# Spaces of non-resultant systems of bounded multiplicity with real
coefficients

Andrzej Kozlowski111Institute of Applied Mathematics and Mechanics, University
of Warsaw, Banacha 2, 02-097 Warsaw, Poland (E-mail<EMAIL_ADDRESS> and
Kohhei Yamaguchi222Department of Mathematics, University of Electro-
Communications, Chofu, Tokyo 182-8585, Japan (E-mail<EMAIL_ADDRESS>
2010 Mathematics Subject Classification. Primary 55P15; Secondly 55R80, 55P35.

###### Abstract

For each pair $(m,n)$ of positive integers with $(m,n)\not=(1,1)$ and an
arbitrary field $\mathbb{F}$ with algebraic closure $\overline{\mathbb{F}}$,
let $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})$ denote the space of $m$-tuples
$(f_{1}(z),\cdots,f_{m}(z))\in\mathbb{F}[z]^{m}$ of $\mathbb{F}$-coefficients
monic polynomials of the same degree $d$ such that the polynomials
$\\{f_{k}(z)\\}_{k=1}^{m}$ have no common root in $\overline{\mathbb{F}}$ of
multiplicity $\geq n$. These spaces $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})$
were first defined and studied by B. Farb and J. Wolfson as generalizations of
spaces first studied by Arnold, Vassiliev and Segal and others in several
different contexts. In previous we determined explicitly the homotopy type of
this space in the case $\mathbb{F}=\mathbb{C}$. In this paper, we investigate
the case $\mathbb{F}=\mathbb{R}$.

##  1 Introduction

### The motivation.

The motivation of this paper comes from the works of V. I. Arnold [2], V.
Vassiliev [29], G. Segal [25], and B. Farb and J. Wolfson [7].

Arnold considered the space $\mbox{{\rm SP}}^{d}_{n}(\mathbb{C})$ of complex
coefficients monic polynomials of degree $d$ without roots of multiplicity
$\geq n$, which plays a role in the theory of singularities. For example, if
$n=2$, this space is the same as the space of complex monic polynomials of
degree $d$ without repeated roots, whose fundamental group is the Braid group
$\mbox{Br}(d)$ of $d$-strings and whose homology is the homology of the braid
group $\mbox{Br}(d)$. Arnold computed the homology of these spaces and
established their homology stability. His results were much extended and
generalized by V. Vassiliev [29].

Analogous results were discovered by G. Segal [25] in a different context
inspired by control theory. Segal considered the space $\mbox{{\rm
Hol}}^{*}_{d}(S^{2},\mathbb{C}{\rm P}^{m-1})$ of bases holomorphic maps of
degree $d$ from the Riemann sphere $S^{2}$ to the $(m-1)$-dimensional complex
projective space $\mathbb{C}{\rm P}^{m-1}$, and its inclusion into the space
$\mbox{{\rm Map}}^{*}_{d}(S^{2},\mathbb{C}{\rm
P}^{m-1})=\Omega^{2}_{d}\mathbb{C}{\rm P}^{m-1}$ of corresponding space of
based continuous maps. Intuitive considerations based on Morse theory suggest
that the homotopy type of the first space should approximate that of the
second space more and more closely as the degree $d$ increases. Segal proved
that this is true by observing that the space $\mbox{{\rm
Hol}}^{*}_{d}(S^{2},\mathbb{C}{\rm P}^{m-1})$ can be identified with the space
of $m$-tuples $(f_{1}(z),\cdots,f_{m}(z))\in\mathbb{C}[z]^{m}$ of monic
polynomials of the same degree $d$ without common roots. He defined a
stabilization map $\mbox{{\rm Hol}}_{d}^{*}(S^{2},\mathbb{C}{\rm
P}^{m-1})\to\mbox{{\rm Hol}}_{d+1}^{*}(S^{2},\mathbb{C}{\rm P}^{m-1})$ and
proved that the induced maps on homotopy groups are isomorphisms up to some
dimension increasing with $d$. Using a different technique he also proved that
there is a homotopy equivalence $S:\varinjlim\mbox{{\rm
Hol}}_{d}^{*}(S^{2},\mathbb{C}{\rm
P}^{m-1})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega^{2}_{0}\mathbb{C}{\rm
P}^{m-1}$ defined by a “scanning of particles”, and that this equivalence is
homotopic to the inclusion of the space of all holomorphic maps into the space
of all continuous maps.

Inspired by the classical theory of resultants and the algebraic nature of
Arnold’s and Segal’s arguments, B. Farb and J. Wolfson ([7], [8]) defined
algebraic varieties $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})$, which
generalize the spaces considered by Arnold and Segal. These varieties
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})$ are defined as follows.

For a field $\mathbb{F}$ with its algebraic closure $\bar{\mathbb{F}}$,
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})$ is the space of of $m$-tuples
$(f_{1}(z),\cdots,f_{m}(z))\in\mathbb{F}[z]^{m}$ of monic
$\mathbb{F}$-coefficients polynomials of the same degree $d$ with no common
root in $\overline{\mathbb{F}}$ of multiplicity $\geq n$. For example, if
$\mathbb{F}=\mathbb{C}$, $\mbox{{\rm Poly}}^{d,1}_{n}(\mathbb{C})=\mbox{{\rm
SP}}^{d}_{n}(\mathbb{C})$ and $\mbox{{\rm Poly}}^{d,m}_{1}(\mathbb{C})$ can be
identified with the space $\mbox{{\rm Hol}}_{d}^{*}(S^{2},\mathbb{C}{\rm
P}^{m-1})$. Note that the space $\text{Poly}_{n}^{d,m}(\mathbb{C})$ is an
affine variety defined by systems of polynomial equations with integer
coefficients by the classical theory of resultants. Thus this variety can be
defined over $\mathbb{Z}$ and over any filed $\mathbb{F}$.

Farb and Wolfson computed various algebraic and geometric invariants of these
varieties (such as the number of points for a finite field $\mathbb{F}_{q}$,
étale cohomology etc.) and found for the varieties $\mbox{{\rm
Poly}}^{dn,m}_{n}(\mathbb{F})$ and $\mbox{{\rm Poly}}^{d,mn}_{1}(\mathbb{F})$
that they were equal. They conjectured that these varieties were algebraically
isomorphic. Although this conjecture was disproved in [27], analogous results
(e.g. (2.14), (2.20)) hold in the homotopy category. So this analogy between
the algebraic and the topological situation remains intriguing.

From the topological point of view, there has been a lot of work on the
homotopy type of $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})$ for
$\mathbb{F}=\mathbb{C}$. The space $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{C})$
has been studied by V. Vassiliev [29], G. Segal [25], M. Guest and the present
authors [11], and the present authors [18].

In this article we study the homotopy type of the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{F})$ for $\mathbb{F}=\mathbb{R}$. Note that the space
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ has already been considered in the
case $n=1$ ([25], [15]).

The main purpose of this paper is to investigate the homotopy type of the
space $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ for the case $mn\geq 3$. In
particular, we prove that an Atiyah-Jones-Segal type result holds for this
space and determine its stable homotopy type explicitly. More precisely, our
main results can be summarized as follows (see Theorems 2.7, 2.11 and
Corollary 2.12 for further details).

Theorem (Theorems 2.7, 2.11 and Corollary 2.12).

  1. (i)

The natural map

$i^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}\simeq\Omega^{2}S^{2mn-1}\times\Omega S^{mn-1}$

is a homotopy equivalence through dimension $D(d;m,n)$ if $mn\geq 4$, and a
homology equivalence through dimension $D(d;m,n)$ if $mn=3$.

Here, $\lfloor x\rfloor$ denotes the integer part of a real number $x$ and the
number $D(d;m,n)$ is given by $D(d;m,n)=(mn-2)(\lfloor d/n\rfloor+1)-1.$

  2. (ii)

If $mn\geq 3$, there is a stable homotopy equivalence

$\displaystyle\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$
$\displaystyle\simeq_{s}\mbox{{\rm Poly}}^{\lfloor
d/2\rfloor,m}_{n}(\mathbb{C})\vee B^{d,m}_{n}\vee\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})$
$\displaystyle\simeq_{s}\Big{(}\bigvee_{i=1}^{\lfloor
d/n\rfloor}S^{(mn-2)i}\Big{)}\vee\Big{(}\bigvee_{i\geq 0,j\geq
1,i+2j\leq\lfloor d/n\rfloor}\Sigma^{(mn-2)(i+2j)}D_{j}\Big{)},$

where $B^{d,m}_{n}$ and $D_{j}$ denote the spaces defined by (2.19) and
$($4.6$)$, respectively.

##  2 Basic definitions and the main results

Before describing the main results of this paper precisely, we shall recall
several definitions and facts needed to give the precise statements of the
main results.

### Basic definitions and notations.

For connected spaces $X$ and $Y,$ let $\mbox{{\rm Map}}(X,Y)$ (resp.
$\mbox{{\rm Map}}^{*}(X,Y)$) denote the space consisting of all continuous
maps (resp. base-point preserving continuous maps) from $X$ to $Y$ with the
compact-open topology. For each element $D\in\pi_{0}(\mbox{{\rm
Map}}^{*}(X,Y))$, let $\mbox{{\rm Map}}^{*}_{D}(X,Y)$ denote the path-
component of $\mbox{{\rm Map}}^{*}(X,Y)$ which corresponds to $D$. When $X$
and $Y$ are complex manifolds, let $\mbox{{\rm
Hol}}_{D}^{*}(X,Y)\subset\mbox{{\rm Map}}_{D}^{*}(X,Y)$ denote the subspace of
all based holomorphic maps from $X$ to $Y$.

Let $\mathbb{R}{\rm P}^{N}$ (resp. $\mathbb{C}{\rm P}^{N}$) denote the
$N$-dimensional real projective space (resp. $N$-dimensional complex
projective space). Note that the based loop space $\mbox{{\rm
Map}}^{*}(S^{1},\mathbb{R}{\rm P}^{N})=\Omega\mathbb{R}{\rm P}^{N}$ has two
path-components $\Omega_{\epsilon}\mathbb{R}{\rm P}^{N}$
($\epsilon\in\\{0,1\\}$) for $N\geq 2$, where the space
$\Omega_{0}\mathbb{R}{\rm P}^{N}$ is the path-component of null homotopic maps
and $\Omega_{1}\mathbb{R}{\rm P}^{N}$ is the path-component which contains the
natural inclusion of the bottom cell $S^{1}$ into $\mathbb{R}{\rm P}^{N}$.
Similarly, for each integer $d\in\mathbb{Z}=\pi_{0}(\mbox{{\rm
Map}}^{*}(S^{2},\mathbb{C}{\rm P}^{N}))$, let $\Omega^{2}_{d}\mathbb{C}{\rm
P}^{N}=\mbox{{\rm Map}}^{*}_{d}(S^{2},\mathbb{C}{\rm P}^{N})$ denote the path
component of $\Omega^{2}\mathbb{C}{\rm P}^{N}$ of based maps from $S^{2}$ to
$\mathbb{C}{\rm P}^{N}$ of degree $d$.

######  Definition 2.1.

Let $\mathbb{N}$ be the set of all positive integers. From now on, let
$d\in\mathbb{N}$, let $(m,n)\in\mathbb{N}^{2}$ be a pair of positive integers
such that $(m,n)\not=(1,1)$, and let $\mathbb{F}$ be a field with its
algebraic closure $\overline{\mathbb{F}}$.

(i) Let ${\rm P}_{d}(\mathbb{F})$ denote the space of all
$\mathbb{F}$-coefficients monic polynomials
$f(z)=z^{d}+a_{1}z^{d-1}+\cdots+a_{d-1}z+a_{d}\in\mathbb{F}[z]$ of degree $d$.
Note that there is a natural homeomorphism ${\rm
P}_{d}(\mathbb{F})\cong\mathbb{F}^{d}$ given by

(2.1) $f(z)=z^{d}+\sum_{k=1}^{d}a_{k}z^{d-k}\mapsto(a_{1},\cdots,a_{d}).$

(ii) For each $m$-tuple $D=(d_{1},\cdots,d_{m})\in\mathbb{N}^{m}$ of positive
integers, we denote by $\mbox{{\rm Poly}}^{D;m}_{n}(\mathbb{F})=\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{m};m}_{n}(\mathbb{F})$ the space consisting of all
$m$-tuples $(f_{1}(z),\cdots,f_{m}(z))\in{\rm P}_{d_{1}}(\mathbb{F})\times{\rm
P}_{d_{2}}(\mathbb{F})\times\cdots\times{\rm P}_{d_{m}}(\mathbb{F})$ of monic
polynomials such that the polynomials $\\{f_{j}(z)\\}_{j=1}^{m}$ have no
common root in $\overline{\mathbb{F}}$ of multiplicity $\geq n$. We call the
space $\mbox{{\rm Poly}}^{D;m}_{n}(\mathbb{F})$ as the space of non-resultant
system of bounded multiplicity $n$ with coefficients in $\mathbb{F}$.333Recall
that the classical resultant of a systems of polynomials vanishes if and only
if they have a common solution in an algebraically closed field containing the
coefficients. Systems which have no common roots are called “non-resultant”.
This is the intuition behind our choice of the term “non-resultant system of
bounded multiplicity.”

In particular, when $D_{m}=(d,d,\cdots,d)\in\mathbb{N}^{m}$ ($m$-times), we
write

(2.2) $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{F})=\mbox{{\rm
Poly}}^{D_{m};m}_{n}(\mathbb{F})=\mbox{{\rm
Poly}}^{d,d,\cdots,d;m}_{n}(\mathbb{F}).$

######  Definition 2.2.

From now on, let us suppose that $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.

(i) Let $\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{K})$ denote the space of all
$m$-tuples $(f_{1}(z),\cdots,f_{m}(z))\in{\rm P}_{d}(\mathbb{K})^{m}$ of
$\mathbb{K}$-coefficients monic polynomials of the same degree $d$ such that
the polynomials $\\{f_{j}(z)\\}_{j=1}^{m}$ have no common real root of
multiplicity $\geq n$ (but may have complex common roots of any multiplicity).
Note that there are two inclusions $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\supset\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\subset\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R}).$ Thus
there are the following two inclusion maps

(2.3) $\begin{CD}\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})@<{\iota^{d,m}_{n,\mathbb{C}}}<{\supset}<\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})@>{\iota^{d,m}_{n,\mathbb{R}}}>{\subset}>\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R}).\end{CD}$

(ii) For a monic polynomial $f(z)\in{\rm P}_{d}(\mathbb{K})$, we define the
$n$-tuple $F_{n}(f)=F_{n}(f)(z)\in{\rm P}_{d}(\mathbb{K})^{n}$ of the monic
polynomials of the same degree $d$ by

(2.4)
$F_{n}(f)(z)=(f(z),f(z)+f^{\prime}(z),f(z)+f^{\prime\prime}(z),\cdots,f(z)+f^{(n-1)}(z)).$

Note that $f(z)\in{\rm P}_{d}(\mathbb{K})$ is not divisible by
$(z-\alpha)^{n}$ for some $\alpha\in\mathbb{K}$ if and only if
$F_{n}(f)(\alpha)\not={\bf 0}_{n}$, where we set ${\bf
0}_{n}=(0,0,\cdots,0)\in\mathbb{K}^{n}.$

(iii) When $\mathbb{K}=\mathbb{C}$, by identifying
$S^{2}=\mathbb{C}\cup\infty$ we define the natural map

$\displaystyle i^{d,m}_{n,\mathbb{C}}$ $\displaystyle:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\to\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1}\simeq\Omega^{2}S^{2mn-1}\quad\mbox{by}$ (2.5) $\displaystyle
i^{d,m}_{n,\mathbb{C}}(f)(\alpha)$
$\displaystyle=\begin{cases}[F_{n}(f_{1})(\alpha):F_{n}(f_{2})(\alpha):\cdots:F_{n}(f_{m})(\alpha)]&\mbox{if
}\alpha\in\mathbb{C}\\\ [1:1:\cdots:1]&\mbox{if }\alpha=\infty\end{cases}$

for $f=(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{C})$
and $\alpha\in\mathbb{C}\cup\infty=S^{2}$, where we choose the points $\infty$
and $*=[1:1:\cdots:1]$ as the base-points of $S^{2}$ and $\mathbb{C}{\rm
P}^{mn-1}$, respectively.

######  Definition 2.3.

Let $\mathbb{Z}_{2}=\\{\pm 1\\}$ denote the (multiplicative) cyclic group of
order $2$. From now on, we will regard the two spaces
$S^{2}=\mathbb{C}\cup\infty$ and $\mathbb{C}{\rm P}^{mn-1}$ as
$\mathbb{Z}_{2}$-spaces with actions induced by the complex conjugation on
$\mathbb{C}$.

(i) Let $(\Omega^{2}_{d}\mathbb{C}{\rm P}^{mn-1})^{\mathbb{Z}_{2}}$ denote the
space consisting of all $\mathbb{Z}_{2}$-equivariant based maps
$f:(S^{2},\infty)\to(\mathbb{C}{\rm P}^{mn-1},*)$.

(ii) Since $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})\subset\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})$ and $i^{d,m}_{n,\mathbb{C}}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}))\subset(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}$, we also define the natural map

$\displaystyle\quad i^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to(\Omega_{d}^{2}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}\qquad\mbox{by the restriction}$ (2.6)
$\displaystyle i^{d,m}_{n,\mathbb{R}}$
$\displaystyle=i^{d,m}_{n,\mathbb{C}}|\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}):\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}.$

(iii) When $mn\geq 3$, by identifying $S^{1}=\mathbb{R}\cup\infty$ we define a
natural map

$\displaystyle i^{d,m}_{n}:$ $\displaystyle\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\to\Omega_{[d]_{2}}\mathbb{R}{\rm
P}^{mn-1}\simeq\Omega S^{mn-1}\qquad\mbox{by}$ (2.7) $\displaystyle
i^{d,m}_{n}(f)(\alpha)$
$\displaystyle=\begin{cases}[F_{n}(f_{1})(\alpha):F_{n}(f_{2})(\alpha):\cdots:F_{n}(f_{m})(\alpha)]&\mbox{if
}\alpha\in\mathbb{R}\\\ [1:1:\cdots:1]&\mbox{if }\alpha=\infty\end{cases}$

for $f=(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})$ and
$\alpha\in\mathbb{R}\cup\infty=S^{1}$, where $[d]_{2}\in\\{0,1\\}$ is the
integer $d$ mod $2$ and we choose the points $\infty$ and $*=[1:1:\cdots:1]$
as the base-points of $S^{1}=\mathbb{R}\cup\infty$ and $\mathbb{R}{\rm
P}^{mn-1}$, respectively.

(iv) For positive integer $n\geq 3$, we define the jet embedding

$\displaystyle j^{d}_{n}:\mbox{{\rm Poly}}^{d,1}_{n}(\mathbb{R})\to\mbox{{\rm
Poly}}^{d,n}_{1}(\mathbb{R})\qquad\mbox{ by }$ (2.8) $\displaystyle
j^{d}_{n}(f(z))$
$\displaystyle=(f(z),f(z)+f^{\prime}(z),f(z)+f^{\prime\prime}(z),\cdots,f(z)+f^{(n-1)}(z))$

for $f(z)\in\mbox{{\rm Poly}}^{d,1}_{n}(\mathbb{R})$.

######  Definition 2.4.

(i) Let $f:X\to Y$ be a based map between based spaces $X$ and $Y$. Then the
map $f$ is called a homotopy equivalence (resp. a homology equivalence)
through dimension $N$ if the induced homomorphism

$f_{*}:\pi_{k}(X)\to\pi_{k}(Y)\quad(\mbox{resp. }f_{*}:H_{k}(X;\mathbb{Z})\to
H_{k}(Y;\mathbb{Z}))$

is an isomorphism for any integer $k\leq N$.

(ii) Similarly, let $G$ be a group and $f:X\to Y$ be a $G$-equivariant based
map between $G$-spaces $X$ and $Y$. Then the map $f$ is called a
$G$-equivariant homotopy equivalence through dimension $N$ (resp. a
$G$-equivariant homology equivalence through dimension $N$) if the restriction
map

$f^{H}=f|X^{H}:X^{H}\to Y^{H}$

is a homotopy equivalence through dimension $N$ (resp. a homology equivalence
through dimension $N$) for any subgroup $H\subset G$, where $W^{H}$ denotes
the $H$-fixed subspace of a $G$-space $W$ given by

(2.9) $W^{H}=\\{x\in W:h\cdot x=x\mbox{ for any }h\in H\\}.$

(iii) Recall [12] that there is a following homotopy equivalence for $N\geq 2$
obtained by using the reduced product

(2.10) $\Omega S^{N+1}\simeq S^{N}\cup e^{2N}\cup e^{3N}\cup\cdots\cup
e^{(k-1)N}\cup e^{kN}\cup e^{(k+1)N}\cup\cdots$

We denote by $J_{k}(S^{N})$ the $kN$-skeleton of $\Omega S^{N+1}$, i.e.

(2.11) $J_{k}(S^{N})=S^{N}\cup e^{2N}\cup e^{3N}\cup\cdots\cup e^{(k-1)N}\cup
e^{kN},$

which is usually called the $k$-th stage James filtration of $\Omega S^{N+1}$.

### Some known results.

Remark that there are homeomorphisms

(2.12) $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{K})\cong\mathbb{K}^{dm}\ \
(\mathbb{K}=\mathbb{R},\ \mathbb{C}),\ \mbox{and}\quad\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\cong\mathbb{R}^{dm}\quad\mbox{if }d<n.$

Thus, these spaces are contractible if $d<n$. From now on, in this paper we
always assume that $d$ is the positive integer such that $d\geq n$.

Now recall the following known two results.

######  Theorem 2.5 ([18]).

Let $m,n\geq 1$ be positive integers such that $mn\geq 3$.

  1. (i)

The natural map

$i^{d,m}_{n,\mathbb{C}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\to\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1}\simeq\Omega^{2}S^{2mn-1}$

is a homotopy equivalence through dimension $D(d;m,n;\mathbb{C})$, where
$\lfloor x\rfloor$ denotes the integer part of a real number $x$ and the
positive integer $D(d;m,n;\mathbb{C})$ is defined by

(2.13) $D(d;m,n;\mathbb{C})=(2mn-3)(\lfloor d/n\rfloor+1)-1.$

  2. (ii)

There is a homotopy equivalence

(2.14) $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{C})\simeq\mbox{{\rm
Poly}}^{\lfloor d/n\rfloor,mn}_{1}(\mathbb{C}).$

  3. (iii)

There is a stable homotopy equivalence

(2.15) $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\simeq_{s}\bigvee_{j=1}^{\lfloor
d/n\rfloor}\Sigma^{2(mn-2)k}D_{j},$

where $\Sigma^{j}$ and
$D_{j}=D_{j}(S^{1})=F(\mathbb{C},j)_{+}\wedge_{S_{j}}(S^{1})^{\wedge j}$
denote the $j$-fold reduced suspension and the space defined by $($4.6$)$,
respectively. ∎

######  Theorem 2.6 ([20], [29], [31]).

  1. (i)

The natural map

$i^{d,m}_{n}:\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\to\Omega_{[d]_{2}}\mathbb{R}{\rm
P}^{mn-1}\simeq\Omega S^{mn-1}$

is a homotopy equivalence through dimension $D(d;m,n)$ if $mn\geq 4$ and a
homology equivalence through dimension $D(d;m,n)$ if $mn=3$,444If $mn\geq 4$,
the assertion (i) follows from [20]. The assertion (i) follows from [28] and
[29, Theorem 3 (page 88)] if $(m,n)=(1,3)$, and it follows from [31] if
$(m,n)=(3,1)$.  where the positive integer $D(d;m,n)$ is defined by

(2.16) $D(d;m,n)=(mn-2)(\lfloor d/n\rfloor+1)-1.$

  2. (ii)

If $mn\geq 4$, there is a homotopy equivalence

(2.17) $\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})\simeq J_{\lfloor
d/n\rfloor}(S^{mn-2}).\qquad\qed$

### The main results.

The main purpose of this paper is to investigate the homotopy type of the
space $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ and to prove an Atiyah-Jones-
Segal type result for it.

First, we consider the unstable homotopy type of the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$. For this purpose, the key observation will be
the homotopy equivalence (Lemma 3.3 below)

(2.18) $(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{N})^{\mathbb{Z}_{2}}\simeq\Omega^{2}S^{2N+1}\times\Omega
S^{N}\qquad\mbox{for }N\geq 2.$

Using it, we shall prove the following result, closely related to Theorems 2.5
and 2.6.

######  Theorem 2.7.

The natural map

$i^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}\simeq\Omega^{2}S^{2mn-1}\times\Omega S^{mn-1}$

is a homotopy equivalence through dimension $D(d;m,n)$ if $mn\geq 4$, and a
homology equivalence through dimension $D(d;m,n)$ if $mn=3$.

Since $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{C})^{\mathbb{Z}_{2}}=\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$ and
$(i^{d,m}_{n,\mathbb{C}})^{\mathbb{Z}_{2}}=i^{d,m}_{n,\mathbb{R}}$, by using
Theorems 2.5 and 2.7 we also obtain the following:

######  Corollary 2.8.

The natural map

$i^{d,m}_{n,\mathbb{C}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\to\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1}\simeq\Omega^{2}S^{2mn-1}$

is a $\mathbb{Z}_{2}$-equivariant homotopy equivalence through dimension
$D(d;m,n)$ if $mn\geq 4$, and a $\mathbb{Z}_{2}$-equivariant homology
equivalence through dimension through dimension $D(d;m,n)$ if $mn=3$. ∎

######  Corollary 2.9.

Let $mn\geq 3$, let $I^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\Omega^{2}S^{2mn-1}\times\Omega S^{mn-1}$
denote the map defined by $($8.5$)$, and $q_{k}$ is the projection to the
$k$-th factor for $k=1,2$ given by $($8.2$)$.

  1. (i)

The map $q_{1}\circ I^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\Omega^{2}S^{2mn-1}$ induces an epimorphism on
the homology group $H_{k}(\ ;\mathbb{Z})$ for any $k\leq D(\lfloor
d/2\rfloor;m,n;\mathbb{C})$.

  2. (ii)

The map $q_{2}\circ I^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\Omega S^{mn-1}$ induces an epimorphism on the
homology group $H_{k}(\ ;\mathbb{Z})$ for any $k\leq D(d;m,n)$.

######  Corollary 2.10.

The jet embedding

$j^{d}_{n}:\mbox{{\rm Poly}}^{d,1}_{n}(\mathbb{R})\to\mbox{{\rm
Poly}}^{d,n}_{1}(\mathbb{R})$

is a homotopy equivalence through dimension $(n-2)(\lfloor d/n\rfloor+1)-1$ if
$n\geq 4$, and a homology equivalence through dimension $\lfloor d/3\rfloor$
if $n=3$, where $D(d;1,n)=(n-2)(\lfloor d/n\rfloor+1)-1$.

Next, we consider the stable homotopy type of the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$.

######  Theorem 2.11.

If $mn\geq 3$, there is a stable homotopy equivalence

$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})\simeq_{s}\mbox{{\rm Poly}}^{\lfloor
d/2\rfloor,m}_{n}(\mathbb{C})\vee B^{d,m}_{n}\vee\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R}),$

where the space $B^{d,m}_{n}$ is defined by

(2.19) $\displaystyle B^{d,m}_{n}$ $\displaystyle=\bigvee_{i,j\geq
1,i+2j\leqq\lfloor d/n\rfloor}\Sigma^{(mn-2)(i+2j)}D_{j}.$

By using the stable homotopy equivalences (2.15), (4.26) and the equality
$\lfloor\lfloor d/n\rfloor/2\rfloor=\lfloor\lfloor d/2\rfloor/n\rfloor$ (by
(iii) of Lemma 3.10), we obtain the following result.

######  Corollary 2.12.

If $mn\geq 3$, there is a stable homotopy equivalence

$\displaystyle\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$
$\displaystyle=\Big{(}\bigvee_{i=1}^{\lfloor
d/n\rfloor}S^{(mn-2)i}\Big{)}\vee\Big{(}\bigvee_{i\geq 0,j\geq
1,i+2j\leq\lfloor d/n\rfloor}\Sigma^{(mn-2)(i+2j)}D_{j}\Big{)}.\qquad\qed$

In particular, we obtain the following result, analogous to (2.14).

######  Corollary 2.13.

If $mn\geq 3$, there is a stable homotopy equivalence

(2.20) $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})\simeq_{s}\mbox{{\rm
Poly}}^{\lfloor d/n\rfloor,mn}_{1}(\mathbb{R}).$

By using Theorems 2.11 and 5.6, we also have the following result.

######  Corollary 2.14.

Let $mn\geq 3$.

  1. (i)

The inclusion map $\iota^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})$ induces a split epimorphism on the homology group
$H_{*}(\ ;\mathbb{Z})$.

  2. (ii)

The inclusion map $\iota^{d,m}_{n,\mathbf{\rm H}_{+}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$ induces a split monomorphism on the homology
group $H_{*}(\ ;\mathbb{Z})$.555 Here $\mathbf{\rm H}_{+}$ denotes the upper-
half plane in the complex plane $\mathbb{C}$. The map
$\iota^{d,m}_{n,\mathbf{\rm H}_{+}}$ and the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ will be defined in Definition
3.7.

The organization of this paper is as follows. In §2 we recall several
definitions, notations, and known results. After then we give the main results
of this paper (Theorems 2.7 and 2.11). In §3 we investigate the homotopy type
of the spaces $(\Omega^{2}_{d}\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_{2}}$ and
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$. In §4 we
construct the Vassiliev spectral sequence converging to the homology of
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$, and compute its $E^{1}$-terms. In
§5, we construct loop products and stabilization maps for the spaces
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{K})$ $(\mathbb{K}=\mathbb{R}$ or
$\mathbb{C}$) and $\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R}),$ and use them to
prove Theorem 5.6. In §6 we prove the homology stability theorem for the space
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ (Theorem 6.2). In §7 we consider
configuration space models for $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{K})$
($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) and $\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})$, define corresponding stabliization maps and use
them to prove the stable theorems (Theorems 7.5 and 7.9). In §8 we study about
the homotopy type of $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ and give the
proofs of our main results (Theorems 2.7 and 2.11) and their corollaries
(Corollaries 2.9, 2.10, 2.13 and 2.14).

##  3 The spaces $(\Omega^{2}_{d}\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_{2}}$ and
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$

In this section, we shall investigate the homotopy type of the spaces
$(\Omega^{2}_{d}\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_{2}}$ and $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$.

We first consider the space $(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{N})^{\mathbb{Z}_{2}}$. By identifying $S^{2}=\mathbb{C}\cup\infty$,
$S^{2}$ acquires a $\mathbb{Z}_{2}$-action induced by complex conjugation on
$\mathbb{C}$. The space $\mathbb{C}{\rm P}^{N}$ also has a natural
$\mathbb{Z}_{2}$-action induced by complex conjugation. Thus, we can consider
the space $(\Omega^{2}_{d}\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_{2}}$ of based
$\mathbb{Z}_{2}$-equivariant maps from $S^{2}$ to $\mathbb{C}{\rm P}^{N}$.

######  Definition 3.1.

Let $(X,A)$ and $(Y,B)$ be pairs of connected based spaces.

(i) We denote by $\mbox{{\rm Map}}^{*}(X,A;Y,B)$ the subspace of $\mbox{{\rm
Map}}^{*}(X,Y)$ consisting of all based maps $f\in\mbox{{\rm Map}}^{*}(X,Y)$
such that $f(A)\subset B$.

(ii) Let $\mbox{{\rm Map}}^{*}_{0}(X,Y)$ denote the path component of
$\mbox{{\rm Map}}^{*}(X,Y)$ containing null-homotopic maps. Then for a based
map $g\in\mbox{{\rm Map}}^{*}_{0}(A,Y)$, let $F(X,A;Y;g)\subset\mbox{{\rm
Map}}^{*}_{0}(X,Y)$ denote the subspace

(3.1) $F(X,A;Y;g)=\\{f\in\mbox{{\rm Map}}^{*}_{0}(X,Y):f|A=g\\}.$

(iii) The pair $(X,A)$ is called a NDR-pair if the inclusion
$i_{A}:A\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}X$ is a
cofibration. In this situation, we have a cofibration sequence

(3.2) $A\stackrel{{\scriptstyle
i_{A}}}{{\longrightarrow}}X\stackrel{{\scriptstyle
q_{A}}}{{\longrightarrow}}X/A,$

where $q_{A}:X\to X/A$ denotes the natural pinching map.

######  Lemma 3.2.

If $(X,A)$ is a NDR-pair, for each pair of spaces $(Y,B)$ the following is a
fibration sequence

(3.3) $\mbox{{\rm
Map}}^{*}(X/A,Y)\stackrel{{\scriptstyle}}{{\longrightarrow}}\mbox{{\rm
Map}}^{*}(X,A;Y,B)\stackrel{{\scriptstyle r_{A}}}{{\longrightarrow}}\mbox{{\rm
Map}}^{*}(A,B),$

where the two maps $q^{\\#}_{A}$ and $r_{A}$ are defined by
$q^{\\#}_{A}(f)=f\circ q_{A}$ and $r_{A}(g)=g\circ i_{A}=g|A$ for
$(f,g)\in\mbox{{\rm Map}}^{*}(X/A,Y)\times\mbox{{\rm Map}}^{*}(X,A;Y,B).$

###### Proof.

By (3.2) we obtain a fibration sequence

(3.4) $\mbox{{\rm Map}}^{*}(X/A,Y)\stackrel{{\scriptstyle
q_{A}^{\\#}}}{{\longrightarrow}}\mbox{{\rm
Map}}^{*}(X,Y)\stackrel{{\scriptstyle re_{A}}}{{\longrightarrow}}\mbox{{\rm
Map}}^{*}(A,Y).$

Let $i_{B}:B\to Y$ be an inclusion map and let $i_{B\\#}:\mbox{{\rm
Map}}^{*}(A,B)\to\mbox{{\rm Map}}^{*}(A,Y)$ denote the map given by
$i_{B\\#}(h)=i_{B}\circ h$ for $h\in\mbox{{\rm Map}}^{*}(A,Y).$

Now consider the following fibration induced from (3.4) by the map $i_{B\\#}$

(3.5) $\begin{CD}\mbox{{\rm Map}}^{*}(X/A,Y)@>{}>{}>E@>{r_{A}}>{}>\mbox{{\rm
Map}}^{*}(A,B)\\\ \|@V{}V{}V@V{i_{B\\#}}V{}V\\\ \mbox{{\rm
Map}}^{*}(X/A,Y)@>{q_{A}^{\\#}}>{}>\mbox{{\rm
Map}}^{*}(X,Y)@>{re_{A}}>{}>\mbox{{\rm Map}}^{*}(A,Y),\end{CD}$
$\displaystyle\mbox{ where}\quad\quad E$ $\displaystyle=$
$\displaystyle\\{(f,g)\in\mbox{{\rm Map}}^{*}(X,Y)\times\mbox{{\rm
Map}}^{*}(A,B):r_{A}(f)=i_{B\\#}(g)\\}$ $\displaystyle=$
$\displaystyle\\{(f,g)\in\mbox{{\rm Map}}^{*}(X,Y)\times\mbox{{\rm
Map}}^{*}(A,B):f|A=g\\}$ $\displaystyle=$ $\displaystyle\mbox{{\rm
Map}}^{*}(X,A;Y,B)$

Thus, we have obtained the fibration sequence (3.3). ∎

######  Lemma 3.3.

If $N\geq 2$, there is a homotopy equivalence

(3.6) $(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{N})^{\mathbb{Z}_{2}}\simeq\Omega^{2}S^{2N+1}\times\Omega S^{N}.$

###### Proof.

By applying (3.3) to the case $(X,A;Y,B)=(D^{2},S^{1};\mathbb{C}{\rm
P}^{N},\mathbb{R}{\rm P}^{N})$, we obtain the following fibration sequence

(3.7) $\Omega^{2}_{d}\mathbb{C}{\rm P}^{N}\to\mbox{{\rm
Map}}^{*}_{d}(D^{2},S^{1};\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm
P}^{N})\stackrel{{\scriptstyle
r_{S^{1}}}}{{\longrightarrow}}\Omega_{[d]_{2}}\mathbb{R}{\rm P}^{N}.$

It follows from the proof of Lemma 3.2 that (3.7) is the induced fibration
from the map $\Omega j:\Omega_{[d]_{2}}\mathbb{R}{\rm
P}^{N}\to\Omega_{[d]_{2}}\mathbb{C}{\rm P}^{N}$, where $j:\mathbb{R}{\rm
P}^{N}\to\mathbb{C}{\rm P}^{N}$ denotes the inclusion map. Let
$\tilde{j}:S^{N}\to S^{2N+1}$ denote the natural inclusion, and let
$\gamma_{N}:S^{2N+1}\to\mathbb{C}{\rm P}^{N}$ and
$\gamma_{N,\mathbb{R}}:S^{N}\to\mathbb{R}{\rm P}^{N}$ be the Hopf fibering and
the usual double covering, respectively. Consider the following commutative
diagram

$\begin{CD}\Omega S^{N}@>{\Omega\tilde{j}}>{}>\Omega S^{2N+1}\\\
@V{\Omega\gamma_{N,\mathbb{R}}}V{\simeq}V@V{}V{\Omega\gamma_{N}}V\\\
\Omega_{0}\mathbb{R}{\rm P}^{N}@>{\Omega j}>{}>\Omega_{0}\mathbb{C}{\rm
P}^{N}\end{CD}$

Since the map $\tilde{j}$ is null-homotopic, the map $\Omega
j:\Omega_{0}\mathbb{R}{\rm P}^{N}\to\Omega_{0}\mathbb{C}{\rm P}^{N}$ is also
null-homotopic and hence also the map $j$. Thus, (3.7) is a trivial fibration
and there is a homotopy equivalence

$\mbox{{\rm Map}}^{*}_{d}(D^{2},S^{1};\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm
P}^{N})\simeq\Omega^{2}_{d}\mathbb{C}{\rm
P}^{N}\times\Omega_{[d]_{2}}\mathbb{R}{\rm
P}^{N}\simeq\Omega^{2}S^{2N+1}\times\Omega S^{N}.$

Hence, we have a homotopy equivalence

(3.8) $\mbox{{\rm Map}}^{*}_{d}(D^{2},S^{1};\mathbb{C}{\rm
P}^{N},\mathbb{R}{\rm P}^{N})\simeq\Omega^{2}S^{2N+1}\times\Omega
S^{N}\quad\mbox{for any }d\in\mathbb{Z}.$

Now we regard $S^{2}$ as the union $S^{2}=D^{2}_{+}\cup D^{2}_{-}$, where let
$D^{2}_{+}$ and $D^{2}_{-}$ denote the northern hemisphere and the southern
one given by

$\displaystyle D^{2}_{+}$
$\displaystyle=\\{\textbf{{x}}=(x,y)\in\mathbb{R}^{2}:\|\textbf{{x}}\|\leq
1,y\geq 0\\},$ $\displaystyle D^{2}_{-}$
$\displaystyle=\\{\textbf{{x}}=(x,y)\in\mathbb{R}^{2}:\|\textbf{{x}}\|\leq
1,y\leq 0\\}.$

Then each $\mathbb{Z}_{2}$-equivariant map $f:S^{2}\to\mathbb{C}{\rm P}^{N}$
can be identified with the map $(D^{2}_{+},S^{1})\to(\mathbb{C}{\rm
P}^{N},\mathbb{R}{\rm P}^{N}).$ Then by identifying $D^{2}\cong D^{2}_{+}$
there is a homeomorphism

(3.9) $(\Omega^{2}_{d}\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_{2}}\cong\mbox{{\rm
Map}}^{*}_{d}(D^{2},S^{1};\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm P}^{N}).$

Hence, the assertion easily follows from (3.8) and (3.9). ∎

######  Remark 3.4.

It follows from (3.6) and (3.8) that the homotopy types of the spaces
$(\Omega^{2}_{d}\mathbb{C}{\rm P}^{N})^{\mathbb{Z}_{2}}$ and $\mbox{{\rm
Map}}^{*}_{d}(D^{2},S^{1};\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm P}^{N})$ do not
depend on the choice of the integer $d$. ∎

######  Lemma 3.5.

If $N\geq 1$, there is a homotopy equivalence

(3.10) $F(D^{2},S^{1};\mathbb{C}{\rm
P}^{N};g)\simeq\Omega^{2}_{0}\mathbb{C}{\rm
P}^{N}\simeq\Omega^{2}S^{2N+1}\quad\mbox{for any $g\in\Omega_{0}\mathbb{C}{\rm
P}^{N}$.}$

###### Proof.

Let $r:\mbox{{\rm Map}}^{*}_{0}(D^{2},\mathbb{C}{\rm P}^{N})\to\mbox{{\rm
Map}}_{0}^{*}(S^{1},\mathbb{C}{\rm P}^{N})=\Omega_{0}\mathbb{C}{\rm P}^{N}$
denote the restriction map given by $r(f)=f|S^{1}$. It is easy to see that $r$
is a fibration. Moreover, since $D^{2}$ is contractible, the space $\mbox{{\rm
Map}}^{*}_{0}(D^{2},\mathbb{C}{\rm P}^{N})$ is contractible. Thus, each fiber
of $r$ is homotopy equivalent to the space $\Omega^{2}_{0}\mathbb{C}{\rm
P}^{N}\simeq\Omega^{2}S^{2N+1}$. Hence, there is a homotopy equivalence

$F(D^{2},S^{1};\mathbb{C}{\rm
P}^{N};g)=r^{-1}(g)\simeq\Omega^{2}_{0}\mathbb{C}{\rm
P}^{N}\simeq\Omega^{2}S^{2N+1},$

and this completes the proof. ∎

######  Lemma 3.6.

If $N\geq 2$, there is a following homotopy commutative diagram:

(3.11) $\begin{CD}\mbox{{\rm Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm
P}^{N},*)@>{\iota_{\mathbb{C}}}>{\simeq}>\Omega^{2}_{0}\mathbb{C}{\rm
P}^{N}\simeq\Omega^{2}S^{2N+1}\\\ @V{\hat{j}}V{\cap}V\ \ \ @A{q_{1}}A{}A\\\
\mbox{{\rm Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm
P}^{N})@>{\iota_{\mathbb{C}\mathbb{R}}}>{\simeq}>\Omega^{2}S^{2N+1}\times\Omega
S^{N}\\\ @V{r_{S^{1}}}V{}V\ \ \ @V{q_{2}}V{}V\\\ \Omega_{0}\mathbb{R}{\rm
P}^{N}@>{\iota_{\mathbb{R}}}>{\simeq}>\ \ \ \Omega S^{N}\end{CD}$

where $\hat{j}:\mbox{{\rm Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm
P}^{N},*)\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}\mbox{{\rm
Map}}^{*}_{0}(D^{2},S^{1}:\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm P}^{N})$
denotes the natural inclusion, $r_{S^{1}}:\mbox{{\rm
Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm P}^{N},\mathbb{R}{\rm
P}^{N})\to\Omega_{0}\mathbb{R}{\rm P}^{N}$ is the restriction map given by
$r_{S^{1}}(f)=f|S^{1}$, $\iota_{\mathbb{C}\mathbb{R}}$ and
$\iota_{\mathbb{C}}$ are homotopy equivalences given by (3.8) and (3.10), and
$q_{1}$ and $q_{2}$ are the projections to the first factors and the second
factor, respectively.

###### Proof.

The assertion easily follows from the naturality of the homotopy equivalences
$\iota_{\mathbb{C}\mathbb{R}}$ and $\iota_{\mathbb{C}}$. ∎

Next, we define the space $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})$.

######  Definition 3.7.

Let $\mathbf{\rm H}_{+}=\\{\alpha\in\mathbb{C}:\mbox{ Im }\alpha>0\\}$ denote
the upper half plane in the complex plane $\mathbb{C}$.

(i) For each even positive integer $d=2d_{0}\in\mathbb{N}$, let ${\rm
P}_{d}(\mathbb{R};\mathbf{\rm H}_{+})$ denote the space of monic polynomials
$f(z)\in\mathbb{R}[z]$ of degree $d$ such that it has just $d_{0}$ roots in
$\mathbf{\rm H}_{+}$. Thus, if $f(z)\in{\rm P}_{d}(\mathbb{R};\mathbf{\rm
H}_{+})$, it is represented as the form

(3.12)
$f(z)=\displaystyle\prod_{k=1}^{d_{0}}(z-\alpha_{k})(z-\overline{\alpha_{k}})\quad\mbox{for
some }\alpha_{k}\in\mathbf{\rm H}_{+}\ (1\leq k\leq d_{0}).$

(ii) For each $d\in\mathbb{N}$, we define the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ as follows.

  1. (ii-0)

When $d\in\mathbb{N}$ is an even integer, the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ is

(3.13) $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})=\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\cap({\rm P}_{d}(\mathbb{R};\mathbf{\rm
H}_{+}))^{m}.$

  2. (ii-1)

When $d\in\mathbb{N}$ is an odd integer, the space $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ is the subspace of
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ consisting of all elements of the
form

(3.14) $\big{(}(z-1)f_{1}(z),(z-2)f_{2}(z),\cdots,(z-m)f_{m}(z)\big{)}$

where $(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm
Poly}}^{d-1,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+}).$

(iii) It is easy to see that there is a homeomorphism

(3.15) $\mbox{{\rm Poly}}^{2d_{0}+1,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})\cong\mbox{{\rm Poly}}^{2d_{0},m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})\quad\mbox{for any }d_{0}\in\mathbb{N},$

and that there is an inclusion $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\subset\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$ for any $d\in\mathbb{N}$. We denote this natural
inclusion map by

(3.16) $\iota^{d,m}_{n,\mathbf{\rm H}_{+}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}).$

(iv) By making the identification $S^{2}=\mathbf{\rm H}_{+}\cup\infty$, we
obtain a map

(3.17) $\displaystyle i^{d,m}_{n,\mathbf{\rm H}_{+}}$
$\displaystyle:\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})\to\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1}\simeq\Omega^{2}S^{2mn-1}\quad\mbox{given by}$ $\displaystyle
i^{d,m}_{n,\mathbf{\rm H}_{+}}(f)(\alpha)$
$\displaystyle=\begin{cases}[F_{n}(f_{1})(\alpha):F_{n}(f_{2})(\alpha):\cdots:F_{n}(f_{m})(\alpha)]&\mbox{if
}\alpha\in\mathbf{\rm H}_{+}\\\ [1:1:\cdots:1]&\mbox{if
}\alpha=\infty\end{cases}$

for $f=(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ and $\alpha\in
S^{2}=\mathbf{\rm H}_{+}\cup\infty$.

The next lemma is a simple but crucial observation.

######  Lemma 3.8.

(i) If $d\in\mathbb{N}$ is an even positive integer, there is a homeomorphism
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\cong\mbox{{\rm
Poly}}^{\lfloor d/2\rfloor,m}_{n}(\mathbb{C}).$

(ii) There is a homeomorphism $\mbox{{\rm Poly}}^{\lfloor
d/2\rfloor,m}_{n}(\mathbb{C})\cong\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ for any $d\geq 2$.

###### Proof.

(i) Let $d=2d_{0}$ and let
$\psi:\mathbb{C}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbf{\rm
H}_{+}$ be any fixed homeomorphism. Then we have the homeomorphism
$\psi_{d}:{\rm
P}_{d_{0}}(\mathbb{C})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\rm
P}_{d}(\mathbb{R};\mathbf{\rm H}_{+})$ given by
$\psi_{d}\big{(}\prod_{k=1}^{d_{0}}(z-\alpha_{k})\big{)}=\prod_{k=1}^{d_{0}}(z-\psi(\alpha_{k}))(z-\overline{\psi(\alpha_{k})})$
for $\alpha_{k}\in\mathbb{C}$. This naturally extends to the desired
homeomorphism

(3.18) $\mbox{{\rm
Poly}}^{d_{0},m}_{n}(\mathbb{C})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$

given by
$\big{(}f_{1}(z),\cdots,f_{m}(z)\big{)}\mapsto\big{(}\psi_{d}(f_{1}(z)),\cdots,\psi_{d}(f_{m}(z))\big{)}$.

(ii) The assertion (ii) easily follows from (3.15) and the assertion (i). ∎

######  Theorem 3.9.

If $mn\geq 3$, the natural map

$i^{d,m}_{n,\mathbf{\rm H}_{+}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\to\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1}\simeq\Omega^{2}S^{2mn-1}$

is a homotopy equivalence through dimension $D(\lfloor
d/2\rfloor;m,n;\mathbb{C})$.

###### Proof.

Since there is a homeomorphism $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\cong\mbox{{\rm Poly}}^{\lfloor
d/2\rfloor,m}_{n}(\mathbb{C})$ (by Lemma 3.8), the proof of [18, Theorem 1.8]
works verbatim by replacing $\mathbb{C}$ by $\mathbf{\rm H}_{+}$ in the case
when $d$ is even. The case of odd $d$ can be easily reduced to the even one by
(3.15). ∎

######  Lemma 3.10.

If $d\in\mathbb{N}$ and $d_{0}=\lfloor d/2\rfloor$, the following assertions
hold:

  1. (i)

$\lfloor d/n\rfloor=2\lfloor d_{0}/n\rfloor$ or $\lfloor d/n\rfloor=2\lfloor
d_{0}/n\rfloor+1.$

  2. (ii)

$\lfloor d/(2n)\rfloor=\lfloor d_{0}/n\rfloor.$

  3. (iii)

$\displaystyle\lfloor\lfloor d/n\rfloor/2\rfloor=\lfloor d_{0}/n\rfloor$, and
$D(d;m,n)<D(\lfloor d/2\rfloor;m,n;\mathbb{C})$.

  4. (iv)

Define a finite subset $\mathcal{F}^{d,m}_{n}\subset\mathbb{N}^{2}$ by

(3.19) $\mathcal{F}^{d,m}_{n}=\\{(i,j)\in\mathbb{N}^{2}:i+2j\leq\lfloor
d/n\rfloor\\}.$

Then if $(i,j)\in\mathcal{F}^{d,m}_{n}$, $1\leq i<\lfloor d/n\rfloor$ and
$1\leq j\leq\lfloor d_{0}/n\rfloor,$ and

(3.20) $\mathcal{F}^{d,m}_{n}\subset\\{(i,j)\in\mathbb{N}^{2}:1\leq i<\lfloor
d/n\rfloor,1\leq j\leq\lfloor d_{0}/n\rfloor,i+2j\leq\lfloor d/n\rfloor\\}.$

###### Proof.

(i) Let us write $q=\lfloor d_{0}/n\rfloor$. Then we can also write

$\displaystyle d$ $\displaystyle=2d_{0}+\epsilon_{0}\ \mbox{ with }\
\epsilon_{0}\in\\{0,1\\},\ \ d_{0}=nq+\epsilon\ \mbox{ with }\
0\leq\epsilon\leq n-1.$ Since $\displaystyle\begin{cases}0&\leq
2\epsilon+\epsilon_{0}\leq 2(n-1)+1=2n-1<2n,\mbox{ and}\\\
d&=2d_{0}+\epsilon_{0}=2(nq+\epsilon)+\epsilon_{0}=2nq+(2\epsilon+\epsilon_{0}),\end{cases}$
$\displaystyle 2nq$ $\displaystyle\leq d<(2q+2)n.\quad\mbox{So }2q\leq\lfloor
d/n\rfloor<2q+2.$

Hence, $\lfloor d/n\rfloor=2q=2\lfloor d_{0}/2\rfloor\ \mbox{or}\ \lfloor
d/n\rfloor=2q+1=2\lfloor d_{0}/2\rfloor+1.$ Thus, we have proved (i).

(ii) Since $2nq\leq d<(2q+2)n=2n(q+1)$ by (i), $q\leq d/(2n)<q+1$. Thus, we
also have $\lfloor d/(2n)\rfloor=q=\lfloor d_{0}/n\rfloor$ and (ii) was
obtained.

(iii) By using (i), we have

$\lfloor d/n\rfloor/2=q=\lfloor d_{0}/2\rfloor\ \ \mbox{ or }\ \ \lfloor
d/n\rfloor/2=q+\frac{1}{2}=\lfloor d_{0}/2\rfloor+\frac{1}{2}.$

Thus, $\lfloor\lfloor d/n\rfloor/2\rfloor=q=\lfloor d_{0}/2\rfloor$. Moreover,
since $\lfloor d/n\rfloor\leqq 2q+1$ (by (i)),

$\displaystyle\delta$ $\displaystyle=D(d;m,n)-D(\lfloor
d/2\rfloor;m,n;\mathbb{C})$ $\displaystyle=\big{\\{}(mn-2)(\lfloor
d/n\rfloor+1)-1\big{\\}}-\big{\\{}(2mn-3)(q+1)-1\big{\\}}$
$\displaystyle\leqq(mn-2)(2q+2)-(2mn-3)(q+1)\leqq-(q+1)<0$

and the assertion (iii) follows.

(iv) Suppose that $(i,j)\in\mathcal{F}^{d,m}_{n}$. Since we can see that the
first assertion holds and it remains show the second one. Since $i\geq 1$,
$1\leq 2j<\lfloor d/n\rfloor$.

Hence, by using (iii), we have $1\leq j\leq\lfloor\lfloor
d/n\rfloor/2\rfloor=\lfloor d_{0}/n\rfloor$, and we obtain the assertion (iv).
∎

##  4 The Vassiliev spectral sequence

In this section we construct a Vassiliev type spectral sequence converging to
the homology of $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ by means of a non-
degenerate simplicial resolutions of discriminants, and compute its
$E^{1}$-terms.

First, we summarize the basic facts of the theory of non-degenerate simplicial
resolutions ([29], [30]; cf. [24]) and the spectral sequences associated with
them.

######  Definition 4.1.

For a finite set $\textbf{{v}}=\\{v_{1},\cdots,v_{l}\\}\subset\mathbb{R}^{N}$,
let $\sigma(\textbf{{v}})$ denote the convex hull spanned by $\textbf{{v}}.$
Suppose that $h:X\to Y$ is a surjective map such that $h^{-1}(y)$ is a finite
set for any $y\in Y$, and let $i:X\to\mathbb{R}^{N}$ be an embedding.

(i) Then let $\mathcal{X}^{\Delta}$ and $h^{\Delta}:{\mathcal{X}}^{\Delta}\to
Y$ denote the space and the map defined by

(4.1) $\mathcal{X}^{\Delta}=\big{\\{}(y,\textbf{{u}})\in
Y\times\mathbb{R}^{N}:\textbf{{u}}\in\sigma(i(h^{-1}(y)))\big{\\}}\subset
Y\times\mathbb{R}^{N},\ h^{\Delta}(y,\textbf{{u}})=y.$

The pair $(\mathcal{X}^{\Delta},h^{\Delta})$ is called the simplicial
resolution of $(h,i)$. In particular, $(\mathcal{X}^{\Delta},h^{\Delta})$ is
called a non-degenerate simplicial resolution if for each $y\in Y$ any $k$
points of $i(h^{-1}(y))$ span $(k-1)$-dimensional simplex of $\mathbb{R}^{N}$.

(ii) For each $k\geq 0$, let
$\mathcal{X}^{\Delta}_{k}\subset\mathcal{X}^{\Delta}$ be the subspace given by

(4.2)
$\mathcal{X}_{k}^{\Delta}=\big{\\{}(y,\textbf{{u}})\in\mathcal{X}^{\Delta}:\textbf{{u}}\in\sigma(\textbf{{v}}),\textbf{{v}}=\\{v_{1},\cdots,v_{l}\\}\subset
i(h^{-1}(y)),l\leq k\big{\\}}.$

We make identification $X=\mathcal{X}^{\Delta}_{1}$ by identifying $x\in X$
with $(h(x),i(x))\in\mathcal{X}^{\Delta}_{1}$, and we note that there is an
increasing filtration

$\emptyset=\mathcal{X}^{\Delta}_{0}\subset
X=\mathcal{X}^{\Delta}_{1}\subset\mathcal{X}^{\Delta}_{2}\subset\cdots\subset\mathcal{X}^{\Delta}_{k}\subset\mathcal{X}^{\Delta}_{k+1}\subset\cdots\subset\bigcup_{k=0}^{\infty}\mathcal{X}^{\Delta}_{k}=\mathcal{X}^{\Delta}.$

Since the map $h^{\Delta}$ is a proper map, it extends the map
$h^{\Delta}_{+}:\mathcal{X}^{\Delta}_{+}\to Y_{+}$ between one-point
compactifications, where $X_{+}$ denotes the one-point compactification of a
locally compact space $X$.

######  Theorem 4.2 ([29], [30] (cf. [17], [24])).

Let $h:X\to Y$ be a surjective map such that $h^{-1}(y)$ is a finite set for
any $y\in Y,$ $i:X\to\mathbb{R}^{N}$ an embedding, and let
$(\mathcal{X}^{\Delta},h^{\Delta})$ denote the simplicial resolution of
$(h,i)$.

  1. (i)

If $X$ and $Y$ are semi-algebraic spaces and the two maps $h$, $i$ are semi-
algebraic maps, then
$h^{\Delta}_{+}:\mathcal{X}^{\Delta}_{+}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}Y_{+}$
is a homology equivalence.

  2. (ii)

If there is an embedding $j:X\to\mathbb{R}^{M}$ such that its associated
simplicial resolution $(\tilde{\mathcal{X}}^{\Delta},\tilde{h}^{\Delta})$ is
non-degenerate, the space $\tilde{\mathcal{X}}^{\Delta}$ is uniquely
determined up to homeomorphism and there is a filtration preserving homotopy
equivalence
$q^{\Delta}:\tilde{\mathcal{X}}^{\Delta}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}{\mathcal{X}}^{\Delta}$
such that $q^{\Delta}|X=\mbox{id}_{X}$.

  3. (iii)

A non-degenerate simplicial resolution exists even if the map $h$ is not
finite to one.

###### Proof.

See the proof of [20, Theorem 2.2] for the details. ∎

######  Remark 4.3.

It is known that the map $h^{\Delta}_{+}$ is a homotopy equivalence [30, page
156]. However this stronger assertion is not needed in this paper. ∎

Now recall several basic definitions and notations.

######  Definition 4.4.

(i) For connected space $X$, let $F(X,d)$ denote the ordered configuration
space of distinct $d$ points of $X$ defined by

(4.3) $F(X,d)=\\{(x_{1},\cdots,x_{d})\in X^{d}:x_{i}\not=x_{j}\mbox{ if
}i\not=j\\}.$

(ii) Let $S_{d}$ denote the symmetric group of $d$-letters. Then the group
$S_{d}$ acts on $F(X,d)$ by the coordinate permutation and let $C_{d}(X)$
denote the unordered configuration space of $d$-distinct points of $X$ defined
by the orbit space

(4.4) $C_{d}(X)=F(X,d)/S_{d}.$

(iii) For connected space $X$, let $D_{j}(X)$ denote the equivariant half-
smash product of $X$ defined by

(4.5) $D_{j}(X)=F(X,j)_{+}\wedge_{S_{j}}X^{\wedge j},$

where we set $F(X,j)_{+}=F(X,j)\cup\\{*\\}$ (disjoint union), $X^{\wedge
j}=X\wedge X\wedge\cdots\wedge X$ ($j$-times) and the $j$-th symmetric group
$S_{j}$ acts on $X^{\wedge j}$ by the coordinate permutation. In particular,
for $X=S^{1}$, we set

(4.6) $D_{j}=D_{j}(S^{1})=F(\mathbb{C},j)_{+}\wedge_{S_{j}}(S^{1})^{\wedge
j}.$

Let $mn\geq 3$ and we shall construct the Vassiliev-type spectral sequence.

######  Definition 4.5.

(i) Let $\Sigma^{d,m}_{n}$ denote _the discriminant_ of $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$ in ${\rm P}_{d}(\mathbb{R})^{m}$ given by the
complement

(4.7) $\Sigma^{d,m}_{n}={\rm P}_{d}(\mathbb{R})^{m}\setminus\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}).$

(ii) For each $m$-tuple $(f_{1}(z),\cdots,f_{m}(z))\in{\rm
P}_{d}(\mathbb{R})^{m}$ let $F^{m}_{n}(f_{1},\cdots,f_{m})$ denote the
$mn$-tuple of $\mathbb{R}$-coefficients monic polynomials of the same degree
$d$ defined by

(4.8)
$F^{m}_{n}(f_{1},\cdots,f_{m})=F^{m}_{n}(f_{1},\cdots,f_{m})(z)=(F_{n}(f_{1}),\cdots,F_{n}(f_{m})).$

Since $F^{m}_{n}(f_{1},\cdots,f_{m})$ is an $mn$-tuple of
$\mathbb{R}$-coefficients polynomials,

$F^{m}_{n}(f_{1},\cdots,f_{m})(\alpha)=0\Leftrightarrow
F^{m}_{n}(f_{1},\cdots,f_{m})(\overline{\alpha})=0\quad\mbox{for
}\alpha\in\mathbf{\rm H}_{+}.$

Thus, the space $\Sigma^{d,m}_{n}$ is given by

(4.9) $\displaystyle\Sigma^{d,m}_{n}$ $\displaystyle=\\{f\in{\rm
P}_{d}(\mathbb{R})^{m}:F^{m}_{n}(f)(x)={\bf 0}\mbox{ for some
}x\in\overline{\mathbf{\rm H}}_{+}\\},$

where ${\bf 0}\in\mathbb{C}^{mn}$,
$F^{m}_{n}(f)=F^{m}_{n}(f_{1},\cdots,f_{m})$ for
$f=(f_{1}(z),\cdots,f_{m}(z))\in{\rm P}_{d}(\mathbb{R})^{m}$ and
$\overline{\mathbf{\rm H}}_{+}$ denotes the space defined by

(4.10) $\overline{\mathbf{\rm H}}_{+}=\mathbf{\rm
H}_{+}\cup\mathbb{R}=\\{\alpha\in\mathbb{C}:\mbox{Im }\alpha\geq 0\\}.$

(iii) Let $Z^{d,m}_{n}\subset\Sigma^{d,m}_{n}\times\mathbb{C}$ denote the
tautological normalization of $\Sigma^{d,m}_{n}$ given by

$Z^{d,m}_{n}=\\{((f_{1}(z),\cdots,f_{m}(z)),x)\in\Sigma^{d,m}_{n}\times\overline{\mathbf{\rm
H}}_{+}:F^{m}_{n}(f_{1},\cdots,f_{m})(x)={\bf 0}\\}.$

Projection on the first factor gives the surjective map

(4.11) $\pi^{d,m}_{n}:Z^{d,m}_{n}\to\Sigma^{d,m}_{n}.$

(iv) Let $\varphi_{\mathbb{R}}:{\rm
P}_{d}(\mathbb{R})^{nm}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}\mathbb{R}^{dmn}$
be any fixed homeomorphism. We identify $\mathbb{C}=\mathbb{R}^{2}$ and define
the embedding
$i_{\mathbb{R}}:Z^{d,m}_{n}\to\mathbb{R}^{dmn}\times\mathbb{C}=\mathbb{R}^{dmn+2}$
by

(4.12)
$i_{\mathbb{R}}((f_{1},\cdots,f_{m}),x)=(\varphi_{\mathbb{R}}(F_{n}(f_{1}),\cdots,F_{n}(f_{m})),x)$

for $((f_{1},\cdots,f_{m}),x)\in Z^{d,m}_{n}.$

(v) Let
$({\mathcal{X}}^{d},\pi^{\Delta}:{\mathcal{X}}^{d}\to\Sigma^{d,m}_{n})$ be
non-degenerate simplicial resolution of $(\pi^{d,m}_{n},i_{\mathbb{R}})$. Then
it is easy to see that there is a natural increasing filtration

$\displaystyle\emptyset$ $\displaystyle=$
$\displaystyle{\mathcal{X}}^{d}_{0}\subset{\mathcal{X}}^{d}_{1}\subset{\mathcal{X}}^{d}_{2}\subset\cdots\cdots\subset\bigcup_{k=0}^{\infty}{\mathcal{X}}^{d}_{k}={\mathcal{X}}^{d},$

such that $\mathcal{X}^{d}_{k}=\mathcal{X}^{d}$ if $k>\lfloor d/n\rfloor.$ ∎

By Theorem 4.2, the map
$\pi_{+}^{\Delta}:{\mathcal{X}}^{d}_{+}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}{\Sigma^{d,m}_{n+}}$
is a homology equivalence. The filtration on ${\mathcal{X}^{d}_{+}}$ gives
rise to a spectral sequence

(4.13) $\big{\\{}E_{t;d}^{k,s},d_{t}:E_{t;d}^{k,s}\to
E_{t;d}^{k+t,s+1-t}\big{\\}}\Rightarrow
H^{k+s}_{c}(\Sigma_{n}^{d,m};\mathbb{Z}),$

where $H_{c}^{k}(X;\mathbb{Z})$ denotes the cohomology group with compact
supports of a locally compact space $X$ given by
$H_{c}^{k}(X;\mathbb{Z})=\tilde{H}^{k}(X_{+};\mathbb{Z})$ and

(4.14)
$E_{1;d}^{k,s}=\tilde{H}^{k+s}({\mathcal{X}_{k}^{d}}_{+}/{{\mathcal{X}}^{d}_{k-1}}_{+};\mathbb{Z})=H^{k+s}_{c}({\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1};\mathbb{Z})$

(since
${\mathcal{X}_{k}^{d}}_{+}/{{\mathcal{X}}^{d}_{k-1+}}\cong({\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1})_{+}$).

Since there is a homeomorphism ${\rm
P}_{d}(\mathbb{R})^{m}\cong\mathbb{R}^{dm}$, by Alexander duality there is a
natural isomorphism

(4.15) $\tilde{H}_{k}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})\cong
H_{c}^{dm-k-1}(\Sigma_{n}^{d,m};\mathbb{Z})\quad\mbox{for any }k.$

By reindexing we obtain the Vassiliev-type spectral sequence

(4.16) $\displaystyle\big{\\{}E^{t;d}_{k,s},\ d^{t}:E^{t;d}_{k,s}\to
E^{t;d}_{k+t,s+t-1}\big{\\}}\Rightarrow\tilde{H}_{s-k}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z}),$

where we set

(4.17)
$E^{1;d}_{k,s}=H^{dm+k-s-1}_{c}({\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1};\mathbb{Z}).$

######  Definition 4.6.

Let $\mathbf{\rm H}_{+}=\\{\alpha\in\mathbb{C}:\mbox{Im }\alpha>0\\}$ denote
the upper half space, and let $(d,k)\in\mathbb{N}^{2}$ be a pair of positive
integers such that $1\leq k\leq\lfloor d/n\rfloor$.

(i) Let $\Sigma^{d,m}_{n}(k)\subset\Sigma^{d,m}_{n}$ denote the subspace
consisting of all $m$-tuples $(f_{1}(z),\cdots,f_{m}(z))\in\Sigma^{d,m}_{n}$
such that the polynomials $\\{f_{t}(z)\\}_{t=1}^{m}$ have exactly $k$ common
roots of multiplicity $\geq n$ in $\overline{\mathbf{\rm H}}_{+}$.

(ii) Let $(f_{1}(z),\cdots,f_{m}(z))\in\Sigma^{d,m}_{n}(k)$. Since each of
these polynomials has real coefficients, these common roots of multiplicity
$\geq n$ can be uniquely represented as a set

(4.18)
$\tilde{c}=\\{x_{1},\cdots,x_{i};\alpha_{1},\overline{\alpha}_{1},\cdots.\alpha_{j},\overline{\alpha}_{j}\\}\quad(i+j=k,0\leq
i\leq\lfloor k/2\rfloor)$

of $(k+j)$-complex numbers which satisfy the following two conditions:

  1. (4.18.1)

$x_{s}\in\mathbb{R}$ for each $1\leq s\leq i$ and $x_{i}\not=x_{l}$ if
$i\not=l$.

  2. (4.18.2)

$\alpha_{t}\in\mathbf{\rm H}_{+}$ for each $1\leq t\leq j$ and
$\alpha_{t}\not=\alpha_{l}$ if $t\not=l$.

We define a subspace $\Sigma^{d,m}_{n}(i,j)\subset\Sigma^{d,m}_{n}(k)$ as the
space of all $m$-tuples $(f_{1}(z),\cdots,f_{m}(z))\in\Sigma^{d,m}_{n}(k)$
whose common roots of multiplicity $\geq n$ satisfy the two above conditions
(4.18.1) and (4.18.2).

(iii) Note that
$(\pi^{\Delta})^{-1}(\Sigma^{d,m}_{n}(k))={\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1}$.
For each pair $(i,j)\in(\mathbb{Z}_{\geq 0})^{2}$ with $i+j=k$, define a
subspace
${\mathcal{X}}^{d}_{k}(i,j)\subset{\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1}$
by

(4.19)
${\mathcal{X}}^{d}_{k}(i,j)=(\pi^{\Delta})^{-1}(\Sigma^{d,m}_{n}(i,j)).$

It is easy to see that

(4.20)
$\displaystyle\displaystyle{\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1}$
$\displaystyle=\coprod_{i+j=k}{\mathcal{X}}^{d}_{k}(i,j)\quad\mbox{(disjoint
union)},$

where $\\{{\mathcal{X}}^{d}_{k}(i,j):i\geq 0,j\geq 0,i+j=k\\}$ are the path
components of ${\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1}$.

######  Lemma 4.7.

If $mn\geq 3$, $1\leq k\leq\lfloor d/n\rfloor$ and $(i,j)\in(\mathbb{Z}_{\geq
0})^{2}$ with $i+j=k$, the space ${\mathcal{X}}^{d}_{k}(i,j)$ is homeomorphic
to the total space of a real affine bundle $\xi^{d}_{k}(i,j)$ over
$C_{i}(\mathbb{R})\times C_{j}(\mathbf{\rm H}_{+})$ with real rank
$l_{d,k}(i,j)=m(d-nk-nj)+k-1$.

###### Proof.

The argument is exactly analogous to the one used in the proof of [20, Lemma
3.2] or [18, Lemma 3.3]. Namely, an element of ${\mathcal{X}}^{d}_{k}(i,j)$ is
represented by the $(m+1)$-tuple $(f_{1}(z),\cdots,f_{m}(z),\textbf{{u}})$,
where $(f_{1}(z),\cdots,f_{m}(z))$ is an $m$-tuple of monic polynomials of the
same degree $d$ in $\Sigma^{d,m}_{n}$ and u is an element of the interior of
the span of the images of $k$ distinct points

$c=(\\{x_{1},\cdots,x_{i}\\},\\{\alpha_{1},\cdots,\alpha_{j}\\})\in
C_{i}(\mathbb{R})\times C_{j}(\mathbf{\rm H}_{+})$

under a suitable embedding $i_{k}.$666This means that the embedding $i_{k}$
satisfies the condition [20, (2.3)].  Note that the following $(k+j)$-points

$\tilde{c}=\\{x_{1},\cdots,x_{i},\alpha_{1},\overline{\alpha}_{1},\alpha_{2},\overline{\alpha}_{2},\cdots,\alpha_{j},\overline{\alpha}_{j}\\}$

are common roots of $\\{f_{s}(z)\\}_{s=1}^{m}$ of multiplicity $n$.

By the definition of the non-degenerate simplicial resolution and [20, Lemma
2.5], the $k$ distinct points $c$ are uniquely determined by u. Thus, there is
the projection map

(4.21) $\pi_{k;i,j}^{d}:{\cal X}^{d}_{k}(i,j)\to C_{i}(\mathbb{R})\times
C_{j}(\mathbf{\rm H}_{+})$

defined by
$((f_{1},\cdots,f_{m}),\textbf{{u}})\mapsto(\\{x_{1},\cdots,x_{k}\\},\\{\alpha_{1},\cdots,\alpha_{j}\\})$.

Now suppose that $1\leq k\leq\lfloor d/n\rfloor$, $i,j\geq 0$ with $i+j=k$,
and let $c=(\\{x_{1},\cdots,x_{i}\\},\\{\alpha_{1},\cdots,\alpha_{j}\\})\in
C_{i}(\mathbb{R})\times C_{j}(\mathbf{\rm H}_{+})$ be any fixed element.
Consider the fibre $(\pi^{d}_{k;i,j})^{-1}(c)$. It is easy to see that the
condition that a polynomial $f_{s}(z)\in{\rm P}_{d}(\mathbb{R})$ is divisible
by $\prod_{u=1}^{i}(z-x_{u})^{n}$, is equivalent to the following the
condition:

(4.22) $f^{(t)}_{s}(x_{u})=0\quad\mbox{for all }0\leq t<n,\ 1\leq u\leq i.$

In general, for each $0\leq t<n$ and $1\leq u\leq i$, the condition
$f^{(t)}_{s}(x_{u})=0$ gives one linear condition on the coefficients of
$f_{s}(z)$, and this determines an affine hyperplane in ${\rm
P}_{d}(\mathbb{R})\cong\mathbb{R}^{d}$. For example, if
$f_{s}(z)=z^{d}+\sum_{l=1}^{d}a_{l}z^{d-l}$, then $f_{s}(x_{u})=0$ for all
$1\leq u\leq i$ if and only if

$\begin{bmatrix}1&x_{1}&x_{1}^{2}&x_{1}^{3}&\cdots&x_{1}^{d-1}\\\
1&x_{2}&x_{2}^{2}&x_{2}^{3}&\cdots&x_{2}^{d-1}\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
1&x_{i}&x_{i}^{2}&x_{i}^{3}&\cdots&x_{i}^{d-1}\end{bmatrix}\cdot\begin{bmatrix}a_{d}\\\
a_{d-1}\\\ \vdots\\\ a_{1}\end{bmatrix}=-\begin{bmatrix}x_{1}^{d}\\\
x_{2}^{d}\\\ \vdots\\\ x_{i}^{d}\end{bmatrix}.$

Similarly, $f^{\prime}_{s}(x_{u})=0$ for all $1\leq u\leq i$ if and only if

$\begin{bmatrix}0&1&2x_{1}&3x_{1}^{2}&\cdots&(d-1)x_{1}^{d-2}\\\
0&1&2x_{2}&3x_{2}^{2}&\cdots&(d-1)x_{2}^{d-2}\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&1&2x_{i}&3x_{i}^{2}&\cdots&(d-1)x_{i}^{d-2}\end{bmatrix}\cdot\begin{bmatrix}a_{d}\\\
a_{d-1}\\\ \vdots\\\ a_{1}\end{bmatrix}=-\begin{bmatrix}dx_{1}^{d-1}\\\
dx_{2}^{d-1}\\\ \vdots\\\ dx_{i}^{d-1}\end{bmatrix}$

and $f^{\prime\prime}_{s}(x_{u})=0$ for all $1\leq u\leq i$ if and only if

$\begin{bmatrix}0&0&2&6x_{1}&\cdots&(d-1)(d-2)x_{1}^{d-3}\\\
0&0&2&6x_{2}&\cdots&(d-1)(d-2)x_{2}^{d-3}\\\
\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&2&6x_{i}&\cdots&(d-1)(d-2)x_{i}^{d-3}\end{bmatrix}\cdot\begin{bmatrix}a_{d}\\\
a_{d-1}\\\ \vdots\\\ a_{1}\end{bmatrix}=-\begin{bmatrix}d(d-1)x_{1}^{d-2}\\\
d(d-1)x_{2}^{d-1}\\\ \vdots\\\ d(d-1)x_{i}^{d-2}\end{bmatrix}$

and so on. Since $1\leq i\leq k\leq\lfloor d/n\rfloor$ and
$\\{x_{u}\\}_{u=1}^{i}\in C_{i}(\mathbb{R})$, it follows from the properties
of Vandermonde matrices and Gaussian elimination as in the proof of [20, Lemma
3.2] that the the condition (4.22) is equivalent to exactly $ni$ affinely
independent conditions on the coefficients of $f_{s}(z)$. Hence, we see that
the space of $m$-tuples $(f_{1}(z),\cdots,f_{m}(z))\in{\rm
P}_{d}(\mathbb{R})^{m}$ which satisfy the condition (4.22) for each $1\leq
s\leq m$ is the intersection of $mni$ real affine hyperplanes in general
position, and has real codimension $mni$ in ${\rm P}_{d}(\mathbb{R})^{m}$.

Arguing in exactly the same manner, we see that the condition that each
polynomial $f_{s}(z)$ is divisible by $\prod_{t=1}^{j}(z-\alpha_{t})^{n}$ for
each $1\leq s\leq m$, gives a subspace of complex codimension $mnj$ in ${\rm
P}_{d}(\mathbb{C})^{m}$. Since $f_{s}(z)$ is a real coefficient polynomial,
$z=\alpha_{t}$ is a root of $f_{s}(z)$ of multiplicity $n$ if and only if the
same holds for $z=\overline{\alpha}_{t}$. Thus, the fibre
$(\pi_{k;i,j}^{d})^{-1}(c)$ is homeomorphic to the product of an open
$(k-1)$-simplex with the real affine space of dimension $dm-(mni+2mnj)=dm-
mn(k+j)=m(d-nk-nj)$. We can check that local triviality holds. Hence, we see
that ${\mathcal{X}}^{d}_{k}(i,j)$ is a real affine bundle over
$C_{i}(\mathbb{R})\times C_{j}(\mathbf{\rm H}_{+})$ of rank
$l_{d,k}(i,j)=m(d-nk-nj)+k-1$. ∎

######  Lemma 4.8.

If $1\leq k\leq\lfloor d/n\rfloor$ and $mn\geq 3$, there is a natural
isomorphism

$E^{1;d}_{k,s}\cong\Big{(}\bigoplus_{j=1}^{k}\tilde{H}_{s-(mn-1)k}(\Sigma^{(mn-2)j}D_{j};\mathbb{Z})\Big{)}\oplus\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z}).$

###### Proof.

First, consider the case $j=0$. Then $1\leq k=i\leq\lfloor d/n\rfloor$. Since
$C_{k}(\mathbb{R})\cong\mathbb{R}^{k}$, the affine bundle $\xi^{d}_{k}(k,0)$
is trivial. Hence, there is a homeomorphism
${\mathcal{X}}^{d}_{k}(k,0)_{+}\cong(\mathbb{R}^{k}\times\mathbb{R}^{l_{d,k}(k,0)})_{+}=S^{dm-(mn-1)k+k-1}.$
Hence, there is an isomorphism

$\displaystyle H^{dm+k-s-1}_{c}({\mathcal{X}}^{d}_{k}(k,0);\mathbb{Z})$
$\displaystyle\cong\tilde{H}^{dm+k-s-1}(S^{dm-(mn-1)k+k-1};\mathbb{Z})$
$\displaystyle\cong\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z}).$

Next consider the case $j\geq 1$.

$\mbox{Since }\ \begin{cases}dm+k-s-1-l_{d,k}(i,j)&=mn(k+j)-s,\\\
2j-\\{mn(k+j)-s-i\\}&=s-(mn-1)k-(mn-1)j,\end{cases}$

by the Thom isomorphism and Poincare duality, there are isomorphisms

$\displaystyle H^{dm+k-s-1}_{c}({\mathcal{X}}^{d}_{k}(i,j);\mathbb{Z})\cong
H_{c}^{dm+k-s-1-l_{d,k}(i,j)}(C_{i}(\mathbb{R})\times C_{j}(\mathbf{\rm
H}_{+});\pm\mathbb{Z})$
$\displaystyle=H_{c}^{mn(k+j)-s}(C_{i}(\mathbb{R})\times C_{j}(\mathbf{\rm
H}_{+});\pm\mathbb{Z})\cong H_{c}^{mn(k+j)-s-i}(C_{j}(\mathbf{\rm
H}_{+});\pm\mathbb{Z})$
$\displaystyle\cong\tilde{H}_{s-(mn-1)k-(mn-1)j}(C_{j}(\mathbf{\rm
H}_{+});\pm\mathbb{Z})\cong\tilde{H}_{s-(mn-1)k-(mn-1)j}(C_{j}(\mathbb{C});\pm\mathbb{Z}).$

Hence, by (4.20) we have the following isomorphisms

$\displaystyle E^{1;d}_{k,s}$
$\displaystyle=H_{c}^{dm+k-s-1}({\mathcal{X}}^{d}_{k}\setminus{\mathcal{X}}^{d}_{k-1};\mathbb{Z})=\bigoplus_{i+j=k}H_{c}^{dm+k-s-1}({\mathcal{X}}^{d}_{k}(i,j);\mathbb{Z})$
$\displaystyle\cong\Big{(}\bigoplus_{j=1}^{k}\tilde{H}_{s-(mn-1)k-(mn-1)j}(C_{j}(\mathbb{C});\pm\mathbb{Z})\Big{)}\oplus\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z}).$

It follows from [6] that $D_{j}=D_{j}(S^{1})$ is the Thom space of the
following $j$-dimensional vector bundle over $C_{j}(\mathbb{C})$,

(4.23) $F(\mathbb{C},j)\times_{S_{j}}\mathbb{R}^{j}\to
F(\mathbb{C},j)\times_{S_{j}}\\{*\\}=F(\mathbb{C},j)/S_{j}=C_{j}(\mathbb{C}).$

Thus, by the Thom isomorphism theorem, there is an isomorphism

(4.24)
$\tilde{H}_{*+j}(D_{j};\mathbb{Z})\cong\tilde{H}_{*}(C_{j}(\mathbb{C});\pm\mathbb{Z}).$

Hence, we have the isomorphisms

$\displaystyle E^{1;d}_{k,s}$
$\displaystyle\cong\Big{(}\bigoplus_{j=1}^{k}\tilde{H}_{s-(mn-1)k-(mn-1)j}(C_{j}(\mathbb{C});\pm\mathbb{Z})\Big{)}\oplus\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z})$
$\displaystyle\cong\Big{(}\bigoplus_{j=1}^{k}\tilde{H}_{s-(mn-1)k-(mn-2)j}(D_{j};\mathbb{Z})\Big{)}\oplus\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z})$
$\displaystyle\cong\Big{(}\bigoplus_{j=1}^{k}\tilde{H}_{s-(mn-1)k}(\Sigma^{(mn-2)j}D_{j};\mathbb{Z})\Big{)}\oplus\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z}),$

and this completes the proof. ∎

######  Corollary 4.9.

If $mn\geq 3$, there is a natural isomorphism

$E^{1;d}_{k,s}=\begin{cases}\displaystyle
A_{k,s}\oplus\tilde{H}_{s-(mn-1)k}(S^{0};\mathbb{Z})&\mbox{if }1\leq
k\leq\lfloor d/n\rfloor\mbox{ and }s\geq(mn-1)k,\\\ \quad
0&\mbox{otherwise,}\end{cases}$

where $A_{k,s}$ denotes the abelian group defined by

(4.25)
$A_{k,s}=\bigoplus_{j=1}^{k}\tilde{H}_{s-(mn-1)k}(\Sigma^{(mn-2)j}D_{j};\mathbb{Z}).$

###### Proof.

The assertion easily follows from Lemma 4.8. ∎

The following results will be needed in the proof of Theorem 2.11.

######  Lemma 4.10.

If $mn\geq 3$, there is a stable homotopy equivalence

(4.26) $\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})\simeq_{s}\bigvee_{k=1}^{\lfloor
d/n\rfloor}S^{k(mn-2)}.$

###### Proof.

First, consider the case $mn\geq 4$. By (2.17), there is a homotopy
equivalence $\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})\simeq J_{\lfloor
d/n\rfloor}(S^{mn-2})$. From [12] it follows that there is a homotopy
equivalence $\displaystyle\Sigma\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\simeq\bigvee_{k=1}^{\lfloor
d/n\rfloor}S^{k(mn-2)+1},$ and we obtain the stable homotopy equivalence
(4.26).

Next, consider the case $mn=3$, i.e. $(m,n)=(3,1)$ or $(1,3)$.

If $(m,n)=(3,1)$, the assertion easily follows from [31, Theorems A and B]. If
$(m,n)=(1,3)$, then by using Theorem 2.6, [28] and [29, Theorem 3 (page 88)],
we obtain the following result:

  1. $(**)$

The natural map $i^{d,1}_{3}:\mbox{{\rm Q}}^{d,1}_{3}(\mathbb{R})\to\Omega
S^{2}$ is a homology equivalence through dimension $D(d;1,3)=\lfloor
d/3\rfloor$, and $H_{i}(\mbox{{\rm Q}}^{d,1}_{3}(\mathbb{R});\mathbb{Z})=0$
for any $i>\lfloor d/3\rfloor$.

Let us consider the stable map given by the composite of stable maps

$\mbox{{\rm Q}}^{d,1}_{3}(\mathbb{R})\stackrel{{\scriptstyle
i^{d,1}_{3}}}{{\longrightarrow}}\Omega
S^{2}\stackrel{{\scriptstyle\simeq_{s}}}{{\longrightarrow}}\bigvee_{i=1}^{\infty}S^{i}\stackrel{{\scriptstyle
q}}{{\longrightarrow}}\bigvee_{i=1}^{\lfloor d/3\rfloor}S^{i},$

where $q$ is the pinching map. It is easy to see that this map induces an
isomorphism on the homology groups $H_{*}(\ ;\mathbb{Z})$ and hence gives a
stable homotopy equivalence $\displaystyle\mbox{{\rm
Q}}^{d,1}_{3}(\mathbb{R})\simeq_{s}\bigvee_{i=1}^{\lfloor d/3\rfloor}S^{i}$. ∎

######  Definition 4.11.

Let ${\rm P}^{d,m}_{n}$ denote the space given by

(4.27) $\displaystyle{\rm P}^{d,m}_{n}$ $\displaystyle=\mbox{{\rm
Poly}}^{\lfloor d/2\rfloor,m}_{n}(\mathbb{C})\vee B^{d,m}_{n}\vee\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R}),$

where the space $B^{d,m}_{n}$ is given by

(4.28) $\displaystyle B^{d,m}_{n}$ $\displaystyle=\bigvee_{i,j\geq
1,i+2j\leq\lfloor d/n\rfloor}\Sigma^{(mn-2)(i+2j)}D_{j}$

as in (2.19).

######  Lemma 4.12.

If $mn\geq 3$, there is an isomorphism

$E^{1}_{s}:=\bigoplus_{k\in\mathbb{Z}}E^{1;d}_{k,k+s}\cong H_{s}({\rm
P}^{d,m}_{n};\mathbb{Z})\quad\mbox{for any }s\geq 1.$

###### Proof.

Since, in general, the total number of roots of multiplicity $n$ is
$k+j=i+2j$, we only need to consider the case:

(4.29) $i\geq 0,\ j\geq 0,\ i+2j\leq\lfloor d/n\rfloor.$

Suppose that $s\geq 1$. Since $k=i+j$, by (3.19), we have

$\displaystyle\bigoplus_{k\in\mathbb{Z}}A_{k,k+s}$
$\displaystyle=\bigoplus_{i\geq 0,j\geq 1,i+2j\leq\lfloor
d/n\rfloor}\tilde{H}_{s-(mn-2)(i+j)}(\Sigma^{(mn-2)j}D_{j};\mathbb{Z})$
$\displaystyle\cong\bigoplus_{(i,j)\in\mathcal{G}^{d,m}_{n}}\tilde{H}_{s}(\Sigma^{(mn-2)(i+2j)}D_{j};\mathbb{Z}),$

where we set $\mathcal{G}^{d,m}_{n}=\\{(i,j):i\geq 0,j\geq 1,i+2j\leq\lfloor
d/n\rfloor\\}.$

Thus, by Corollary 4.9 and Lemma 4.10,

$\displaystyle E^{1}_{s}$
$\displaystyle=\bigoplus_{k\in\mathbb{Z}}E^{1;d}_{k,k+s}=\bigoplus_{k=1}^{\lfloor
d/n\rfloor}\big{(}A_{k,s}\oplus\tilde{H}_{s-(mn-2)k}(S^{0};\mathbb{Z})\big{)}$
$\displaystyle\cong\big{(}\bigoplus_{(i,j)\in\mathcal{G}^{d,m}_{n}}\tilde{H}_{s}(\Sigma^{(mn-2)(i+2j)}D_{j};\mathbb{Z})\big{)}\oplus\big{(}\bigoplus_{k=1}^{\lfloor
d/n\rfloor}\tilde{H}_{s-(mn-2)k}(S^{0};\mathbb{Z})\big{)}$
$\displaystyle\cong\big{(}\bigoplus_{(i,j)\in\mathcal{G}^{d,m}_{n}}\tilde{H}_{s}(\Sigma^{(mn-2)(i+2j)}D_{j};\mathbb{Z})\big{)}\oplus\tilde{H}_{s}(\bigvee_{k=1}^{\lfloor
d/n\rfloor}S^{(mn-2)k};\mathbb{Z})$
$\displaystyle\cong\Big{(}\bigoplus_{(i,j)\in\mathcal{G}^{d,m}_{n}}\tilde{H}_{s}(\Sigma^{(mn-2)(i+2j)}D_{j};\mathbb{Z})\Big{)}\oplus\tilde{H}_{s}(\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})$
$\displaystyle\cong\tilde{H}_{s}\Big{(}\big{(}\bigvee_{(i,j)\in\mathcal{G}^{d,m}_{n}}\Sigma^{(mn-2)(i+2j)}D_{j}\big{)}\vee\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R});\mathbb{Z}\Big{)}.$

Let $d_{0}=\lfloor d/2\rfloor$ and let $\displaystyle
C^{d,m}_{n}=\bigvee_{(i,j)\in\mathcal{G}^{d,m}_{n}}\Sigma^{(mn-2)(i+2j)}D_{j}.$

Then by Lemma 3.10 and (2.15),

$\displaystyle C^{d,m}_{n}=\bigvee_{i\geq 0,j\geq 1,i+2j\leq\lfloor
d/n\rfloor}\Sigma^{(mn-2)(i+2j)}D_{j}$ $\displaystyle=\Big{(}\bigvee_{i,j\geq
1,i+2j\leq\lfloor
d/n\rfloor}\Sigma^{(mn-2)(i+2j)}D_{j}\Big{)}\vee\Big{(}\bigvee_{2j\leq\lfloor
d/n\rfloor,\ j\geq 1}\Sigma^{2(mn-2)j}D_{j}\Big{)}$
$\displaystyle=B^{d,m}_{n}\vee\Big{(}\bigvee_{j=1}^{\lfloor
d_{0}/n\rfloor}\Sigma^{2(mn-2)j}D_{j}\Big{)}\simeq_{s}B^{d,m}_{n}\vee\mbox{{\rm
Poly}}^{d_{0},m}_{n}(\mathbb{C})$ $\displaystyle=B^{d,m}_{n}\vee\mbox{{\rm
Poly}}^{\lfloor d/2\rfloor,m}_{n}(\mathbb{C}).$

Hence, for $s\geq 1$, there are isomorphisms

$\displaystyle E^{1}_{s}$
$\displaystyle=\bigoplus_{k\in\mathbb{Z}}E^{1;d}_{k,k+s}\cong
H_{s}(C^{d,m}_{n}\vee\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})$
$\displaystyle=H_{s}(B^{d,m}_{n}\vee\mbox{{\rm Poly}}^{\lfloor
d/2\rfloor,m}_{n}(\mathbb{C})\vee\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})=H_{s}({\rm P}^{d,m}_{n};\mathbb{Z})$

and the assertion follows. ∎

##  5 Loop products and stabilization maps

In this section we construct loop products and stabilization maps on the
spaces $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{K})$ and $\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})$, and use them to prove Theorem 5.6.

######  Definition 5.1.

Let
$\varphi:\mathbb{C}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(0,\infty)\times\mathbb{R}$
and
$\psi:\mathbb{C}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(-\infty,0)\times\mathbb{R}$
be any fixed homeomorphisms satisfying the following two conditions:

  1. (5.1.1)

$\varphi(\mathbb{R})=(0,\infty)\times\\{0\\}$ and
$\psi(\mathbb{R})=(-\infty,0)\times\\{0\\}.$

  2. (5.1.2)

$\varphi(\overline{\alpha})=\overline{\varphi(\alpha)}$ and
$\psi(\overline{\alpha})=\overline{\psi(\alpha)}$ for any
$\alpha\in\mathbb{C}$.

For each monic polynomial $f(z)=\prod_{k=1}^{d}(z-x_{k})\in{\rm
P}_{d}(\mathbb{C})$, let $\varphi(f)$ and $\psi(f)$ denote the monic
polynomials of the same degree $d$ given by

(5.1)
$\widetilde{\varphi}(f)=\prod_{k=1}^{d}(z-\varphi(x_{k}))\quad\mbox{and}\quad\widetilde{\psi}(f)=\prod_{k=1}^{d}(z-\psi(x_{k})).$

######  Remark 5.2.

It is easy to see that the following hold:

  1. (i)

If $f(z)\in{\rm P}_{d}(\mathbb{R})$, then $\widetilde{\varphi}(f)\in{\rm
P}_{d}(\mathbb{R})$ and $\widetilde{\psi}(f)\in{\rm P}_{d}(\mathbb{R}).$

  2. (ii)

If $f=(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{K})$ and
$g=(g_{1}(z),\cdots,g_{m}(z))\in\mbox{{\rm Poly}}^{d_{2},m}_{n}(\mathbb{K})$,
$\big{(}\widetilde{\varphi}(f_{1})\widetilde{\psi}(g_{1}),\cdots,\widetilde{\varphi}(f_{m})\widetilde{\psi}(g_{m})\big{)}\in\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{K})$ for $\mathbb{K}=\mathbb{R}$ or
$\mathbb{C}$.

  3. (iii)

If $f=(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm Q}}^{d_{1},m}_{n}(\mathbb{R})$
and $g=(g_{1}(z),\cdots,g_{m}(z))\in\mbox{{\rm Q}}^{d_{2},m}_{n}(\mathbb{R})$,
$\big{(}\widetilde{\varphi}(f_{1})\widetilde{\psi}(g_{1}),\cdots,\widetilde{\varphi}(f_{m})\widetilde{\psi}(g_{m})\big{)}\in\mbox{{\rm
Q}}^{d_{1}+d_{2},m}_{n}(\mathbb{R}).$ ∎

Using an idea from [4, Definition 4.9] and Remark 5.2 we will now define loop
products.

######  Definition 5.3.

(i) Define the loop product

(5.2) $\mu_{d_{1},d_{2}}^{\mathbb{C}}:\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{C})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{C})\to\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{C})\quad\mbox{by}$
$\mu_{d_{1},d_{2}}^{\mathbb{C}}(f,g)=\big{(}\widetilde{\varphi}(f_{1})\widetilde{\psi}(g_{1}),\cdots,\widetilde{\varphi}(f_{m})\widetilde{\psi}(g_{m})\big{)}$

for $(f,g)\in\mbox{{\rm Poly}}^{d_{1},m}_{n}(\mathbb{C})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{C})$, where we write

$(f,g)=((f_{1}(z),\cdots,f_{m}(z)),(g_{1}(z),\cdots,g_{m}(z))).$

It is easy to see that

(5.3) $\mu_{d_{1},d_{2}}^{\mathbb{C}}(\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{R})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R}))\subset\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{R}).$

Hence, one can define the loop product

(5.4) $\mu_{d_{1},d_{2}}^{\mathbb{R}}:\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{R})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R})\to\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{R})$

as the restriction
$\mu_{d_{1},d_{2}}^{\mathbb{R}}=\mu_{d_{1},d_{2}}^{\mathbb{C}}|\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{R})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R}).$

Since the following relation also holds

(5.5) $\mu_{d_{1},d_{2}}^{\mathbb{R}}\big{(}\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\big{)}\subset\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+}),$

define the loop product

(5.6) $\mu_{d_{1},d_{2}}^{\mathbf{\rm H}_{+}}:\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\to\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$

as the restriction $\mu^{\mathbf{\rm
H}_{+}}_{d_{1},d_{2}}=\mu^{\mathbb{R}}_{d_{1},d_{2}}|\mbox{{\rm
Poly}}^{d_{1},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R};\mathbf{\rm H}_{+}).$

(iv) Similarly, we define the loop product

(5.7) $\mu_{d_{1},d_{2}}:\mbox{{\rm
Q}}^{d_{1},m}_{n}(\mathbb{R})\times\mbox{{\rm
Q}}^{d_{2},m}_{n}(\mathbb{R})\to\mbox{{\rm
Q}}^{d_{1}+d_{2},m}_{n}(\mathbb{R})\quad\mbox{by}$
$\mu_{d_{1},d_{2}}(f,g)=\big{(}\widetilde{\varphi}(f_{1})\widetilde{\psi}(g_{1}),\cdots,\widetilde{\varphi}(f_{m})\widetilde{\psi}(g_{m})\big{)}$

for
$(f,g)=((f_{1}(z),\cdots,f_{m}(z)),(g_{1}(z),\cdots,g_{m}(z)))\in\mbox{{\rm
Q}}^{d_{1},m}_{n}(\mathbb{R})\times\mbox{{\rm Q}}^{d_{2},m}_{n}(\mathbb{R})$.

Next, recall the definitions of stabilization maps.

######  Definition 5.4.

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For each integer $d\geq 1$, let
$\\{x_{d,i}:1\leq i\leq m\\}\subset(d,d+1)$ be any fixed real numbers such
that $x_{i}\not=x_{k}$ if $i\not=k$, and let
$\varphi_{d}:\mathbb{C}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbb{C}_{d}=\\{\alpha\in\mathbb{C}:\mbox{Re
}(\alpha)<d\\}$ be any homeomorphism satisfying the following condition:

  1. (5.7.1)

$\varphi_{d}(\mathbb{R})=(-\infty,d)\times\mathbb{R}$ and
$\varphi_{d}(\overline{\alpha})=\overline{\varphi_{d}(\alpha)}$ for any
$\alpha\in\mathbb{C}$, where we identify $\mathbb{C}=\mathbb{R}^{2}$ in a
usual way.

(i) Define the stabilization map $s^{d,m}_{n,\mathbb{K}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{K})\to\mbox{{\rm Poly}}^{d+1,m}_{n}(\mathbb{K})$ by

(5.8)
$s^{d,m}_{n,\mathbb{K}}(f_{1}(z),\cdots,f_{m}(z))=\big{(}(z-x_{d,1})\widetilde{\varphi_{d}}(f_{1}),\cdots,(z-x_{d,m})\widetilde{\varphi_{d}}(f_{m})\big{)}$

for $(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{K})$,
where we set
$\widetilde{\varphi_{d}}(f)=\prod_{k=1}^{d}(z-\varphi_{d}(x_{k}))$ if
$f=f(z)=\prod_{k=1}^{d}(z-x_{k})\in{\rm P}_{d}(\mathbb{K}).$

Note that the map $s^{d,m}_{n,\mathbb{K}}$ depends on the choice of points
$\\{x_{d,k}\\}_{k=1}^{m}$ and the homeomorphism $\varphi_{d}$, but its
homotopy type does not, as in [19, Def. 3.11].

(ii) Let $s^{d,m}_{n}:\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})\to\mbox{{\rm
Q}}^{d+1,m}_{n}(\mathbb{R})$ be the stabilization map given by

(5.9)
$s^{d,m}_{n}(f_{1}(z),\cdots,f_{m}(z))=\big{(}(z-x_{d,1})\widetilde{\varphi_{d}}(f_{1}),\cdots,(z-x_{d,m})\widetilde{\varphi_{d}}(f_{m})\big{)}$

for $(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})$.

(iii) Since one can easily see that $s^{d,m}_{n,\mathbb{R}}(\mbox{{\rm
Poly}}^{d,m}_{n,\mathbf{\rm H}_{+}})\subset\mbox{{\rm
Poly}}^{d+1,m}_{n,\mathbf{\rm H}_{+}}$, we define the stabilization map
$s^{d,m}_{n,\mathbf{\rm H}_{+}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\to\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})$ by the restriction

(5.10) $s^{d,m}_{n,\mathbf{\rm H}_{+}}=s^{d,m}_{n,\mathbb{R}}|\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+}).$

######  Remark 5.5.

It follows from the definitions of (5.8), (5.9) and (5.10) that the following
diagram is commutative:

(5.11) $\begin{CD}\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})@>{\iota^{d,m}_{n,\mathbf{\rm H}_{+}}}>{\subset}>\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})@>{\iota^{d,m}_{n,\mathbb{R}}}>{\subset}>\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\\\ @V{s^{d,m}_{n,\mathbf{\rm
H}_{+}}}V{}V@V{s^{d,m}_{n,\mathbb{R}}}V{}V@V{s^{d,m}_{n}}V{}V\\\ \mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})@>{\iota^{d+1,m}_{n,\mathbf{\rm H}_{+}}}>{\subset}>\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R})@>{\iota^{d+1,m}_{n,\mathbb{R}}}>{\subset}>\mbox{{\rm
Q}}^{d+1,m}_{n}(\mathbb{R})\end{CD}$

Moreover, by using the method invented by C. Boyer and B. Mann [4, Def. 4.9]
we obtain the following two homotopy commutative diagrams:

(5.12) $\begin{CD}\mbox{{\rm Poly}}^{d_{1}.m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})\times\mbox{{\rm Poly}}^{d_{2},m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})@>{\mu_{d_{1},d_{2}}^{\mathbf{\rm H}_{+}}}>{}>\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{R})\\\ @V{\iota^{d_{1},m}_{n,\mathbf{\rm
H}_{+}}\times\iota^{d_{2},m}_{n,\mathbf{\rm
H}_{+}}}V{\bigcap}V@V{\iota^{d_{1}+d_{2},m}_{n,\mathbf{\rm
H}_{+}}}V{\bigcap}V\\\ \mbox{{\rm
Poly}}^{d_{1}.m}_{n}(\mathbb{R})\times\mbox{{\rm
Poly}}^{d_{2},m}_{n}(\mathbb{R})@>{\mu_{d_{1},d_{2}}^{\mathbb{R}}}>{}>\mbox{{\rm
Poly}}^{d_{1}+d_{2},m}_{n}(\mathbb{R})\\\
@V{\iota^{d_{1},m}_{n,\mathbb{R}}\times\iota^{d_{2},m}_{n,\mathbb{R}}}V{\bigcap}V@V{\iota^{d_{1}+d_{2},m}_{n,\mathbb{R}}}V{\bigcap}V\\\
\mbox{{\rm Q}}^{d_{1},m}_{n}(\mathbb{R})\times\mbox{{\rm
Q}}^{d_{2},m}_{n}(\mathbb{R})@>{\mu_{d_{1},d_{2}}}>{}>\mbox{{\rm
Q}}^{d_{1}+d_{2},m}_{n}(\mathbb{R})\\\ @V{i^{d_{1},m}_{n}\times
i^{d_{2},m}_{n}}V{}V@V{i^{d_{1}+d_{2},m}_{n}}V{}V\\\ \Omega
S^{mn-1}\times\Omega S^{mn-1}@>{l_{S}}>{}>\Omega S^{mn-1}\end{CD}$

where $l_{S}$ denotes the loop product on the loop space $\Omega S^{mn-1}$. ∎

Now we are ready to give a proof of the following result.

######  Theorem 5.6.

If $mn\geq 3$, the inclusion map

$\iota^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})$

induces a split epimorphism on the homology group $H_{*}(\ ;\mathbb{Z})$.

###### Proof.

Recall that there is an isomorphism $H_{*}(\Omega
S^{mn-1};\mathbb{Z})=\mathbb{Z}[\iota_{mn-2}]$ for the generator
$\iota_{mn-2}\in H_{mn-2}(\Omega S^{mn-1};\mathbb{Z})\cong\mathbb{Z}$. It
follows from (2.17) that there is a generator $\iota_{Q}\in
H_{mn-2}(\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})\cong\mathbb{Z}$
which satisfies the equality

(5.13) $(i^{mn-2,m}_{n})_{*}(\iota_{Q})=\iota_{mn-2},$

where $n_{0}=\lfloor d/n\rfloor$ and $H_{*}(\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})=\mathbb{Z}[\iota_{Q}]/((\iota_{Q})^{n_{0}+1}).$

Note that if a polynomial $f(z)\in\mathbb{R}[z]$ has a complex root
$\alpha\in\mathbb{C}\setminus\mathbb{R}$, its conjugate $\overline{\alpha}$ is
also a root of $f(z)$. Hence, we see that

(5.14) $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})=\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\quad\mbox{ if $d<2n$.}$

Thus, the inclusion map $\iota^{d,m}_{n,\mathbb{R}}$ is the identity map if
$d<2n$.

Since there is a homotopy equivalence $\mbox{{\rm
Poly}}^{n,m}_{n}(\mathbb{R})\cong\mathbb{R}^{mn}\setminus\mathbb{R}\simeq
S^{mn-2}$, there is a generator $\iota_{P}\in H_{mn-2}(\mbox{{\rm
Poly}}^{n,m}_{n}(\mathbb{R});\mathbb{Z})\cong\mathbb{Z}$ such that

(5.15) $(\iota^{mn-2,m}_{n,\mathbb{R}})_{*}(\iota_{P})=\iota_{Q}.$

Thus, for any $1\leq k\leq d_{0}=\lfloor d/n\rfloor$, by using the diagrams
(5.12) and (5.11) we see that
$(\iota_{Q})^{k}=((\iota^{mn-2,m}_{n,\mathbb{R}})_{*}(\iota_{P}))^{k}=(\iota^{k(mn-2),m}_{n,\mathbb{R}})_{*}((\iota_{P})^{k}).$
However, since $k(mn-2)\leq d$,

$(\iota_{Q})^{k}=(\iota^{k(mn-2),m}_{n,\mathbb{R}})_{*}((\iota_{P})^{k})\in(\iota^{d,m}_{n,\mathbb{R}})_{*}(H_{k(mn-2)}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})).$

Hence, the map $\iota^{d,m}_{n,\mathbb{R}}$ induces a split epimorphism on the
homology group $H_{k}(\ ;\mathbb{Z})$ for any $k$. ∎

##  6 The homology stability theorem

In this section we consider the homology stability of the stabilization maps
$s^{d,m}_{n,\mathbb{R}}$ and prove the homology stability theorem (Theorem
6.2).

Consider the stabilization map $s^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\mbox{{\rm Poly}}^{d+1,m}_{n}(\mathbb{R})$
given by (5.8). Note that the map $s^{d,m}_{n,\mathbb{R}}$ clearly extends to
a map

(6.1) ${\rm P}_{d}(\mathbb{R})^{m}\to{\rm P}_{d+1}(\mathbb{R})^{m}$

and its restriction gives a stabilization map
$\tilde{s}^{d,m}_{n,\mathbb{R}}:\Sigma^{d,m}_{n}\to\Sigma^{d+1,m}_{n}$ between
discriminants. It is easy to see that it also extends to an open embedding

(6.2)
$\tilde{s}^{d,m}_{n,\mathbb{R}}:\Sigma^{d,m}_{n}\times\mathbb{R}^{m}\to\Sigma^{d+1,m}_{n}.$

Since one-point compactification is contravariant for open embeddings, it
induces the map

(6.3)
$(\tilde{s}^{d,m}_{n,\mathbb{R}})_{+}:(\Sigma^{d+1,m}_{n})_{+}\to(\Sigma^{d,m}_{n}\times\mathbb{R}^{m})_{+}=\Sigma^{d,m}_{n+}\wedge
S^{m}$

between one-point compactifications. Thus we see that the following diagram is
commutative

(6.4) $\begin{CD}\tilde{H}_{k}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})@>{{(s^{d,m}_{n,\mathbb{R}})}_{*}}>{}>\tilde{H}_{k}(\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R});\mathbb{Z})\\\
@V{Al}V{\cong}V@V{Al}V{\cong}V\\\
H^{dm-k-1}_{c}(\Sigma^{d,m}_{n};\mathbb{Z})@>{(\tilde{s}^{d,m}_{n,\mathbb{R}+})^{*}}>{}>H^{(d+1)m-k-1}_{c}(\Sigma^{d+1,m}_{n};\mathbb{Z})\end{CD}$

where $Al$ denotes the Alexander duality isomorphism and the homomorphism
$(\tilde{s}^{d,m}_{n,\mathbb{R}+})^{*}$ denotes the composite of the
suspension isomorphism with the homomorphism
$\tilde{s}^{d,m*}_{n,\mathbb{R}+}$,

$H^{dm-k-1}_{c}(\Sigma^{d,m}_{n};\mathbb{Z})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{(d+1)m-k-1}_{c}(\Sigma^{d,m}_{n}\times\mathbb{R}^{m};\mathbb{Z})\stackrel{{\scriptstyle\tilde{s}^{d,m*}_{n,\mathbb{R}+}}}{{\longrightarrow}}H^{(d+1)m-k-1}_{c}(\Sigma^{d+1,m}_{n};\mathbb{Z}).$

The map $\tilde{s}^{d,m}_{n,\mathbb{R}}$ naturally extends to a filtration
preserving open map

(6.5)
$\hat{s}^{d,m}_{n,\mathbb{R}}:{\mathcal{X}}^{d}\times\mathbb{R}^{m}\to\mathcal{X}^{d+1}$

and this extends to a filtration preserving map
$(\hat{s}^{d,m}_{n,\mathbb{R}})_{+}:\mathcal{X}^{d+1}_{+}\to{\mathcal{X}}^{d}_{+}\wedge
S^{m}.$ This map induces a homomorphism of spectral sequences

(6.6) $\\{\theta_{k,s}^{t}:E^{t;d}_{k,s}\to E^{t;d+1}_{k,s}\\}.$

It is easy to see that
$\hat{s}^{d,m}_{n,\mathbb{R}}({\mathcal{X}}^{d}_{k}(i,j)\times\mathbb{R}^{m})\subset\mathcal{X}^{d+1}_{k}(i,j)$
if $1\leq k\leq\lfloor d/n\rfloor$ and $i+j=k$. Hence, if $i+j=k$ and $1\leq
k\leq\lfloor d/n\rfloor$, one can define a map

(6.7)
$\hat{s}^{d,m}_{n;i,j}:{\mathcal{X}}^{d}_{k}(i,j)\times\mathbb{R}^{m}\to\mathcal{X}^{d+1}_{k}(i,j)$

as the restriction
$\hat{s}^{d,m}_{n;i,j}=\hat{s}^{d,m}_{n,\mathbb{R}}|{\mathcal{X}}^{d}_{k}(i,j)$.
Since $\hat{s}^{d,m}_{n;i,j}$ is an open embedding, it induces a map

(6.8)
$\hat{s}^{d,m}_{n;i,j+}:\mathcal{X}^{d+1}_{k}(i,j)_{+}\to\mathcal{X}^{d}_{k}(i,j)_{+}\wedge
S^{m}$

such that the following equality holds:

(6.9) $\displaystyle\theta^{1}_{k,s}$
$\displaystyle=\sum_{i+j=k}(\hat{s}^{d,m}_{n;i,j+})^{*}:E^{1;d}_{k,s}\to
E^{1;d+1}_{k,s}\quad\mbox{if $1\leq k\leq\lfloor\frac{d}{n}\rfloor$, }$ where
$\displaystyle(\hat{s}^{d,m}_{n;i,j+})^{*}$
$\displaystyle:H_{c}^{dm+k-s-1}({\mathcal{X}}^{d}_{k}(i,j);\mathbb{Z})\to
H_{c}^{(d+1)m+k-s-1}(\mathcal{X}^{d+1}_{k}(i,j);\mathbb{Z}),$ and
$\displaystyle\quad\begin{cases}E^{1;d}_{k,s}&=\displaystyle\bigoplus_{i+j=k}H_{c}^{dm+k-s-1}({\mathcal{X}}^{d}_{k}(i,j);\mathbb{Z}),\\\
E^{1;d+1}_{k,s}&=\displaystyle\bigoplus_{i+j=k}H_{c}^{(d+1)m+k-s-1}(\mathcal{X}^{d+1}_{k}(i,j);\mathbb{Z}).\end{cases}$

######  Lemma 6.1.

If $1\leq k\leq\lfloor d/n\rfloor$,
$\theta^{1}_{k,s}:E^{1;d}_{k,s}\stackrel{{\scriptstyle\cong}}{{\rightarrow}}{E}^{1;d+1}_{k,s}$
is an isomorphism for any $s$.

###### Proof.

Suppose that $1\leq k\leq\lfloor d/n\rfloor$ with $i+j=k.$ By (6.9) it
suffices to show that $(\hat{s}^{d,m}_{n;i,j+})^{*}$ is an isomorphism. Note
that the projection $\pi^{d}_{k;i,j}$ (defined in (4.21)) also naturally
extends to a map
$\hat{\pi}^{d}_{k;i,j}:{\mathcal{X}}^{d}_{k}(i,j)\times\mathbb{R}^{m}\to
C_{i}(\mathbb{R})\times\mathbb{C}_{j}(\mathbf{\rm H}_{+})$ and it is easy to
check that the following diagram is commutative.

$\begin{CD}{\mathcal{X}}^{d}_{k}(i,j)\times\mathbb{R}^{m}@>{\hat{\pi}_{k;i,j}^{d}}>{}>C_{i}(\mathbb{R})\times\mathbb{C}_{j}(\mathbf{\rm
H}_{+})\\\ @V{\hat{s}^{d,m}_{n;i,j}}V{}V\|\\\
\mathcal{X}^{d+1}_{k}(i,j)@>{\pi_{k;i,j}^{d+1}}>{}>C_{i}(\mathbb{R})\times
C_{j}(\mathbf{\rm H}_{+})\end{CD}$

It follows from the naturality of the Thom isomorphism that the homomorphism
$(\hat{s}^{d,m}_{n;i,j+})^{*}$ is indeed an isomorphism if $1\leq k\leq\lfloor
d/n\rfloor$. ∎

Now we can prove the following result.

######  Theorem 6.2.

If $mn\geq 3$, the stabilization map

$s^{d,m}_{n,\mathbb{R}}:\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})\to\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R})$

is a homology equivalence for $\lfloor d/n\rfloor=\lfloor(d+1)/n\rfloor$, and
a homology equivalence through dimension $D(d;m,n)$ for $\lfloor
d/n\rfloor<\lfloor(d+1)/n\rfloor$, where $D(d;m,n)$ denotes the positive
integer given by (2.16).

###### Proof.

First, consider the case $\lfloor d/n\rfloor=\lfloor(d+1)/n\rfloor$. In this
case, by using Corollary 4.9 and Lemma 6.1 it is easy to show that
$\theta^{1}_{k,s}:E^{1;d}_{k,s}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}E^{1;d+1}_{k,s}$
is an isomorphism for any $(k,s)$. Hence, $\theta^{\infty}_{k,s}$ is an
isomorphism for any $(k,s).$ Since $\theta^{t}_{k,s}$ is induced from
$\hat{s}^{d,m}_{n,\mathbb{R}}$, it follows from (6.4) that the map
$s^{d,m}_{n,\mathbb{R}}$ is a homology equivalence.

Next assume that $\lfloor d/n\rfloor<\lfloor(d+1)/n\rfloor$, i.e.
$\lfloor(d+1)/n\rfloor=\lfloor d/n\rfloor+1$. and let $\epsilon\in\\{0,1\\}$.
In this case, by considering the differential $d^{t}:E^{t;d+\epsilon}_{k,s}\to
E^{t;d+\epsilon}_{k+t,s+t-1}$, Lemma 6.1 and Corollary 4.9, one can show that
$\theta^{t}_{k,s}:E^{t;d}_{k,s}\to E^{t;d+1}_{k,s}$ is an isomorphism for any
$(k,s)$ and any $t\geq 1$ as long as the condition $s-t\leq D(d;m,n)$ is
satisfied. Hence, if $s-t\leq D(d;m,n)$, $\theta^{\infty}_{k,s}$ is always an
isomorphism and so the map $s^{d,m}_{n,\mathbb{R}}$ is a homology equivalence
through dimension $D(d;m,n)$. ∎

The following two results will be needed for the proof of Theorem 7.11.

######  Lemma 6.3.

The space $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ is simply connected if
$mn\geq 4$, and $\pi_{1}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R}))=\mathbb{Z}$
if $mn=3$ and $d\geq n$.

###### Proof.

We use the description of the fundamental group in terms of braids as in [20,
Lemma 3.5]. We represent elements of the group as strings of $m$-different
colors, which can move continuously, with only the following case not allowed
to occur:

  1. $(*)$

All strings of multiplicity $\geq n$ of $m$-different colors pass through a
single point.

Since $mn\geq 3$, this representation shows that $\pi_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}))$ is an abelian group. Hence, there is an
isomorphism $\pi_{1}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R}))\cong
H_{1}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})$. Now consider the
spectral sequence (4.16). Then, by Corollary 4.9 , we see that

$E^{1;d}_{k,k+1}=\begin{cases}\mathbb{Z}&\mbox{ if }k=1\mbox{ and }mn=3,\\\
0&\mbox{ otherwise}.\end{cases}$

First, consider the case $mn\geq 4$. We see that $H_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})=0$, and the space $\mbox{{\rm
Poly}}^{d,m}_{n}(|\mathbb{R})$ is simply connected.

Next, consider the case $mn=3$. Since $E^{1;d}_{2,2}=0$ by Corollary 4.9, by
considering the differential $d^{t}:E^{t;d}_{k,s}\to E^{t;d}_{k+t,s+t+1}$, we
see that
$\bigoplus_{k\in\mathbb{Z}}E^{1;d}_{k,k+1}=E^{1;d}_{1,2}=\mathbb{Z}=E^{\infty;d}_{1,2}.$
Hence, if $mn=3$,

$\pi_{1}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R}))\cong H_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})\cong\mathbb{Z}.$

This completes the proof. ∎

######  Corollary 6.4.

If $mn\geq 4$, the stabilization map

$\mbox{s}^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\mbox{{\rm Poly}}^{d+1,m}_{n}(\mathbb{R})$

is a homotopy equivalence if $\lfloor d/n\rfloor=\lfloor(d+1)/n\rfloor$ and a
homotopy equivalence through dimension $D(d;m,n)$ otherwise.

###### Proof.

This follows from Theorem 6.2 and Lemma 6.3. ∎

######  Definition 6.5.

Let $D=(d_{1},\cdots,d_{m})\in\mathbb{N}^{m}$ be an $m$-tuple of positive
integers. Let $x_{d}\in(d,d+1)$ be any fixed real number and let
$\varphi_{d}:\mathbb{C}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbb{C}_{d}=\\{\alpha\in\mathbb{C}:\mbox{Re
}(\alpha)<d\\}$ be any fixed homeomorphism satisfying the condition (5.4.1).

Then for each $1\leq i\leq m$ and $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, let

(6.10) $s^{D,i;m}_{n,\mathbb{K}}:\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{m};m}_{n}(\mathbb{K})\to\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{i-1},d_{i}+1,d_{i+1},\cdots,d_{m};m}_{n}(\mathbb{K})$

denote the stabilization map defined by

$s^{D,i;m}_{n,\mathbb{K}}(f)=\big{(}\widetilde{\varphi_{d}}(f_{1}),\cdots,\widetilde{\varphi_{d}}(f_{i-1}),(z-x_{d})\widetilde{\varphi_{d}}(f_{i}),\widetilde{\varphi_{d}}(f_{i+1}),\cdots,\widetilde{\varphi_{d}}(f_{m})\big{)}$

for $f=(f_{1}(z),\cdots,f_{m}(z))\in\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{m};m}_{n}(\mathbb{K})$.

######  Theorem 6.6.

Let $1\leq i\leq m$, $mn\geq 3$ and $D=(d_{1},\cdots,d_{m})\in\mathbb{N}^{m}$
be an $m$-tuple of positive integers.

(i) The stabilization map

$s^{D,i;m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{m};m}_{n}(\mathbb{R})\to\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{i-1},d_{i}+1,d_{i+1},\cdots,d_{m};m}_{n}(\mathbb{R})$

is a homology equivalence if $\lfloor
d_{i}/n\rfloor=\lfloor(d_{i}+1)/n\rfloor$ and a homology equivalence through
dimension $D(d_{i};m,n)$ otherwise.

(ii) The stabilization map

$s^{D,i;m}_{n,\mathbb{C}}:\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{m};m}_{n}(\mathbb{C})\to\mbox{{\rm
Poly}}^{d_{1},\cdots,d_{i-1},d_{i}+1,d_{i+1},\cdots,d_{m};m}_{n}(\mathbb{C})$

is a homotopy equivalence if $\lfloor
d_{i}/n\rfloor=\lfloor(d_{i}+1)/n\rfloor$ and a homotopy equivalence through
dimension $D(d_{i};m,n;\mathbb{C})$ otherwise.

###### Proof.

The assertion (i) can be proved completely same way as that of Theorem 6.2 and
the assertion (ii) can be proved analogously to [18, Theorem 1.8]. ∎

##  7 Configuration spaces and scanning maps

In this section we define the “horizontal scanning maps”and then use them to
prove our stable results (Theorems 7.5 and 7.9).

######  Definition 7.1.

For a space $X$, let $\mbox{{\rm SP}}^{d}(X)$ denote the $d$-th symmetric
product defined by the quotient space

(7.1) $\mbox{{\rm SP}}^{d}(X)=X^{d}/S_{d},$

where the symmetric group $S_{d}$ of $d$ letters acts on $X^{d}$ by the
permutation of coordinates. Since $F(X,d)$ is an $S_{d}$-invariant subspace of
$X^{d}$ and $C_{d}(X)=F(X,d)/S_{d}$, there is a natural inclusion
$C_{d}(X)\subset\mbox{{\rm SP}}^{d}(X).$

Note that an element $\alpha\in\mbox{{\rm SP}}^{d}(X)$ may be identified with
the formal linear combination

(7.2) $\alpha=\sum_{i=1}^{k}d_{i}x_{i},\quad\quad\mbox{where
}\\{x_{i}\\}_{i=1}^{k}\in C_{k}(X)\ \ \mbox{and }\ \sum_{i=1}^{k}d_{i}=d.$

We shall refer to $\alpha$ as a configuration (or $0$-cycle) having a
multiplicity $d_{i}$ at the point $x_{i}$.

######  Remark 7.2.

(i) If we use the notation (7.2), the space ${\rm P}_{d}(\mathbb{C})$ can be
easily identified with the space $\mbox{{\rm SP}}^{d}(\mathbb{C})$ by the
homeomorphism $\Phi_{d}:{\rm
P}_{d}(\mathbb{C})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mbox{{\rm
SP}}^{d}(\mathbb{C})$ defined by

(7.3)
$\Phi_{d}\Big{(}\prod_{i=1}^{k}(z-\alpha_{i})^{d_{i}}\Big{)}=\sum_{i=1}^{k}d_{i}\alpha_{i}.$

(ii) Since there is a natural inclusion ${\rm P}_{d}(\mathbb{R})\subset{\rm
P}_{d}(\mathbb{C})$, one can define a subspace $\mbox{{\rm
SP}}^{d}_{\mathbb{R}}\subset\mbox{{\rm SP}}^{d}(\mathbb{C})$ as the image

(7.4) $\mbox{{\rm SP}}^{d}_{\mathbb{R}}=\Phi_{d}({\rm P}_{d}(\mathbb{R})).$

Then the restriction gives a homeomorphism

(7.5) $\Phi_{d}|{\rm P}_{d}(\mathbb{R}):{\rm
P}_{d}(\mathbb{R})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mbox{{\rm
SP}}^{d}_{\mathbb{R}}.$

Note that any element $\alpha\in\mbox{{\rm SP}}^{d}_{\mathbb{R}}$ can be
represented as the formal sum

(7.6)
$\alpha=\sum_{k=1}^{s}d_{k}x_{k}+\sum_{j=1}^{t}e_{j}(y_{j}+\overline{y_{j}})\quad(d_{k},e_{j}\in\mathbb{N},\
x_{k}\in\mathbb{R},\ y_{j}\in\mathbf{\rm H}_{+}),$

where $x_{k}\not=x_{l}$ if $k\not=l$, $y_{j}\not=y_{i}$ if $j\not=i$ and
$\sum_{k=1}^{s}d_{k}+2\sum_{j=1}^{t}e_{j}=d.$

######  Definition 7.3.

Let $X$ be a space and $D=(d_{1},\cdots,d_{m})\in(\mathbb{Z}_{\geq 0})^{m}$ be
an $m$-tuple of non-negative integers.

(i) Let $\mbox{{\rm SP}}^{0}(X)=\\{\emptyset\\}$ and let $\mbox{{\rm
SP}}^{D}(X)$ denote the space given by

(7.7) $\mbox{{\rm SP}}^{D}(X)=\mbox{{\rm
SP}}^{d_{1},\cdots,d_{m}}(X)=\mbox{{\rm SP}}^{d_{1}}(X)\times\mbox{{\rm
SP}}^{d_{2}}(X)\times\cdots\times\mbox{{\rm SP}}^{d_{m}}(X).$

We define the space $\mbox{{\rm Pol}}^{D;m}_{n}(X)=\mbox{{\rm
Pol}}^{d_{1},\cdots,d_{m};m}_{n}(X)\subset\mbox{{\rm SP}}^{D}(X)$ by

(7.8) $\mbox{{\rm Pol}}^{D;m}_{n}(X)=\mbox{{\rm
Pol}}^{d_{1},\cdots,d_{m};m}_{n}(X)=\\{(\xi_{1},\cdots,\xi_{m})\in\mbox{{\rm
SP}}^{D}(X):(*)_{n}\\},$

where the condition $(*)_{n}$ is given by

  1. $(*)_{n}$:

$\cap_{i=1}^{m}\xi_{i}$ does not contain any element of multiplicity $\geq n$.

(ii) When $X\subset\mathbb{C}$, let $\mbox{{\rm SP}}^{D}_{\mathbb{R}}(X)$
denote the space defined by

(7.9) $\mbox{{\rm SP}}^{D}_{\mathbb{R}}(X)=(\mbox{{\rm
SP}}^{d_{1}}_{\mathbb{R}}\times\mbox{{\rm
SP}}^{d_{2}}_{\mathbb{R}}\times\cdots\times\mbox{{\rm
SP}}^{d_{m}}_{\mathbb{R}})\cap\mbox{{\rm SP}}^{D}(X).$

We define the space
$\mathcal{Q}^{D;m}_{n}(X)=\mathcal{Q}^{d_{1},\cdots,d_{m};m}_{n}(X)\subset\mbox{{\rm
SP}}^{D}_{\mathbb{R}}(X)$ by

(7.10)
$\mathcal{Q}^{D;m}_{n}(X)=\mathcal{Q}^{d_{1},\cdots,d_{m};m}_{n}(X)=\\{(\xi_{1},\cdots,\xi_{m})\in\mbox{{\rm
SP}}^{D}_{\mathbb{R}}(X):(*)_{n}^{\mathbb{R}}\\},$

where the condition $(*)_{n}^{\mathbb{R}}$ is given by

  1. $(*)_{n}^{\mathbb{R}}:$

$(\cap_{i=1}^{m}\xi_{i})\cap\mathbb{R}$ does not contain any element of
multiplicity $\geq n$.

In particular, when $D_{m}=(d,\cdots,d)\in\mathbb{N}^{m}$ $(m$-times), we
write

(7.11) $\mbox{{\rm Pol}}^{d,m}_{n}(X)=\mbox{{\rm Pol}}^{D_{m};m}_{n}(X)\ \
\mbox{ and }\ \ \mathcal{Q}^{d,m}_{n}(X)=\mathcal{Q}^{D_{m};m}_{n}(X).$

(iii) If $A\subset X$ is a closed subspace, we define the space $\mbox{{\rm
Pol}}^{D;m}_{n}(X,A)$ by

(7.12) $\mbox{{\rm Pol}}^{D;m}_{n}(X,A)=\mbox{{\rm
Pol}}^{d_{1},\cdots,d_{m};m}_{n}(X,A)=\mbox{{\rm Pol}}^{D;m}_{n}(X)/\sim,$

where the equivalence relation “$\sim$”is defined by

$(\xi_{1},\cdots,\xi_{m})\sim(\eta_{1},\cdots,\eta_{m})\ \mbox{ if }\
\xi_{i}\cap(X\setminus A)=\eta_{i}\cap(X\setminus A)$

for all $1\leq i\leq m$. Therefore, points in $A$ are ignored. When
$A\not=\emptyset$, there is a natural inclusion

$\mbox{{\rm
Pol}}^{d_{1},\cdots,d_{i},\cdots,d_{m};m}_{n}(X,A)\subset\mbox{{\rm
Pol}}^{d_{1},\cdots,d_{i}+1,\cdots,d_{m};m}_{n}(X,A)$

by adding a point of $A$ into the $i$-th part. Let $\mbox{{\rm
Pol}}^{0,\cdots,0;m}_{n}(X,A)=\\{\emptyset\\}$ and define the space
$\mbox{{\rm Pol}}^{m}_{n}(X,A)$ by the union

(7.13) $\mbox{{\rm Pol}}^{m}_{n}(X,A)=\bigcup_{d_{1}\geq 0,\cdots,d_{m}\geq
0}\mbox{{\rm Pol}}^{d_{1},\cdots,d_{m};m}_{n}(X,A).$

(iv) If $X\subset\mathbb{C}$ and $A\subset X$ is a closed subspace, we also
define the space $\mathcal{Q}^{D;m}_{n}(X,A)$ by

(7.14)
$\mathcal{Q}^{D;m}_{n}(X,A)=\mathcal{Q}^{d_{1},\cdots,d_{m};m}_{n}(X,A)=\mathcal{Q}^{D;m}_{n}(X)/\sim,$

where the equivalence relation “$\sim$”is defined by

$(\xi_{1},\cdots,\xi_{m})\sim(\eta_{1},\cdots,\eta_{m})\ \mbox{ if }\
\xi_{i}\cap(X\setminus A)\cap\mathbb{R}=\eta_{i}\cap(X\setminus
A)\cap\mathbb{R}$

for all $1\leq i\leq m$. Again, when $A\not=\emptyset$, there is a natural
inclusion

$\mathcal{Q}^{d_{1},\cdots,d_{i},\cdots,d_{m};m}_{n}(X,A)\subset\mathcal{Q}^{d_{1},\cdots,d_{i}+1,\cdots,d_{m};m}_{n}(X,A)$

by adding a point of $A$ into the $i$-th part. Let
$\mathcal{Q}^{0,\cdots,0;m}_{n}(X,A)=\\{\emptyset\\},$ and define the space
$\mathcal{Q}^{m}_{n}(X,A)$ by the union

(7.15) $\mathcal{Q}^{m}_{n}(X,A)=\bigcup_{d_{1}\geq 0,\cdots,d_{m}\geq
0}\mathcal{Q}^{d_{1},\cdots,d_{m};m}_{n}(X,A).$

(v) Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and define the stabilized
spaces $\mbox{{\rm Poly}}^{\infty,m}_{n}(\mathbb{K})$ and $\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})$ by the colimits

(7.16) $\mbox{{\rm Poly}}^{\infty,m}_{n}(\mathbb{K})=\displaystyle\
\lim_{d\to\infty}\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{K}),\quad\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})=\displaystyle\ \lim_{d\to\infty}\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})$

taken over the stabilization maps $\\{s^{d,m}_{n,\mathbb{K}}\\}$ and
$\\{s^{d,m}_{n}\\}$ given by (5.8) and (5.9), respectively.

We will need two kinds of horizontal scanning maps. First, we define the
scanning map for a configuration space of particles. From now on, we make the
identification $\mathbb{C}=\mathbb{R}^{2}$ by identifying $\mathbb{C}\ni
x+\sqrt{-1}y$ with $(x,y)\in\mathbb{R}^{2}$ in a usual way.

######  Definition 7.4.

For a rectangle $X$ in $\mathbb{C}=\mathbb{R}^{2}$, let $\sigma X$ denote the
union of the sides of $X$ which are parallel to the $y$-axis, and for a
subspace $Z\subset\mathbb{C}=\mathbb{R}^{2}$, let $\overline{Z}$ denote the
closure of $Z$.

For example, if $X=[a,b]\times[c,d]=\\{(t,s)\in\mathbb{R}^{2}:a\leq t\leq
b,c\leq s\leq d\\}$, then $\sigma
X=\\{a,b\\}\times[c,d]=\\{(t,s)\in\mathbb{R}^{2}:t\in\\{a,b\\},c\leq s\leq
d\\}.$

From now on, let $I$ denote the interval $I=[-1,1]$ and let $\epsilon>0$ be a
fixed positive real number. For each $x\in\mathbb{R}$, let $V(x)$ be the set
defined by

(7.17) $\displaystyle V(x)$
$\displaystyle=\\{w\in\mathbb{C}:|\mbox{Re}(w)-x|<\epsilon,|\mbox{Im}(w)|<1\\}$
$\displaystyle=(x-\epsilon,x+\epsilon)\times(-1,1),$

and let us identify $I\times I=I^{2}$ with the closed unit rectangle

$\\{t+s\sqrt{-1}\in\mathbb{C}:-1\leq t,s\leq 1\\}\subset\mathbb{C}.$

(i) First, we define the horizontal scanning map

(7.18) $sc^{d,m}_{n}:\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\to\Omega\mathcal{Q}^{m}_{n}(I^{2},\partial I\times
I)=\Omega\mathcal{Q}^{m}_{n}(I^{2},\sigma I^{2})$

of the space $\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})$ as follows.

For each $m$-tuple
$\alpha=(\xi_{1},\cdots,\xi_{m})\in\mathcal{Q}^{d,m}_{n}(\mathbb{R})$ of
configurations, let
$sc^{d,m}_{n}(\alpha):\mathbb{R}\to\mathcal{Q}^{d,m}_{n}(I^{2},\partial
I\times I)=\mathcal{Q}^{d,m}_{n}(I^{2},\sigma I^{2})$ denote the map given by

$\displaystyle\mathbb{R}\ni x$
$\displaystyle\mapsto(\xi_{1}\cap\overline{V}(x),\cdots,\xi_{m}\cap\overline{V}(x))\in\mathcal{Q}^{m}_{n}(\overline{V}(x),\sigma\overline{V}(x))\cong\mathcal{Q}^{m}_{n}(I^{2},\sigma
I^{2}),\quad$

where we use the canonical identification
$(\overline{V}(x),\sigma\overline{V}(x))\cong(I^{2},\sigma I^{2}).$

Since
$\displaystyle\lim_{x\to\pm\infty}sc^{d,m}_{n}(\alpha)(x)=(\emptyset,\cdots,\emptyset)$,
by setting $sc^{d,m}_{n}(\alpha)(\infty)=(\emptyset,\cdots,\emptyset)$ we
obtain a based loop
$sc^{d,m}_{n}(\alpha)\in\Omega\mathcal{Q}^{m}_{n}(I^{2},\sigma I^{2}),$ where
we identify $S^{1}=\mathbb{R}\cup\infty$ and we choose the empty configuration
$(\emptyset,\cdots,\emptyset)$ as the base point of $\mbox{{\rm
Q}}^{m}_{n}(I^{2},\sigma I^{2})$.

If we identify $\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\cong\mathcal{Q}^{d,m}_{n}(\mathbb{R})=\mathcal{Q}^{D_{m};m}_{n}(\mathbb{R})$,
finally we obtain the map $sc^{d,m}_{n}:\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\to\Omega\mathcal{Q}^{m}_{n}(I^{2},\sigma I^{2}).$

Since $sc^{d+1,m}_{n}\circ s^{d,m}_{n}\simeq sc^{d,m}_{n}$ (up to homotopy
equivalence), by setting $\displaystyle S=\lim_{d\to\infty}sc^{d,m}_{n}$, we
also obtain the stable horizontal scanning map

(7.19) $S:\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})\to\Omega\mathcal{Q}^{m}_{n}(I^{2},\partial
I\times I)=\Omega\mathcal{Q}^{m}_{n}(I^{2},\sigma I^{2}),$

where $\mbox{{\rm Q}}^{\infty,m}_{n}(\mathbb{R})$ denotes the stabilized space
given by (7.16).

(ii) Next, define the horizontal scanning map

(7.20) $sc^{d,m}_{n,\mathbb{C}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\to\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\partial
I\times I)=\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma I^{2})$

of the space $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ as follows.

For each $m$-tuple $\alpha=(\xi_{1},\cdots,\xi_{m})\in\mbox{{\rm
Pol}}^{d,m}_{n}(\mathbb{C})$ of configurations, let
$sc^{d,m}_{n,\mathbb{C}}(\alpha):\mathbb{R}\to\mbox{{\rm
Pol}}^{d,m}_{n}(I^{2},\partial I\times I)=\mbox{{\rm
Pol}}^{d,m}_{n}(I^{2},\sigma I^{2})$ denote the map given by

$\displaystyle\mathbb{R}\ni x$
$\displaystyle\mapsto(\xi_{1}\cap\overline{V}(x),\cdots,\xi_{m}\cap\overline{V}(x))\in\mbox{{\rm
Pol}}^{m}_{n}(\overline{V}(x),\sigma\overline{V}(x))\cong\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2}),\quad$

where we use the canonical identification
$(\overline{V}(x),\sigma\overline{V}(x))\cong(I^{2},\sigma I^{2}).$

Since
$\displaystyle\lim_{x\to\pm\infty}sc^{d,m}_{n,\mathbb{C}}(\alpha)(x)=(\emptyset,\cdots,\emptyset)$,
by setting
$sc^{d,m}_{n,\mathbb{C}}(\alpha)(\infty)=(\emptyset,\cdots,\emptyset)$ we
obtain a based loop $sc^{d,m}_{n,\mathbb{C}}(\alpha)\in\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2}),$ where we identify
$S^{1}=\mathbb{R}\cup\infty$ and we choose the empty configuration
$(\emptyset,\cdots,\emptyset)$ as the base point of $\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2})$.

If we identify $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{C})\cong\mbox{{\rm
Pol}}^{d,m}_{n}(\mathbb{C})=\mbox{{\rm Pol}}^{D_{m};m}_{n}(\mathbb{C})$,
finally we obtain the map $sc^{d,m}_{n,\mathbb{C}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})\to\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2}).$

Since $sc^{d+1,m}_{n,\mathbb{C}}\circ s^{d,m}_{n,\mathbb{C}}\simeq
sc^{d,m}_{n,\mathbb{C}}$ (up to homotopy equivalence), by setting
$\displaystyle S_{\mathbb{C}}=\lim_{d\to\infty}sc^{d,m}_{n,\mathbb{C}}$ we
obtain the stable horizontal scanning map

(7.21) $S_{\mathbb{C}}:\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{C})\to\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2}),$

where $\mbox{{\rm Poly}}^{\infty,m}_{n}(\mathbb{C})$ denotes the stabilized
space given by (7.16). Although the scanning map itself depends on the choice
of $\epsilon$, its homotopy class does not.

(iii) Let $\mathbb{Z}_{2}=\\{\pm 1\\}$ denote the multiplicative cyclic group
of order $2$. Complex conjugation in $\mathbb{C}$ naturally induces a
$\mathbb{Z}_{2}$-action on $\mbox{{\rm Poly}}^{D;m}_{n}(\mathbb{C})$. It is
easy to see that its fixed point set is $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{C})^{\mathbb{Z}_{2}}=\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$. Since the stabilization maps
$\\{s^{d,m}_{n,\mathbb{C}}\\}_{d\geq 1}$ are $\mathbb{Z}_{2}$-equivariant
maps, $\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{C})^{\mathbb{Z}_{2}}=\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{R})$. Moreover, since the scanning maps
$\\{sc^{d,m}_{n,\mathbb{C}}\\}_{d\geq 1}$ are also
$\mathbb{Z}_{2}$-equivariant maps, by setting
$S_{\mathbb{R}}=S_{\mathbb{C}}|\mbox{{\rm Poly}}^{\infty,m}_{n}(\mathbb{R})$
we also obtain the the stable horizontal scanning map

(7.22) $S_{\mathbb{R}}:\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{R})\to\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2})^{\mathbb{Z}_{2}}.$

(iv) We define the map

(7.23) $r_{I}:\mathcal{Q}^{m}_{n}(I^{2},\sigma
I^{2})\to\mathcal{Q}^{m}_{n}(I,\partial I)$

by the restriction
$r_{I}(\xi_{1},\cdots,\xi_{m})=(\xi_{1}\cap\mathbb{R},\cdots,\xi_{m}\cap\mathbb{R})$
for $(\xi_{1},\cdots,\xi_{m})\in\mathcal{Q}^{m}_{n}(I^{2},\sigma I^{2})$.

######  Theorem 7.5 ([25] (cf. [9]).

If $mn\geq 3$, the stable horizontal scanning maps

$\begin{cases}S\ :&\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega\mathcal{Q}^{m}_{n}(I^{2},\sigma
I^{2})\\\ S_{\mathbb{C}}:&\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{C})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2})\\\ S_{\mathbb{R}}:&\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2})^{\mathbb{Z}_{2}}\end{cases}$

are homotopy equivalences.

###### Proof.

The assertions can be proved by using the idea indicated in [25, Proposition
3.2, Lemma 3.4] (and also in [9, Proposition 2]). However, since the argument
in [25] is very sketchy, we provided a detailed proof for the map $S$ in [20,
Theorem 5.6].777Although the proof of [20, Theorem 5.6] was given under the
condition $mn\geq 4$, the same proof works for the case $mn=3$.  The proof for
the map $S_{\mathbb{C}}$ carries over word for word to the present case, if we
replace the condition “$(*)_{n}^{\mathbb{R}}$ ”by the condition “$(*)_{n}$ ”.
A very similar argument also works the case of the map $S_{\mathbb{R}}$, and
this completes the proof. ∎

######  Corollary 7.6.

If $mn\geq 3$, the map $S_{\mathbb{C}}:\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{C})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2})$ is a $\mathbb{Z}_{2}$-equivariant homotopy
equivalence.

###### Proof.

Consider the $\mathbb{Z}_{2}$-action on the space $\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{C})$ induced from the conjugation on
$\mathbb{C}$. If we consider $S^{1}$ as a $\mathbb{Z}_{2}$-space with trivial
$\mathbb{Z}_{2}$-action, we see that $(\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma
I^{2}))^{\mathbb{Z}_{2}}=\Omega\mathcal{Q}^{m}_{n}(I^{2},\sigma
I^{2})^{\mathbb{Z}_{2}}.$ Since
$(S_{\mathbb{C}})^{\mathbb{Z}_{2}}=S_{\mathbb{R}}$, the assertion easily
follows from Theorem 7.5. ∎

######  Lemma 7.7.

The map $r_{I}:\mathcal{Q}^{m}_{n}(I^{2},\sigma
I^{2})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathcal{Q}^{m}_{n}(I,\partial
I)$ is a deformation retraction.

###### Proof.

We identify $I^{2}=\\{a+b\sqrt{-1}\in\mathbb{C}:-1\leq a,b\leq
1\\}\subset\mathbb{C}$ as before. Let $\Pi\subset I^{2}$ denote the subspace
defined by $\Pi=\\{a+b\sqrt{-1}\in I^{2}:b\in\\{0,\pm\frac{1}{2}\\}\\}.$ For
$b\in\mathbb{R}$, let $\epsilon(b)=\frac{b}{|b|}$ if $b\not=0$ and
$\epsilon(0)=0$. Now consider the homotopy $\varphi:I^{2}\times[0,1]\to I^{2}$
given by $\varphi(\alpha,t)=a+\\{(1-t)b+\frac{\epsilon(b)t}{2}\\}\sqrt{-1}$
for $\alpha=a+b\sqrt{-1}\in I^{2}$ $(a,b\in\mathbb{R})$. By means of this
homotopy, one can define a deformation retraction
$R:\mathcal{Q}^{m}_{n}(I^{2},\sigma
I^{2})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathcal{Q}^{m}_{n}(\Pi,\partial
I\times\\{0,\pm\frac{1}{2}\\}).$

Next, by using the homotopy given by $f_{t}(a+b\sqrt{-1})=ta+(1-t)+b\sqrt{-1}$
if $b=\pm\frac{1}{2}$ and $f_{t}(a+b\sqrt{-1})=a$ if $b=0$, one can also
define a deformation retraction $\varphi:\mathcal{Q}^{m}_{n}(\Pi,\partial
I\times\\{0,\pm\frac{1}{2}\\})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathcal{Q}^{m}_{n}(I,\partial
I).$ Since $r_{I}=\varphi\circ R$, it is also a deformation retraction. ∎

######  Proposition 7.8.

  1. (i)

If $mn\geq 3$, there is a homotopy equivalence

$\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma I^{2})\simeq\Omega^{2}S^{2mn-1}.$

  2. (ii)

There is a map

$\Omega\mathcal{Q}^{m}_{n}(I,\partial
I)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega S^{mn-1}$

which is a homotopy equivalence if $mn\geq 4$ and a homology equivalence if
$mn=3$.

###### Proof.

(i) It follows from [18, Theorem 1.8] that there is a homotopy equivalence
$\mbox{{\rm Poly}}^{\infty,m}_{n}(\mathbb{C})\simeq\Omega^{2}S^{2mn-1}$. Since
the map $S_{\mathbb{C}}$ is a homotopy equivalence by Theorem 7.5, we obtain
the homotopy equivalence $\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2})\simeq\Omega^{2}S^{2mn-1}.$

(ii) Consider the map $\hat{S}:\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle}}{{\longrightarrow}}\Omega\mathcal{Q}^{m}_{n}(I,\partial
I)$ defined as the composite of maps $\hat{S}=(\Omega r_{I})\circ S$. It
follows from Theorem 7.5 and Lemma 7.7 that $\hat{S}$ is a homotopy
equivalence. Moreover, it follows from Theorem 2.6 that there is a map
$\displaystyle i^{\infty,m}_{n}=\lim_{d\to\infty}i^{d,m}_{n}:\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega
S^{mn-1}$ which is a homotopy equivalence for $mn\geq 4$ and is a homology
equivalence for $mn=3$. Now consider the composite map

$\begin{CD}\Omega\mathcal{Q}^{m}_{n}(I,\partial
I)@>{\tilde{S}}>{\simeq}>\mbox{{\rm
Q}}^{\infty,m}_{n}(\mathbb{R})@>{i^{\infty,m}_{n}}>{\simeq}>\Omega
S^{mn-1},\end{CD}$

where $\tilde{S}$ denotes a homotopy inverse of $\hat{S}$. This map is the
desired homotopy equivalence if $mn\geq 4$ and homology equivalence if $mn=3$.
∎

######  Theorem 7.9.

There is a map

$\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1}$

which is a homotopy equivalence if $mn\geq 4$ and a homology equivalence if
$mn=3$.

###### Proof.

Since $S_{\mathbb{R}}$ is a homotopy equivalence, it follows from Proposition
7.8 that it suffices to prove that there is a homotopy equivalence

(7.24) $\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2})^{\mathbb{Z}_{2}}\simeq\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2})\times\Omega\mathcal{Q}^{m}_{n}(I,\partial I).$

Consider the map $\hat{r}_{I}:\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2})^{\mathbb{Z}_{2}}\to\mathcal{Q}^{m}_{n}(I,\partial I)$ given by the
restriction
$\hat{r}_{I}(\xi_{1},\cdots,\xi_{m})=(\xi_{1}\cap\mathbb{R},\cdots,\xi_{m}\cap\mathbb{R}).$
Note that $\hat{r}_{I}$ is a quasifibration with fiber

$\displaystyle F$ $\displaystyle=\mbox{{\rm
Pol}}^{m}_{n}(I\times([-1,0)\cup(0,1]),\partial
I\times([-1,0)\cup(0,1]))^{\mathbb{Z}_{2}}$ $\displaystyle\cong\mbox{{\rm
Pol}}^{m}_{n}(I\times(0,1],\partial
I\times(0,1])=\bigcup_{0<\epsilon<1}\mbox{{\rm
Pol}}^{m}_{n}(I\times[\epsilon,1],\partial I\times[\epsilon,1]).$

Since the space ${\rm P}(\epsilon)=\mbox{{\rm
Pol}}^{m}_{n}(I\times[\epsilon,1],\partial I\times[\epsilon,1])$ is homotopy
equivalent to the space ${\rm P}(1/2)=\mbox{{\rm
Pol}}^{m}_{n}(I\times[1/2,1],\partial I\times[1/2,1])$ for any $0<\epsilon<1$
by radial expansion, there is a homotopy equivalence

$F\simeq{\rm P}(1/2)=\mbox{{\rm Pol}}^{m}_{n}(I\times[1/2,1],\partial
I\times[1/2,1])\cong\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma I^{2}).$

Thus, we obtain a fibration sequence (up to homotopy equivalence)

(7.25) $\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma I^{2})\stackrel{{\scriptstyle
s_{I}}}{{\longrightarrow}}\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2})^{\mathbb{Z}_{2}}\stackrel{{\scriptstyle\hat{r}_{I}}}{{\longrightarrow}}\mathcal{Q}^{m}_{n}(I,\partial
I).$

Let $i:(I,\partial
I)\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}(I^{2},\sigma I^{2})$
denote the natural inclusion map given by $i(x)=(x,0)$ for $x\in I$. This
inclusion naturally extends to the map $i_{\\#}:\mathcal{Q}^{m}_{n}(I,\partial
I)\to\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma I^{2})$. Since $\hat{r}_{I}\circ
i_{\\#}=\mbox{id}_{\mathcal{Q}^{m}_{n}(I,\partial I)}$, the loop sum map

$\Omega s_{I}+\Omega i_{\\#}:\Omega\mbox{{\rm Pol}}^{m}_{n}(I^{2},\sigma
I^{2})\times\Omega\mathcal{Q}^{m}_{n}(I,\partial
I)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega\mbox{{\rm
Pol}}^{m}_{n}(I^{2},\sigma I^{2})^{\mathbb{Z}_{2}}$

is a homotopy equivalence. ∎

######  Corollary 7.10.

There is a map

$j^{\infty,m}_{n}:\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{R})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}(\Omega^{2}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}.$

which is a homotopy equivalence if $mn\geq 4$ and a homology equivalence if
$mn=3$.

###### Proof.

The assertion follows from Theorem 7.9 and (3.6). ∎

######  Corollary 7.11.

There is a map

$f^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}\simeq\Omega^{2}S^{2mn-1}\times\Omega S^{mn-1}$

which is a homotopy equivalence through dimension $D(d;m,n)$ if $mn\geq 4$,
and a homology equivalence through dimension $D(d;m,n)$ if $mn=3$.

###### Proof.

Let $\hat{\iota}^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\mbox{{\rm Poly}}^{\infty,m}_{n}(\mathbb{R})$
denote the natural map, and let us consider the composite of maps

$\begin{CD}f^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})@>{\hat{\iota}^{d,m}_{n}}>{}>\mbox{{\rm
Poly}}^{\infty,m}_{n}(\mathbb{R})@>{j^{\infty,m}_{n}}>{\simeq}>(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}.\end{CD}$

Then by using Theorem 6.2, Corollaries 6.4 and 7.10, we see that the map
$f^{d,m}_{n}$ is a homotopy equivalence through dimension $D(d;m,n)$ if
$mn\geq 4$ and it is a homology equivalence through dimension $D(d;m,n)$. ∎

##  8 The homotopy type of $\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$

In this section we give the proofs of of our main results (Theorems 2.7 and
2.11) and their corollaries (Corollaries 2.9, 2.10, 2.13 and 2.14).

Before giving the proof of Theorem 2.7, we comment about its proof. It seems
plausible that one could give a proof of Theorem 2.7 by a method similar to
that used in the proof of [18, Theorem 1.8]. However, this approach would
require an analysis of the $\mathbb{Z}_{2}$-equivariant homotopy of the spaces
of $\mathbb{Z}_{2}$-equivariant maps, which seems difficult. For this reason
we decided to use an indirect approach, combining the corresponding results
given in [18, Theorem 1.8] and [20, Theorem 1.8].

###### Proof of Theorem 2.7.

Assume that $mn\geq 3$. From now on, we use the notations used in Lemma 3.6.
We denote by

(8.1) $r_{S^{1}}:\mbox{{\rm Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm
P}^{mn-1},\mathbb{R}{\rm P}^{mn-1})\to\Omega_{0}\mathbb{R}{\rm P}^{mn-1}$

the restriction map given by $r_{S^{1}}(f)=f|S^{1}$, and let

(8.2) $\Omega^{2}S^{2mn-1}\stackrel{{\scriptstyle\tiny
q_{1}}}{{\longleftarrow}}\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1}\stackrel{{\scriptstyle q_{2}}}{{\longrightarrow}}\Omega S^{mn-1}$

denote the projections to the first and the second factor, respectively. Then
if we write

(8.3) $E_{0}^{*}=\mbox{{\rm Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm
P}^{mn-1},\mathbb{R}{\rm P}^{mn-1}),$

it follows from the diagram (3.11) and the definitions of three natural maps
$i^{d,m}_{n,\mathbf{\rm H}_{+}}$, $i^{d,m}_{n,\mathbb{R}}$ and $i^{d,m}_{n}$
that we have the following homotopy commutative diagram:

(8.4) $\begin{CD}\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})@>{\iota^{d,m}_{n,\mathbf{\rm H}_{+}}}>{\subset}>\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})@>{\iota^{d,m}_{n}}>{\subset}>\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\\\ @V{i^{d,m}_{n,\mathbf{\rm
H}+}}V{}V@V{i^{d,m}_{n,\mathbb{R}}}V{}V@V{i^{d,m}_{n}}V{}V\\\
\Omega^{2}_{d}\mathbb{C}{\rm P}^{mn-1}(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}}\Omega_{[d]_{2}}\mathbb{R}{\rm P}^{mn-1}\\\
@V{\iota_{\mathbb{C}}^{\prime}}V{\simeq}V@V{\iota^{\prime}}V{\simeq}V@V{\iota_{\mathbb{R}}^{\prime}}V{\simeq}V\\\
\mbox{{\rm Map}}^{*}_{0}(D^{2},S^{1};\mathbb{C}{\rm
P}^{mn-1},*)@>{\hat{j}}>{\subset}>E_{0}^{*}@>{r_{S^{1}}}>{}>\Omega_{0}\mathbb{R}{\rm
P}^{mn-1}\\\
@V{\iota_{\mathbb{C}}}V{\simeq}V@V{\iota_{\mathbb{C}\mathbb{R}}}V{\simeq}V@V{\iota_{\mathbb{R}}}V{\simeq}V\\\
\Omega^{2}S^{2mn-1}@<{q_{1}}<{}<\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1}@>{q_{2}}>{}>\Omega S^{mn-1}\end{CD}$

Now consider the following three maps given by

(8.5)
$\begin{cases}I^{d,m}_{n}&=\iota_{\mathbb{C}\mathbb{R}}\circ\iota^{\prime}\circ
i^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to\Omega^{2}S^{2mn-1}\times\Omega S^{mn-1},\\\
I^{d,m}_{n,\mathbf{\rm
H}_{+}}&=\iota_{\mathbb{C}}\circ\iota_{\mathbb{C}}^{\prime}\circ
i^{d,m}_{n,\mathbf{\rm H}_{+}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\to\Omega^{2}S^{2mn-1},\\\
I^{d,m}_{n,\mathbb{R}}&=\iota_{\mathbb{R}}\circ\iota_{\mathbb{R}}^{\prime}\circ
i^{d,m}_{n}:\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})\to\Omega
S^{mn-1}.\end{cases}$

Now let $\mathbb{F}$ be any fixed field. It follows from Theorems 3.9 and 2.6
that the two maps $I^{d,m}_{n,\mathbf{\rm H}_{+}}$ are
$I^{d,m}_{n,\mathbb{R}}$ are homology equivalences through dimension
$D(\lfloor d/2\rfloor;m,n;\mathbb{C})$ and $D(d;m,n)$, respectively.

Since $D(d;m,n)<D(\lfloor d/2\rfloor;m,n;\mathbb{C})$ (by Lemma 3.10), by
using the diagram (8.4), we see that the induced homomorphism

(8.6) $(I^{d,m}_{n})_{*}:H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F})\to
H_{s}(\Omega^{2}S^{2mn-1}\times\Omega S^{mn-1};\mathbb{F})$

is an epimorphism for any $s\leq D(d;m,n)$. However, since

$\dim_{\mathbb{F}}H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F})=\dim_{\mathbb{F}}H_{s}(\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1};\mathbb{F})<\infty$

for any $s\leq D(d;m,n)$ (by Corollary 7.11), we notice that the homomorphism
$(I^{d,m}_{n})_{*}$ (given by (8.6)) is an isomorphism for any $s\leq
D(d;m,n)$.

Then, by putting $\mathbb{F}=\mathbb{Z}/p$ ($p$: prime) or $\mathbb{Q}$, it
follows from the universal coefficient Theorem that the map $I^{d,m}_{n}$
induces an isomorphism on the homology group $H_{s}(\ ;\mathbb{Z})$ for any
$s\leq D(d;m,n)$. Thus, we see that the map $i^{d,m}_{n,\mathbb{R}}$ is a
homology equivalence through dimension $D(d;m,n)$.

Next, assume that $mn\geq 4$. Then by Lemma 6.3, we see that two spaces
$\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R})$ and $\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1}$ are simply connected. Hence, we obtain that the map
$i^{d,m}_{n,\mathbb{R}}$ is a homotopy equivalence through dimension
$D(d;m,n)$. This completes the proof of Theorem 2.7. ∎

###### Proof of Corollary 2.9.

The assertion follows easily from the proof of Theorem 2.7, so we omit the
details. ∎

###### Proof of Corollary 2.10.

Consider the following commutative diagram:

(8.7) $\begin{CD}\mbox{{\rm
Poly}}^{d,1}_{n}(\mathbb{R})@>{i^{d,1}_{n,\mathbb{R}}}>{}>(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{n-1})^{\mathbb{Z}_{2}}\\\ @V{j^{d}_{n}}V{}V\|\\\ \mbox{{\rm
Poly}}^{d,n}_{1}(\mathbb{R})@>{i^{d,n}_{1,\mathbb{R}}}>{}>(\Omega^{2}_{d}\mathbb{C}{\rm
P}^{n-1})^{\mathbb{Z}_{2}}\end{CD}$

First, consider the case $n\geq 4$. It follows from Theorem 2.7 that the maps
$i^{d,1}_{n,\mathbb{R}}$ and $i^{d,n}_{1,\mathbb{R}}$ are homotopy
equivalences through dimension $(D;1,n)$ and $D(d;n,1)$, respectively. Since
$D(d;1,n)<D(d;n,1)$, by using the diagram (8.7) we see that the map
$j^{d}_{n}$ is a homotopy equivalence through dimension
$D(d;1,n)=(n-2)(\lfloor d/n\rfloor+1)-1.$

Next, if $n=3$, by Theorem 2.7 we see that the maps $i^{d,1}_{3,\mathbb{R}}$
and $i^{d,3}_{1,\mathbb{R}}$ are homology equivalences through dimension
$(D;1,3)=\lfloor d/3\rfloor$ and $D(d;3,1)=d$, respectively. Thus, we see that
the map $j^{d}_{3}$ is a homology equivalence through dimension $\lfloor
d/3\rfloor$. ∎

######  Corollary 8.1.

If $mn=3$ and $d\geq n$, two maps $i^{d,m}_{n,\mathbb{R}}$ and
$s^{d,m}_{n,\mathbb{R}}$ induce isomorphisms

$\begin{cases}(i^{d,m}_{n,\mathbb{R}})_{*}:\pi_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi_{1}((\Omega^{2}_{d}\mathbb{C}{\rm
P}^{2})^{\mathbb{Z}_{2}})\cong\pi_{1}(\Omega^{2}S^{5}\times\Omega
S^{2})\cong\mathbb{Z},\\\ (s^{d,m}_{n,\mathbb{R}})_{*}:\pi_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi_{1}(\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R}))\cong\mathbb{Z}.\end{cases}$

###### Proof.

Let $mn=3$ and consider the following commutative diagram:

$\begin{CD}\pi_{1}(\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R}))@<{(s^{d,m}_{n,\mathbb{R}})_{*}}<{}<\pi_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R}))@>{(i^{d,m}_{n,\mathbb{R}})_{*}}>{}>\pi_{1}((\Omega^{2}_{d}\mathbb{C}{\rm
P}^{2})^{\mathbb{Z}_{2}})\\\
@V{h_{1}}V{\cong}V@V{h_{2}}V{\cong}V@V{h_{3}}V{\cong}V\\\ H_{1}(\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R});\mathbb{Z})@<{(s^{d,m}_{n,\mathbb{R}})_{\\#}}<{\cong}<H_{1}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})@>{(i^{d,m}_{n,\mathbb{R}})_{\\#}}>{\cong}>H_{1}((\Omega^{2}_{d}\mathbb{C}{\rm
P}^{2})^{\mathbb{Z}_{2}};\mathbb{Z})\end{CD}$

where $h_{k}$ $(k=1,2,3)$ are corresponding Hurewicz homomorphisms.

Since $\pi_{1}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R}))\cong\pi_{1}(\mbox{{\rm
Poly}}^{d+1,m}_{n}(\mathbb{R}))\cong\pi_{1}((\Omega^{2}_{d}\mathbb{C}{\rm
P}^{mn-1})^{\mathbb{Z}_{2}})\cong\mathbb{Z}$ by Lemma 6.3, by the Hurewicz
theorem each $h_{k}$ is an isomorphism. It follows from Theorems 2.7 and 6.2
that $(i^{d,m}_{n,\mathbb{R}})_{\\#}$ and $(s^{d,m}_{n,\mathbb{R}})_{\\#}$ are
isomorphisms, and the assertion easily follows. ∎

Next, consider the stable homotopy type of $\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})$. For this purpose, recall the following result:

######  Lemma 8.2.

For any field $\mathbb{F}$ and any $s\geq 1$,

$\dim_{\mathbb{F}}H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F})\leq\dim_{\mathbb{F}}H_{s}({\rm
P}^{d,m}_{n};\mathbb{F})<\infty.$

###### Proof.

Consider the Vassiliev type spectral sequence

$\\{E^{t;d}_{k,s},d^{t}:E^{t;d}_{k,s}\to E^{t;d}_{k+t,s+t-1}\\}\Rightarrow
H_{s-k}(\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F}).$

Since $E^{1}_{s}=\bigoplus_{k\in\mathbb{Z}}E^{1;d}_{k,k+s}\cong H_{s}({\rm
P}^{d,m}_{n};\mathbb{F})$ for any $s\geq 1$ by Lemma 4.12, the assertion
easily follows. ∎

Now we can give the proof of Theorem 2.11.

###### Proof of Theorem 2.11.

From now on, let $d_{0}=\lfloor d/2\rfloor$, and we write

(8.8) $\mbox{{\rm Poly}}^{d_{0},m}_{n}=\mbox{{\rm
Poly}}^{d_{0},m}_{n}(\mathbb{C}),\quad\mbox{{\rm Q}}^{d,m}_{n}=\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R}).$

Let $\mathcal{F}^{d,m}_{n}=\\{(i,j)\in\mathbb{N}^{2}:i+2j\leq\lfloor
d/n\rfloor\\}$ as in (3.19), and recall the spaces $B^{d,m}_{n}$ and ${\rm
P}^{d,m}_{n}(\mathbb{R})$ given by (see (2.19) and (4.27))

$\displaystyle B^{d,m}_{n}$
$\displaystyle=\bigvee_{(i,j)\in\mathcal{F}^{d,m}_{n}}S^{(mn-2)i}\wedge\Sigma^{2(mn-2)j}D_{j},\
{\rm P}^{d,m}_{n}=\mbox{{\rm Poly}}^{d_{0},m}_{n}\vee
B^{d,m}_{n}\vee\mbox{{\rm Q}}^{d,m}_{n}.$

It follows from (2.15) and (4.26) that there are stable homotopy equivalences

(8.9) $\theta_{\mathbb{C}}:\bigvee_{j=1}^{\lfloor
d_{0}/n\rfloor}\Sigma^{2(mn-2)j}D_{j}\stackrel{{\scriptstyle\simeq_{s}}}{{\longrightarrow}}\mbox{{\rm
Poly}}^{d_{0},m}_{n},\quad\theta_{\mathbb{R}}:\bigvee_{i=1}^{\lfloor
d/n\rfloor}S^{(mn-2)i}\stackrel{{\scriptstyle\simeq_{s}}}{{\longrightarrow}}\mbox{{\rm
Q}}^{d,m}_{n}.$

Hence, the space ${\rm P}^{d,m}_{n}$ is stably homotopy equivalent to the
following space

$\Big{(}\bigvee_{j=1}^{\lfloor
d_{0}/n\rfloor}\Sigma^{2(mn-2)j}D_{j}\Big{)}\vee\Big{(}\bigvee_{(i,j)\in\mathcal{F}^{d,m}_{n}}\big{(}S^{(mn-2)i}\wedge\Sigma^{2(mn-2)j}D_{j}\big{)}\Big{)}\vee\Big{(}\bigvee_{i=1}^{\lfloor
d/n\rfloor}S^{(mn-2)i}\Big{)}.$

It follows from [12] and the Snaith splitting [26] that there are two stable
homotopy equivalences

(8.10)
$\begin{cases}\displaystyle\Omega^{2}S^{2mn-1}\simeq_{s}A^{m}_{n}(1):=\bigvee_{j=1}^{\infty}\Sigma^{2(mn-2)j}D_{j},\\\
\displaystyle\Omega S^{mn-1}\ \
\simeq_{s}A^{m}_{n}(3):=\bigvee_{i=1}^{\infty}S^{(mn-2)i}.\end{cases}$

Thus, there is a stable homotopy equivalence

$\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1}\simeq_{s}\bigvee_{k=1}^{3}A^{m}_{n}(k)=A^{m}_{n}(1)\vee
A^{m}_{n}(2)\vee A^{m}_{n}(3),$

where the space $A^{m}_{n}(2)$ is defined by

(8.11) $\displaystyle A^{m}_{n}(2)$ $\displaystyle=A^{m}_{n}(1)\wedge
A^{m}_{n}(3)=\bigvee_{i,j\geq 1}S^{(mn-2)i}\wedge\Sigma^{2(mn-2)j}D_{j}.$

Let $q_{1}^{\prime}$ and $q_{3}^{\prime}$ denote the natural projections to
the first and the third factors given by

(8.12) $\begin{CD}A^{m}_{n}(1)@<{q_{1}^{\prime}}<{}<A^{m}_{n}(1)\vee
A^{m}_{n}(2)\vee A^{m}_{n}(3)@>{q_{3}^{\prime}}>{}>A^{m}_{n}(3).\end{CD}$

Similarly, let $p_{\mathbb{C}}$ and $p_{\mathbb{R}}$ also denote the natural
projections to the first and the third factors given by

(8.13) $\begin{CD}\mbox{{\rm Poly}}^{\lfloor
d_{0}\rfloor,m}_{n}@<{p_{\mathbb{C}}}<{}<{\rm P}^{d,m}_{n}=\mbox{{\rm
Poly}}^{\lfloor d_{0}\rfloor,m}_{n}\vee B^{d,m}_{n}\vee\mbox{{\rm
Q}}^{d,m}_{n}@>{p_{\mathbb{R}}}>{}>\mbox{{\rm Q}}^{d,m}_{n}.\end{CD}$

Let $p_{1}$, $p_{2}$ and $p_{3}$ denote the corresponding natural pinching map
given by

$\displaystyle p_{1}:A^{m}_{n}(1)$
$\displaystyle=\bigvee_{j=1}^{\infty}\Sigma^{2(mn-2)j}D_{j}\to{\rm
P}(1):=\bigvee_{j=1}^{\lfloor d_{0}/n\rfloor}\Sigma^{2(mn-2)j}D_{j},$
$\displaystyle p_{2}:A^{m}_{n}(2)$ $\displaystyle=\bigvee_{i,j\geq
1}S^{(mn-2)i}\wedge\Sigma^{2(mn-2)j}D_{j}\to\bigvee_{(i,j)\in\mathcal{F}^{d,m}_{n}}S^{(mn-2)i}\wedge\Sigma^{2(mn-2)j}D_{j},$
$\displaystyle p_{3}:A^{m}_{n}(3)$
$\displaystyle=\bigvee_{i=1}^{\infty}S^{(mn-2)i}\to{\rm
P}(3):=\bigvee_{i=1}^{\lfloor d/n\rfloor}S^{(mn-2)i}.$

By the diagram (8.4) and (8.5), we obtain the following commutative diagram

(8.14) $\begin{CD}\mbox{{\rm Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm
H}_{+})@>{\iota^{d,m}_{n,\mathbf{\rm H}_{+}}}>{\subset}>\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})@>{\iota^{d,m}_{n}}>{\subset}>\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\\\ @V{I^{d,m}_{n,\mathbf{\rm
H}+}}V{}V@V{I^{d,m}_{n}}V{}V@V{I^{d,m}_{n,\mathbb{R}}}V{}V\\\
\Omega^{2}S^{2mn-1}@<{q_{1}}<{}<\Omega^{2}S^{2mn-1}\times\Omega
S^{mn-1}@>{q_{2}}>{}>\Omega S^{mn-1}\\\
@V{\pi^{\prime}}V{\simeq_{s}}V@V{\pi}V{\simeq_{s}}V@V{\pi^{\prime\prime}}V{\simeq_{s}}V\\\
\displaystyle A^{m}_{n}(1)@<{q_{1}^{\prime}}<{}<A^{m}_{1}(1)\vee
A^{m}_{n}(2)\vee A^{m}_{n}(3)@>{q_{3}^{\prime}}>{}>A^{m}_{n}(3)\\\
@V{p_{1}}V{}V@V{p_{1}\vee p_{2}\vee p_{3}}V{}V@V{p_{3}}V{}V\\\
\displaystyle{\rm P}(1)@<{p_{1}^{\prime}}<{}<{\rm P}(1)\vee
B^{d,m}_{n}\vee{\rm P}(3)@>{p_{3}^{\prime}}>{}>{\rm P}(3)\\\
@V{\theta_{\mathbb{C}}}V{\simeq_{s}}V@V{\theta_{\mathbb{C}}\vee\mbox{\tiny
id}\vee\theta_{\mathbb{R}}}V{\simeq_{s}}V@V{\theta_{\mathbb{R}}}V{\simeq_{s}}V\\\
\mbox{{\rm Poly}}^{d_{0},m}_{n}@<{p_{\mathbb{C}}}<{}<\displaystyle{\rm
P}^{d,m}_{n}=\mbox{{\rm Poly}}^{d_{0},m}_{n}\vee B^{d,m}_{n}\vee\mbox{{\rm
Q}}^{d,m}_{n}@>{p_{\mathbb{R}}}>{}>\mbox{{\rm Q}}^{d,m}_{n}\end{CD}$

Now consider the following three maps

$\begin{cases}J^{d,m}_{n,\mathbb{C}}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+})\to\mbox{{\rm
Poly}}^{d_{0},m}_{n},\quad\quad J^{d,m}_{n,\mathbb{R}}:\mbox{{\rm
Q}}^{d,m}_{n}\to\mbox{{\rm Q}}^{d,m}_{n},\\\ J^{d,m}_{n}:\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R})\to{\rm P}^{d,m}_{n}=\mbox{{\rm
Poly}}^{d_{0},m}_{n}\vee B^{d,m}_{n}\vee\mbox{{\rm Q}}^{d,m}_{n}\end{cases}$

defined by

(8.15) $\begin{cases}J^{d,m}_{n,\mathbb{C}}&=\theta_{\mathbb{C}}\circ
p_{1}\circ\pi^{\prime}\circ I^{d,m}_{n,\mathbf{\rm H}_{+}},\quad
J^{d,m}_{n,\mathbb{R}}=\theta_{\mathbb{R}}\circ
p_{3}\circ\pi^{\prime\prime}\circ I^{d,m}_{n,\mathbb{R}},\\\
J^{d,m}_{n}&=(\theta_{\mathbb{C}}\vee\mbox{id}\vee\theta_{\mathbb{R}})\circ(p_{1}\vee
p_{2}\vee p_{3})\circ\pi\circ I^{d,m}_{n}.\end{cases}$

It suffices to prove that the map $J^{d,m}_{n}$ is a stable homotopy
equivalence. It is easy to see that

(8.16) $J^{d,m}_{n,\mathbb{R}}=\mbox{id}:\mbox{{\rm
Q}}^{d,m}_{n}(\mathbb{R})\to\mbox{{\rm Q}}^{d,m}_{n}(\mathbb{R})\quad\mbox{
(up to homotopy equivalence).}$

Now let $\mathbb{F}$ be any fixed field, and consider the homomorphism

$(J^{d,m}_{n,\mathbb{C}})_{*}:H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R};\mathbf{\rm H}_{+});\mathbb{F})\to
H_{s}(\mbox{{\rm Poly}}^{d_{0},m}_{n};\mathbb{F})\quad\mbox{for }s\geq 1.$

We will need the following lemma.

######  Lemma 8.3.

The induced homomorphism

(8.17) $(J^{d,m}_{n})_{*}:H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F})\to H_{s}({\rm
P}^{d,m}_{m};\mathbb{F})$

is an epimorphism for any $s\geq 1$ and for any field $\mathbb{F}$.

We postpone the proof of Lemma 8.3, and first complete the proof of Theorem
2.11 by using this lemma. By Lemma 8.3 we have

(8.18) $\dim_{\mathbb{F}}H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F})\geq\dim_{\mathbb{F}}H_{s}({\rm
P}^{d,m}_{n};\mathbb{F})\quad\mbox{for any }s\geq 1.$

Combining this with Lemma 8.2 we obtain the equality:

(8.19) $\dim_{\mathbb{F}}H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{F})=\dim_{\mathbb{F}}H_{s}({\rm
P}^{d,m}_{n};\mathbb{F})<\infty\quad\mbox{for any }s\geq 1.$

Since $(J^{d,m}_{n})_{*}$ is an epimorphism, (8.19) implies that it is, in
fact, an isomorphism for any $s\geq 1$. By putting $\mathbb{F}=\mathbb{Z}/p$
($p$: prime) and $\mathbb{F}=\mathbb{Q}$, and using the universal coefficients
theorem, we conclude that

$(J^{d,m}_{n})_{*}:H_{s}(\mbox{{\rm
Poly}}^{d,m}_{n}(\mathbb{R});\mathbb{Z})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H_{s}({\rm
P}^{d,m}_{n};\mathbb{Z})$

is an isomorphism for every $s\geq 1$. Hence, the map $J^{d,m}_{n}$ is a
stable homotopy equivalence. This completes the proof of Theorem 2.11. ∎

###### Proof of Lemma 8.3.

It follows from Corollary 2.9, Lemma 3.8 and (8.16) that the maps
$J^{d,m}_{n,\mathbb{C}}$ and $J^{d,m}_{n,\mathbb{R}}$ induce epimorphisms on