# 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.

The deleted product $\widetilde K=K^{\underline2}$$ {{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. For a subset X\subset \mathbb R^{m}, \widetilde X denotes ''the deleted product'' of X, i.e. \widetilde X:= \{(x, y)\in X\times X, x\neq y\} = X^2 minus the diagonal. This is the configuration space of ordered pairs of distinct points of K. {{beginthm|Definition}}[of the Haefliger-Wu invariant \alpha] \label{DefHaef} The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N}) is induced by the Gauss map, also denoted by \alpha. The Gauss map assigns to an individual embedding f:N\to\R^{k} an equivariant map \widetilde{N}\to S^{k-1} defined by the formula (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N. {{beginthm|Theorem}} The Haefliger-Wu invariant \alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N) is one-to-one for m\ge 3n+4. {{endthm}} == Uniqueness theorems == ; {{beginthm|Theorem}}\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 any two embeddings of N into \R^m are isotopic. {{endthm}} In cases (a-d) inequality m\ge3n+4 is clearly satisfied. Hence by Haefliger-Weber theorem \ref{thm_Haef-Web} we have \mathrm{Emb}_m(N) \cong \pi^{m-1}_{\mathrm{eq}}(\widetilde N). So it is sufficient to show that \pi^{m-1}_{\mathrm{eq}}(\widetilde N) is trivial, i.e. every two every two equivariant maps f, g:\widetilde N\to S^{m-1} are equivariantly homotopic. 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}. == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]\widetilde K=K^{\underline2}$ is $\widetilde K:=\{(x,y)\in K\times K\ :\ x\ne y\}.$$\widetilde K:=\{(x,y)\in K\times K\ :\ x\ne y\}.$ This is the configuration space of ordered pairs of distinct points of $K$$K$.

Suppose that $f:K\to\R^d$$f:K\to\R^d$ is an embedding of a subset $K\subset \mathbb R^m$$K\subset \mathbb R^m$. Then the map $\widetilde f:\widetilde K\to S^{d-1}$$\widetilde f:\widetilde K\to S^{d-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$$\widetilde K$ and the antipodal involution on $S^{d-1}$$S^{d-1}$. Thus the existence of an equivariant map $\widetilde K\to S^{d-1}$$\widetilde K\to S^{d-1}$ is a necessary condition for the embeddability of $K$$K$ in $\R^d$$\R^d$.

Definition 1.1.[of the Haefliger-Wu invariant $\alpha$$\alpha$]

The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{k}K\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{K})$$\alpha:\mathrm{Emb}^{k}K\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{K})$ is induced by the Gauss map, also denoted by $\alpha$$\alpha$.

Theorem 1.2. 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$.

## 2 Uniqueness theorems

Theorem 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 any two embeddings of $N$$N$ into $\R^m$$\R^m$ are isotopic.

In cases (a-d) inequality $2m\ge3n+4$$2m\ge3n+4$ is clearly satisfied. Hence by Haefliger-Weber theorem \ref{thm_Haef-Web} we have $\mathrm{Emb}_m(N) \cong \pi^{m-1}_{\mathrm{eq}}(\widetilde N)$$\mathrm{Emb}_m(N) \cong \pi^{m-1}_{\mathrm{eq}}(\widetilde N)$. So it is sufficient to show that $\pi^{m-1}_{\mathrm{eq}}(\widetilde N)$$\pi^{m-1}_{\mathrm{eq}}(\widetilde N)$ is trivial, i.e. every two every two equivariant maps $f, g:\widetilde N\to S^{m-1}$$f, g:\widetilde N\to S^{m-1}$ are equivariantly homotopic.

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}$.

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