# Some calculations involving configuration spaces of distinct points

## 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.

In introducing notation and definitions we follow slides by A. Skopenkov

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. In introducing notation and definitions we follow [https://www.mccme.ru/circles/oim/eliminat_talk.pdf slides by A. Skopenkov] 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}(K) = [K;S^{m}]_{\mathrm{eq}} be the set of equivariant maps ̃K\to S^m up to equivariant homotopy. '''The Haefliger-Wu invariant''' \alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K}) is induced by the Gauss map. I.e. \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 for m\ge 3n+4. {{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 every two every two equivariant maps f, g:\widetilde N\to S^{m-1} are equivariantly homotopic. {{endthm}} ''Proof.'' Take an arbitrary equivariant triangulation T of \widetilde N. (a) One can easily construct an equivariant homotopy between restrictions of f and g on vertices of T. By general position a homotopy of f, g on the boundary of a k-simplex can be extended to a homotopy on the whole k-simplex since k<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 T. A homotopy on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on \widetilde{N}. QED == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == 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}$$\mathbb R^{m}$ up to isotopy. Let $\pi_{\mathrm{eq}}^{m}(K) = [K;S^{m}]_{\mathrm{eq}}$$\pi_{\mathrm{eq}}^{m}(K) = [K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps ̃$K\to S^m$$K\to S^m$ up to equivariant homotopy.

The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is induced by the Gauss map. I.e. $\alpha([f]) = [\widetilde f]$$\alpha([f]) = [\widetilde f]$.

Theorem 1.1. The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$$\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one for $2m\ge 3n+4$$2m\ge 3n+4$.

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 every two every two equivariant maps $f, g:\widetilde N\to S^{m-1}$$f, g:\widetilde N\to S^{m-1}$ are equivariantly homotopic.

Proof. Take an arbitrary equivariant triangulation $T$$T$ of $\widetilde N$$\widetilde N$.

(a) One can easily construct an equivariant homotopy between restrictions of $f$$f$ and $g$$g$ on vertices of $T$$T$. By general position a homotopy of $f, g$$f, g$ on the boundary of a $k$$k$-simplex can be extended to a homotopy on the whole $k$$k$-simplex since $k<2n+1$$k<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)$$(2n-1)$-subcomplex of $T$$T$. A homotopy on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on $\widetilde{N}$$\widetilde{N}$. QED

...

...

...