence, from (\ref{res-fib})
we easily deduce that $\mbox{{\rm Map}}^*_1(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ is $(2n-m)$-connected.
We have proved (i).
\par
(ii)
The assertion (ii) can be easily
proved by induction on $m$ by making use of the following
fibration sequence:
\begin{equation}\label{fibV}
S^{2n-m+1} \to V_{2n+1,m} \to V_{2n+1,m-1}.
\end{equation}
We omit the details.
\par
(iii)
First, let $m\equiv 0$ $\mbox{{\rm (mod $2$)}}$.
It is known that $H^{2n-m+2}(V_{2n+1,m},\Bbb Z/p)=0$ for any
odd prime $p\geq 3$,
$$
H^*(V_{2n+1,m},\Bbb Z/2)=E[x_j:2n-m+1\leq j\leq 2n]
\quad (\vert x_j\vert =j)
$$
and that $Sq^1(x_{2n-m+1})=x_{2n-m+2}$.
Hence, the $(2n-m+2)$-skeleton of
$V_{2n+1,m}$ is $S^{2n-m+1}\cup_2e^{2n-m+2}$
(up to homotopy equivalence), and we have
$\pi_{2n-m+1}(V_{2n-m+1})= \Bbb Z/2.$
If $m\equiv 1$ $\mbox{{\rm (mod $2$)}}$, (\ref{fibV})
induces the  exact sequence
$$
\pi_{N+1}(V_{2n+1,m-1})=\Bbb Z/2\stackrel{\partial}{\rightarrow}
\pi_{N}(S^{2n-m+1})=\Bbb Z\to
\pi_{N}(V_{2n+1,m}) \to 0,
$$
where $N=2n-m+1$.
Hence, $\pi_{2n-m+1}(V_{2n+1,m})\cong\Bbb Z$.
Thus (iii) has also been proved.
\end{proof}

\begin{cor}\label{cor: pi1}
If $1\leq m \leq 2n$,
$\mbox{{\rm Alg}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$ is $(2n-m)$-connected and
$$
\pi_{2n-m+1}(\mbox{{\rm Alg}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n))=
\begin{cases}
\Bbb Z & \mbox{if }\ m\equiv 1\  \mbox{{\rm (mod $2$)}} ,
\\
\Bbb Z /2 & \mbox{if }\ m\equiv 0\  \mbox{{\rm (mod $2$)}} .
\end{cases}
$$
\end{cor}
\begin{proof}
Since there is a homotopy equivalence
$V_{2n+1,m}\simeq \mbox{{\rm Alg}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$, the assertion
follows from Lemma \ref{lemma: 1-connected}.
\end{proof}

\begin{defi}
{\rm
For $1\leq m\leq 2n$,
define the map $\tilde{i}_m:V_{2n+1,m}\to \mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
by
$\tilde{i}_m=i_1^{\Bbb C}\circ \varphi_{m,n}.$
}
\end{defi}

For $2\leq m\leq 2n$, it is easy to verify that the following diagram is commutative
\begin{equation}\label{fibCD}
\begin{CD}
S^{2n-m+1} @>\tilde{j}>> V_{2n+1,m} @>>> V_{2n+1,m-1}
\\
@V{\hat{s}_m}VV @V{\tilde{i}_m}VV @V{\tilde{i}_{m-1}}VV
\\
\Omega^m \Bbb C {\rm P}^n @>{\hat{j}}>>
\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n) @>r>>
\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^{m-1},\Bbb C {\rm P}^n)
\end{CD}
\end{equation}
where we identify
$S^{2n-m+1}\cong \frac{O(2n+2-m)}{O(2n+1-m)}$ and the two rows are
fibration sequences.

\begin{lemma}\label{lemma: s}
If $2\leq m\leq 2n$,
the map $\hat{s}_m:S^{2n-m+1}\to \Omega^m\Bbb C {\rm P}^n$ is a homotopy equivalence
up to dimension $D_{\Bbb C}(1;m,n)=4n-2m+1$.
\end{lemma}
\begin{proof}
By means of a method similar to the one used in the proof of
[\cite{Y12}, Lemma 3.1]
we can show that
$\hat{s}_{m*}:\pi_{2n-m+1}(S^{2n-m+1})
\stackrel{\cong}{\rightarrow}\pi_{2n-m+1}(\Omega^mS^{2n+1})$
$\cong
\pi_{2n-2m+1}(\Omega^m\Bbb C {\rm P}^n)$
is an isomorphism.
Thus we can identify  $\hat{s}_m$ with the $m$-fold suspension
 $E^m :S^{2n-m+1}\to\Omega^mS^{2n+1}\simeq \Omega^m\Bbb C {\rm P}^n$
(up to homotopy equivalence).
Hence $\hat{s}_m$ is a homotopy equivalence up to dimension
$4n-2m+1$.
\end{proof}

\begin{lemma}\label{lemma: m1}
If $n\geq 2$,
$\tilde{i}_{1*}:\pi_k(V_{2n+1,1})\to \pi_k(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^1,\Bbb C {\rm P}^n))$
is an isomorphism for any $2\leq k <4n-1$ and an epimorphism
for $k=4n-1$.
\end{lemma}
\begin{proof}
After identifications (up to homotopy equivalence) $V_{2n+1,1}=S^{2n}$ and
$\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^1,\Bbb C {\rm P}^n)=\Omega \Bbb C {\rm P}^n= \Omega S^{2n+1}\times S^1$,
the map $\tilde{i}_1$ can be viewed as a map
$\tilde{i}_1:S^{2n}\to\Omega\Bbb C {\rm P}^n$.
Let $q_1 :\Omega\Bbb C {\rm P}^n=\Omega S^{2n+1}\times S^1\to \Omega S^{2n+1}$ denote
the projection onto the first factor.
Note that the composite
$q_1\circ \tilde{i}_1$ can be identified with the map
$S^{2n}\simeq Q^1_{(2n+1)}(\Bbb R)\to \Omega S^{2n+1}$
given in [\cite{KY1}, Corollary 5 (2)].
Hence it follows from [\cite{KY1}, Corollary 5] that $q_1\circ \tilde{i}_1$
 is a homotopy equivalence up to dimension
$N(1,2n+2)=4n-1$.
Recalling  that
 ${q_1}_*:\pi_l(\Omega \Bbb C {\rm P}^n)\stackrel{\cong}{\rightarrow}\pi_l(\Omega S^{2n+1})$
is an isomorphism for all $l\geq 2$,
we see that $\pi_k(\tilde{i}_1)$ is an isomorphism for any $2\leq k <4n-1$ and
an epimorphism for $k=4n-1$.
\end{proof}

\begin{prop}\label{prop: V}
If $2\leq m\leq 2n$, the map
$\tilde{i}_m:V_{2n+1,m}\to \mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is a homotopy equivalence up to dimension $D_{\Bbb C}(1;m,n)=4n-2m+1$
\end{prop}
\begin{proof}
The proof proceeds by  induction on $m$.
First, consider the case $m=2$.
Since
$\pi_1(V_{2n+1,2})=\pi_1(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))=0$
by Lemma \ref{lemma: 1-connected},  the map
$\tilde{i}_2$ induces an isomorphism on $\pi_1(\ )$.
Hence it suffices to show that
$\tilde{i}_{2*}:\pi_k(V_{2n+1,2})\to \pi_k(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^2,\Bbb C {\rm P}^n))$
is an isomorphism for any $2\leq k<D_{\Bbb C}(1;2,n)=4n-1$ and
an epimorphism for $k=D_{\Bbb C}(1;2,n)$.
However, recalling the commutative diagram (\ref{fibCD}) for $m=2$,
and using the Five Lemma,
 Lemma \ref{lemma: s} and Lemma \ref{lemma: m1}, we see that
 $\pi_k(\tilde{i}_2)$ is an isomorphism for any $2\leq k<4n-1=D_{\Bbb C}(1;2,n)$ and
 an epimorphism for $k=4n-1=D_{\Bbb C}(1;2,n)$.
Hence, the case $m=2$ is proved.
\par
Now assume that the map $\tilde{i}_{m-1}$ is a homotopy equivalence up to
dimension $D_{\Bbb C}(1;m-1,n)=4n-2m+3$ for some $m\geq 3$.
Since, by Lemma \ref{lemma: s}, $\hat{s}_m$ is a homotopy equivalence up to dimension $D_{\Bbb C}(1;m,n)=4n-2m+1$, the Five Lemma and the diagram (\ref{fibCD}) imply that
the map $\tilde{i}_m$ is a homotopy equivalence up to dimension $D_{\Bbb C}(1;m,n)$.
\end{proof}

\par
{\it Proof of Theorem \ref{thm: V}. }
(i)
Since $i_1^{\Bbb C}=i_{1,\Bbb C}\circ \Psi_1^{\Bbb C}$ and the projection map
$\Psi_1^{\Bbb C}:A_1^{\Bbb C}(m,n)
\stackrel{\cong}{\rightarrow} \mbox{{\rm Alg}}_1^*(\Bbb R\mbox{{\rm P}}^m,\Bbb C {\rm P}^n)$
is a homeomorphism, it suffices to show that
$i_1^{\Bbb C}$ is a homotopy equivalence up to dimension $D_{\Bbb C}(1;m,n).$
As $\tilde{i}_m=i^{\Bbb C}_1\circ \varphi_{m,n}$ and
$\varphi_{m,n}$ is a homotopy equivalence,
this follows from Proposition \ref{prop: V}.
Hence, (i) has been proved.
\par
(ii)
Since
$\hat{s}_{2n*}:\pi_1(S^1)\to \pi_1(\Omega^{2n}\Bbb C {\rm P}^n)$ is an
epimorphism by Lemma \ref{lemma: s}
and $\pi_1(S^1)\cong\Bbb Z\cong\pi_1(\Omega\Bbb C {\rm P}^n)$,
$\hat{s}_{2n}$ induces an isomorphism on $\pi_1(\ )$.
Consider the following commutative diagram
of the exact sequences induced from
(\ref{fibCD}) for $m=2n$
$$
\begin{CD}
\pi_2(V_{2n+1,2n-1}) @>\partial>>\pi_1(S^1) @>>> \pi_1(V_{2n+1,2n})=\Bbb Z/2@>>> 0
\\
@V{\tilde{i}_{2n-1*}}V{\cong}V @V{\hat{s}_{2n*}}V{\cong}V
@V{\tilde{i}_{2n*}}VV @.
\\
\pi_2(\mbox{{\rm Map}}_1^*) @>\partial^{\prime}>> \pi_1(\Omega\Bbb C {\rm P}^n) @>>> \pi_1(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^{2n},\Bbb C {\rm P}^n)) @>>> 0
\end{CD}
$$
where $\mbox{{\rm Map}}_1^*=\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^{2n-1},\Bbb C {\rm P}^n)$.
Since, by Proposition \ref{prop: V}, the induced homomorphism
$\tilde{i}_{2n-1*}:\pi_2(V_{2n+1,2n-1})
\stackrel{\cong}{\rightarrow}\pi_2(\mbox{{\rm Map}}_1^*(\Bbb R\mbox{{\rm P}}^{2n-1},\Bbb C {\rm P}^n))$
is an isomorphism, $\tilde{i}_{2n}$ induces an isomorphism on $\pi_1(\ )$.
\qed

\begin{proof}[Proof of Corollary \ref{cor: V}]
The assertion (i)  follows from Lemma \ref{lemma: simply}
and Theorem \ref{thm: II}.
The assertion (ii) also easily follows from Corollary \ref{cor: M0-homotopy},
Lemma \ref{lemma: V}, Lemma \ref{lemma: 1-connected},
Proposition \ref{prop: V}
and Theorem \ref{thm: V}.
\end{proof}

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


Land Arts of the American West is featured on the cover of the March edition of Catalyst Magazine. Adele Ardent wrote this insightful article reflecting upon her experiences with LAAW in 2017.
Sevilleta LTER REU in High Country News !!!


When it comes to hot sauces, Julio's chipotle is one of my favorites, especially for tacos. In my search for local Texas products, I found this at HEB and saw it was from Del Rio, TX (which is about 4 hours from Austin on the border). When I found out I needed to go to Del Rio for work, I was excited that I would be able to visit Julio's and also dine at their restaurant.
You can definitely tell you're coming to a Mexican restaurant by the outside of the building.
The restaurant was setup like a fast casual restaurant. Since their special of the day was the red taco plate and I know Julio's chipotle sauce goes great on tacos, that's what I ordered. I got my number and waited at the table.
In the meantime, they brought out some chips in salsa while I was waiting. Just like their hot sauce, the salsa with the chips was great! They also make their own tortilla chips. For me the chips are little too seasoned eaten plain, but having them with salsa there did work out well since it tempered some of the seasoning.
Eventually my tacos came out and they brought me a bottle of chipotle hot sauce. The food didn't disappoint and was great!
While I was waiting for my food and after I ate, I did a little walk around of the restaurant. It has a very authentic feel to it.
Between the food being excellent and it having an authentic atmosphere, I do think they do a great job overall. I was happy that my search for local products ultimately led me to this place. I know if I'm ever passing through Del Rio again, this is on my list of places to stop.
For good measure, I bought some of their salsa and also a bottle of habanero hot sauce. I can definitely say the habanero hot sauce packs a lot more kick than the chipotle sauce does!


Usually a few seconds after the game starts, Arnold will say "don't move" and a 2nd-chance ball is shot into play despite the fact that the original ball is still in play. There doesn't seem to be a specific switch that initiates this. I see nothing wrong in the switch test, and I have reseated 3 chips on the cpu board.
Does the game ever kill the flippers and start the end of ball sequence unexpectedly while a ball is still in play?
Yeah, the outhole switch is spongy. I wedged a small ball of paper to desensitize it, and the game seems to be playing better. Will replace the switch I assume. Thanks again, Pinside.


Public Health and School of Medicine, writes that "DSHEA modifies the regulatory environment so that it becomes possible, even likely, that products will be marketed that inadvertently harm people" (Zeisel, p. 1855). Zeisel believes that the DSHEA legislation makes it easy for small enterprises to market products without investing the time and money needed to prove their product's safety and efficacy. He contrasts the development of a new dietary supplement to that of a new drug or food additive, for which there is a formal process to evaluate safety.
A manufacturer developing a new drug or food additive must conduct safety studies following FDA procedures. Results must be submitted to the FDA for review and approval before the ingredient or drug can be sold to the public. This is not the case for dietary supplements, however, because under DSHEA they are legally in a class by themselves. The FDA must simply be notified of the new product, and the notification must provide information that supports the manufacturer's claim that its product is safe. Once the product is marketed, the FDA is responsible for proving that a dietary supplement is unsafe before it can take action to restrict that product's use or remove it from the marketplace.
Another issue critics of DSHEA cite is the scant quality control of dietary supplements. Quality control is important to assure consumers that a product contains the ingredients stated on the label in the stated amounts. Neither the FDA nor any other federal or state agency routinely tests dietary supplements for quality prior to sale. But some manufacturers of dietary supplements do adhere to Good Manufacturing Practices (GMPs) and make every effort to produce a quality product. Also, the FDA has assisted the industry by proposing GMPs that focus on ensuring the identity, purity, quality, strength, and composition of dietary supplements.
DSHEA supporters fear that increased regulation of dietary supplements will decrease access to beneficial products. National opinion surveys show that many supplement users feel so strongly about the potential health benefits of supplements that they would continue to use them even if the supplements were shown to be ineffective in clinical studies. Consumers value freedom of choice, and many view regulation as an attempt by the government and medical establishment to monopolize treatment options. Clearly, a balance needs to be reached between preserving freedom of choice and ensuring that dietary supplements are safe and effective. see also Alternative Medicines and Therapies; Food Labels; Health Claims; Quackery; Vitamins, Fat-Soluble; Vitamins, Water-Soluble.
Blendon, R. J.; DesRoches, C. M.; Benson, J. M.; Brodie, M.; and Altman, D. E. (2001). "American's Views on the Use and Regulation of Dietary Supplements." Archives of Internal Medicine 161(6):805-10.
Dwyer, Johanna T.; Garceau, Anne O.; Evans, Marguerite; Li, Donglin; Lytle, Leslie; Hoelscher, Deanna; Nicklas, Theresa A.; and Zive, Michelle (2001). "Do Adolescent Vitamin-Mineral Supplement Users Have Better Nutrient Intakes than Nonusers? Observations from the CATCH Tracking Study." Journal of the American Dietetic Association 101(11):1340-1346.
Fleming, Thomas, ed. (1998). PDR for Herbal Medicines. Montvale, NJ: Medical Economics.
Sarubin, Allison (2000). The Health Professional's Guide to Popular Dietary Supplements. Chicago: American Dietetic Association.
Skidmore-Roth, Linda (2001). Mosby's Handbook of Herbs & Natural Supplements. St. Louis, MO: Mosby.
Somer, Elizabeth (1996). The Essential Guide to Vitamins and Minerals, 2nd edition. New York: HarperPerennial.


text:
Lifestyle / TV
Dame Barbara Windsor asks Boris Johnson: Can I have a kiss?
September 2, 2019, 1:35 pm Updated: September 2, 2019, 3:35 pm
Dame Barbara Windsor meeting Prime Minister Boris Johnson after she delivered an Alzheimer's Society open letter to 10 Downing Street in Westminster, London, calling on him to address the 'devastating state' of dementia care. (Simon Dawson/PA)
Dame Barbara Windsor asked Prime Minister Boris Johnson for a kiss during a meeting to discuss dementia.
The Carry On and EastEnders actress was diagnosed with Alzheimer's in 2014, and today delivered a letter to Downing Street pleading for better care for fellow sufferers.
Dame Barbara attended Number 10 with her husband Scott Mitchell, where she was warmly received by an affectionate Prime Minister.
Mr Johnson held the ailing actress' hand as they sat in the garden, discussing the limitations of dementia care in the UK.
At the end of their afternoon chat, Dame Barbara turned to the Conservative leader and asked: "Can I have a kiss?"
The Prime Minister responded "Of course you can," and pecked the cheeks of the star, famed for portraying Peggy Mitchell.
Dame Barbara delivered a letter signed by 100,000 people calling for better care for dementia sufferers.
Mr Johnson assured the actress: "We'll do this. It's very hard. We have to sort it out.
Dame Barbara Windsor and her husband Scott Mitchell (Dominic Lipinski/PA)
"I'm going to do my best for you. It's a big old job."
Mr Johnson reminisced with Dame Barbara about his time as Mayor Of London.
When the actress claimed that "they all loved you", the politician joked that "it was variable".
The Prime Minister grew concerned at one point during the meeting, when he asked the actress, who was wrapped in a shawl: "Are you cold? What shall we do?"
Dame Barbara was joined by representatives from the Alzheimer's Society, who put the case to Mr Johnson for more funding in dementia care.
Dame Barbara Windsor and Prime Minister Boris Johnson holding hands (Simon Dawson/PA)
The letter taken to Number 10 stated: "Our experience is of a care system that too often doesn't care – one that is completely inadequate, unfair, unsustainable and in dire need of more money.
"Across the country dementia care is difficult, an in some places even impossible, to access."
The Prime Minister nodded in agreement with the Alzheimer's Society representatives regarding issues facing care, and gave Dame Barbara assurances he would look into the issue.
The actress was all smiles as she was aided up Downing Street before the meeting, greeting the on-duty police officer and laughing as she knocked on the door of Number 10.
Peggy Mitchell

redpajama_set_name: RedPajamaCommonCrawl


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We are delighted to announce two new members of our Scientific Advisory Board: Professor Richard Edmondson, Clinical Professor in Gynaecological Oncology at the University of Manchester, and Professor Maurizio D'Incalci, Chief of the Laboratory of Cancer Chemotherapy and Chief of the Department of Oncology at the Mario Negri Institute of Milan.
Funding research of the highest quality is imperative to us – research projects that are novel, led by outstanding scientists or clinicians and with the greatest likelihood of finding new treatments or improving existing treatments for ovarian cancer. The Scientific Advisory Board, chaired by Professor Ruth Plummer, is a panel of scientific and clinical experts who scrutinise each research application and recommend to our Board of Trustees which ones should be funded. So far we have funded over £1 million of research and the Scientific Advisory Board, together with vital input from our Research Advocates, makes sure that the generous donations we receive are used to give women with ovarian cancer the best chances of survival.
Professor D'Incalci brings with him a wealth of knowledge and expertise spanning decades. He has been chief of the Laboratory of Cancer Chemotherapy at the Mario Negri Institute since 1986 and became chief of the Department of Oncology in 1996.
Professor Edmondson was awarded a Target Ovarian Cancer research grant in autumn last year for his project on translating our knowledge of the DNA damage response into clinical benefits for patients with ovarian cancer. We are thrilled that in 2018 he is able to bring his expertise in ovarian cancer to the Scientific Advisory Board to help us fund the groundbreaking research projects of the future.


Grout is a thin mortar used for filling joints between tiles, which also adds strength to the tilework. Grout tile joints are necessary due to the natural variations between tile pieces. The quality of grout is dependent on its fade resistance and stain resistance qualities.
A tile pattern can be accentuated by choosing a grout color that is darker, adding contrast like a medium grey for white subway tile.
Some grout is already sealed while other products recommend sealing after installation. Customers concerned about stains or fading should seal their grout or buy a premium grout product.


text:
Anti-Government rebels of Africa's Ivory Coast military overwhelm loyalists
As Ethiopia's army declares daily victories, its people are being plunged into violence
test post news for africa
On Wednesday, a military confrontation erupted in Bouake, the second city of the Ivory Coast. Members of a mutinous faction within the army lay siege to the camp of an elite Ivory Coast military unit called the Coordination Center for Operational Decisions (CCDO) and seized their weapons and supplies after the CCDO soldiers fled.
"I'm hearing very intense shooting from machine-guns,"said Georges Kouame, a resident of Bouake. "There are also explosions from heavy weapons."
Rival army factions fought in the base on the outskirts of the city. Last year, Army mutinies in Bouake kicked off a confrontation which persisted for several months. Last week, members of the CCDO were deployed to Bouake, sparking tension with those former mutineers.
"We entered the CCDO camp around midnight and took all of their arms and ammunitions," one of the mutineer faction's leaders told Reuters. "Their troops fled the camp, but we are looking for them."
Mutineer leaders told Reuters that they suspected the CCDO deployment in Bouake was a cover to spy on them.
In January 2017, soldiers dissatisfied with their pay and working conditions seized control of Bouake. The revolt quickly spread to major urban areas, and by May the government had capitulated to the soldiers'demands for $8,000 bonuses, a recompense for their service in two civil wars.
The catch for the government is that the mutineers aremostly members of the rebel group that was integrated into the military following the civil war. The rebels, with assistance from the French Military, helped to bring the current president to power.
"We have to stop this. I've asked the soldiers to keep their heads. No matter what their demands are, they can't launch such brutal protests," Bouake's mayor Nicolas Djibo said to the state broadcaster RTI.
The Ivory Coast had two successive civil wars (2002-2007, 2010-2011), in which successive rebellions based in Bouake dividedthe country into North-South and Muslim-Christian lines. President Alassane Ouattara was elected in 2010 and gained office in 2011 after the rebels moved south and French forces arrested his predecessor.
Downtown Bouaké (Urbaplan.ch)
Since fighting ended in 2011, the Ivory Coast's economy has gone vertical, growing at thefastest rate of any African country; 8.5%. Ouattara's administration has worked aggressively to integrate the Ivory Coast into the global markets, and 40-45% of capital invested in businesses comes from outside sources.
However, income inequality has outpaced growth, with a 45% jump in the number of millionaires in the last decade, and with an 80% rise in the number of millionaires projected for the next decade. The growth is uneven thought the country, with the port city of Abidjan receiving its bulk. This has enriched the Christian south, while leaving a Muslim north, with a history of successful military ventures into the south, relatively impoverished.
Abidjan, Côte d'Ivoire. Photo: UN Photo/Basile Zoma
The country has tried and failed to reduce the size of its military after the civil wars, with demobilizing soldiers being a frequent source of violence. In December, the state offered its soldiers a $26,000 incentive to retire from service.
Title Photo: Soldiers in the streets of Bouaké. Via beninwebtv.com
LIMA CHARLIE NEWS, with Diego Lynch
Lima Charlie provides global news, insight & analysis by military veterans and service members Worldwide.
For up-to-date news, please follow us on twitter at @LimaCharlieNews
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