# 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.

For a subset $X\subset \mathbb R^{m}$$ {{Stub}} == Introduction == ; ‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics. The Haefliger-Wu invariant is a manifestation of these ideas in the theory of embeddings. The complement to the diagonal idea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem and the Borsuk-Ulam Antipodes Theorem. For a subset X\subset \mathbb R^{m}, \widetilde X denotes ''the deleted product'' of X, i.e. \widetilde X:= \{(x, y)\in X\times X, x\neq y\} = X^2 minus the diagonal. This is the configuration space of ordered pairs of distinct points of K. {{beginthm|Definition}}[of the Haefliger-Wu invariant \alpha] \label{DefHaef} The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N}) is induced by the Gauss map, also denoted by \alpha. The Gauss map assigns to an individual embedding f:N\to\R^{k} an equivariant map \widetilde{N}\to S^{k-1} defined by the formula (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N. {{beginthm|Theorem}} The Haefliger-Wu invariant \alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N) is one-to-one for m\ge 3n+4. {{endthm}} == Construction and examples == ; == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]X\subset \mathbb R^{m}$, $\widetilde X$$\widetilde X$ denotes the deleted product of $X$$X$, i.e. $\widetilde X:= \{(x, y)\in X\times X, x\neq y\} = X^2$$\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.

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

The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$$\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$ is induced by the Gauss map, also denoted by $\alpha$$\alpha$. The Gauss map assigns to an individual embedding $f:N\to\R^{k}$$f:N\to\R^{k}$ an equivariant map $\widetilde{N}\to S^{k-1}$$\widetilde{N}\to S^{k-1}$ defined by the formula

$\displaystyle (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N.$

Theorem 1.2. The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$$\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$ is one-to-one for $2m\ge 3n+4$$2m\ge 3n+4$.

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