Some calculations involving configuration spaces of distinct points

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{{Stub}}
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== Introduction ==
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<wikitex>;
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‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics<!-- [Gl68, Va92]-->. The Haefliger-Wu invariant is a manifestation of these ideas in the theory of embeddings. The complement to the diagonal idea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem and the Borsuk-Ulam Antipodes Theorem. See \cite{Vassiliev1992}.
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<!-- COMMENT:
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In introducing notation and definitions we follow \cite{Skopenkov2020a}.
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To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:
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If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
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- For statements like Theorem, Lemma, Definition etc., use e.g.
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The ''deleted product''
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{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.
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$$\widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.$$
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This is the configuration space of ordered pairs of distinct points of $K$.
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- For references, use e.g. {{cite|Milnor1958b}}.
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Suppose that $f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^N$.
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Then the map $\widetilde f:\widetilde K\to S^{m-1}$ is well-defined by the Gauss formula
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$$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$
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END OF COMMENT
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We have $\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the `exchanging factors' involution
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$(x,y)\mapsto(y,x)$ on $\widetilde K$ and the antipodal involution on $S^{m-1}$.
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Thus the existence of an equivariant map $\widetilde K\to S^{m-1}$ is a necessary condition for the embeddability of $K$ in $\R^m$.
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-->
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Denote by $\mathrm{Emb}^{m}K$ the set embeddings of $K$ into $\mathbb R^{m}$ up to isotopy.
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{{Stub}}
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Let $\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps $\widetilde K \to S^m$ up to equivariant homotopy. By $[·]$ we denote the isotopy class of an embedding or the homotopy class of a map.
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== Introduction ==
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<wikitex>;
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<!--Definition of the Haefliger-Wu invariant $\alpha$-->
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...
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'''The Haefliger-Wu invariant'''
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$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is defined by formula
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<!--is induced by the Gauss map.-->
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$\alpha([f]) = [\widetilde f]$.
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<!-- The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.-->
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{{beginthm|Theorem}}
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The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one either
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(a) $K$ is a compact connected $n$-complex and $2m\ge 3n+4$ or
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<!--(b) $K$ is a compact connected $n$-manifold with nonempty boundary and $2m\ge 3n+4$ or-->
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(b) $K$ is a compact $n$-manifold with nonempty boundary, $(K, \partial K)$ is $k$-connected, $\pi_1(\partial K) = 0$,
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$k + 3 \le n$, $(n, k) \notin \{(5, 2), (4, 1)\}$ and $2m\ge 3n+2-k$.
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{{endthm}}
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See \cite[$\S$ 5]{Skopenkov2006} and \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} for the PL case.
</wikitex>
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== Construction and examples ==
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== Uniqueness theorems ==
<wikitex>;
<wikitex>;
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For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution.
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{{beginthm|Lemma}}\label{th::unknotting}
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{{theorem|Definition}}[of the Haefliger-Wu invariant $\alpha$]
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Assume that $N$ is a compact $n$-manifold and either
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\label{DefHaef}
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The Haefliger-Wu invariant
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(a) $m \ge 2n+2$ or
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$\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$
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is induced by the Gauss map, also denoted by $\alpha$.
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(b) $N$ is connected and $m \ge 2n+1 \ge 5$.
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The Gauss map assigns to an individual embedding $f:N\to\R^{k}$
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an equivariant map $\widetilde{N}\to S^{k-1}$
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Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic.
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defined by the formula
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{{endthm}}
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$$
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(x,y)\mapsto
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Hereafter denote by $\widetilde K$ the product $K\times K$ minus tubular neighborhood of the diagonal.
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\frac{f(x)-f(y)}
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{\|f(x)-f(y)\|},
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''Proof.''
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\quad
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Given two equivariant maps $\phi, \psi\colon\widetilde N \to S^{m-1}$ take an arbitrary equivariant triangulation $T$ of $\widetilde N$.
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(x,y)\in\widetilde{N}\subset N\times N.
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$$
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(a) One can easily construct an equivariant homotopy between restrictions of $\phi$ and $\psi$
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The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.
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on vertices of $T$.
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{{definition}}
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By general position a homotopy between $\phi$ and $\psi$ on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed $2n+1$.
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We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.
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(b) Since $\widetilde{N}$ has non-empty boundary, there exists an equivariant deformational retraction of $\widetilde{N}$ to an equivariant $(2n-1)$-subcomplex of $T$. A homotopy between $\phi$ and $\psi$ on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on $\widetilde{N}$.
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QED
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{{beginthm|Lemma}}\label{th::unknotting}
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Assume that $N$ is a closed $k$-connected $n$-manifold and $m-1 \ge 2n-k$.
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Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic.
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{{endthm}}
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== Invariants ==
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<!--== Invariants ==
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<!-- == Acknowledgments ==
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== Acknowledgments ==
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Latest revision as of 15:42, 8 January 2021

This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics. The Haefliger-Wu invariant is a manifestation of these ideas in the theory of embeddings. The complement to the diagonal idea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem and the Borsuk-Ulam Antipodes Theorem. See [Vassiliev1992].

In introducing notation and definitions we follow [Skopenkov2020a].

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

The deleted product

\displaystyle \widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.

This is the configuration space of ordered pairs of distinct points of K.

Suppose that f:K\to\R^m is an embedding of a subset K\subset \mathbb R^N. Then the map \widetilde f:\widetilde K\to S^{m-1} is well-defined by the Gauss formula

\displaystyle \widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.

We have \widetilde f(y,x)=-\widetilde f(x,y), i.e. this map is equivariant with respect to the `exchanging factors' involution (x,y)\mapsto(y,x) on \widetilde K and the antipodal involution on S^{m-1}. Thus the existence of an equivariant map \widetilde K\to S^{m-1} is a necessary condition for the embeddability of K in \R^m.

Denote by \mathrm{Emb}^{m}K the set embeddings of K into \mathbb R^{m} up to isotopy. Let \pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}} be the set of equivariant maps \widetilde K \to S^m up to equivariant homotopy. By [·] we denote the isotopy class of an embedding or the homotopy class of a map.

The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K}) is defined by formula \alpha([f]) = [\widetilde f].

Theorem 1.1. The Haefliger-Wu invariant \alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K) is one-to-one either

(a) K is a compact connected n-complex and 2m\ge 3n+4 or

(b) K is a compact n-manifold with nonempty boundary, (K, \partial K) is k-connected, \pi_1(\partial K) = 0, k + 3 \le n, (n, k) \notin \{(5, 2), (4, 1)\} and 2m\ge 3n+2-k.

See [Skopenkov2006, \S 5] and [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case.

2 Uniqueness theorems

Lemma 2.1. Assume that N is a compact n-manifold and either

(a) m \ge 2n+2 or

(b) N is connected and m \ge 2n+1 \ge 5.

Then each two equivariant maps from \widetilde N to S^{m-1} are equivariantly homotopic.

Hereafter denote by \widetilde K the product K\times K minus tubular neighborhood of the diagonal.

Proof.

Given two equivariant maps \phi, \psi\colon\widetilde N \to S^{m-1} take an arbitrary equivariant triangulation 
Tex syntax error
 of \widetilde N.

(a) One can easily construct an equivariant homotopy between restrictions of \phi and \psi

on vertices of 
Tex syntax error
.

By general position a homotopy between \phi and \psi on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed 2n+1. We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.

(b) Since \widetilde{N} has non-empty boundary, there exists an equivariant deformational retraction of \widetilde{N} to an equivariant (2n-1)-subcomplex of 
Tex syntax error
. A homotopy between \phi and \psi on the subcomplex can by constructed similarly to case (a). This homotopy can be extended to a homotopy on \widetilde{N}.

QED

Lemma 2.2. Assume that N is a closed k-connected n-manifold and m-1 \ge 2n-k.

Then each two equivariant maps from \widetilde N to S^{m-1} are equivariantly homotopic.


3 References

V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications., Amer. Math. Soc., Providence, RI, (1992).

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