# Some calculations involving configuration spaces of distinct points

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## 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$${{Stub}} == 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 \cite{Vassiliev1992}. In introducing notation and definitions we follow \cite{Skopenkov2020a}. If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category. The ''deleted product'' \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 \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]. {{beginthm|Theorem}} 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 m\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 m\ge 3n+2-k. {{endthm}} 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. == Uniqueness theorems == ; {{beginthm|Lemma}}\label{th::unknotting} 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. {{endthm}} 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 T of \widetilde N. (a) One can easily construct an equivariant homotopy between restrictions of \phi and \psi on vertices of T. 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 n+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 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}. QED {{beginthm|Lemma}}\label{th::unknotting} Assume that N is a closed k-connected n-manifold and m \ge 2n+1-k. Then each two equivariant maps from \widetilde N to S^{m-1} are equivariantly homotopic. {{endthm}} == References == {{#RefList:}} [[Category:Manifolds]]K$.

Suppose that $f:K\to\R^m$$f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^N$$K\subset \mathbb R^N$. Then the map $\widetilde f:\widetilde K\to S^{m-1}$$\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)$$\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the exchanging factors' involution $(x,y)\mapsto(y,x)$$(x,y)\mapsto(y,x)$ on $\widetilde K$$\widetilde K$ and the antipodal involution on $S^{m-1}$$S^{m-1}$. Thus the existence of an equivariant map $\widetilde K\to S^{m-1}$$\widetilde K\to S^{m-1}$ is a necessary condition for the embeddability of $K$$K$ in $\R^m$$\R^m$.

Denote by $\mathrm{Emb}^{m}K$$\mathrm{Emb}^{m}K$ the set embeddings of $K$$K$ into $\mathbb R^{m}$$\mathbb R^{m}$ up to isotopy. Let $\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$$\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps $\widetilde K \to S^m$$\widetilde K \to S^m$ up to equivariant homotopy. By $[·]$$[·]$ 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})$$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is defined by formula $\alpha([f]) = [\widetilde f]$$\alpha([f]) = [\widetilde f]$.

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

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

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

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

## 2 Uniqueness theorems

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

(a) $m \ge 2n+2$$m \ge 2n+2$ or

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

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

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

Proof.

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

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

on vertices of
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$T$.

By general position a homotopy between $\phi$$\phi$ and $\psi$$\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$$2n+1$. We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.

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

QED

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

Then each two equivariant maps from $\widetilde N$$\widetilde N$ to $S^{m-1}$$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).