# Some calculations involving configuration spaces of distinct points

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## 2 Construction and examples

For a manifold $X$$ {{Stub}} == Introduction == ; ... == Construction and examples == ; For a manifold X, \widetilde X denotes ''the deleted product'' of X, i.e. X^2 minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution. {{theorem|Definition}}[of the Haefliger-Wu invariant \alpha] \label{DefHaef} The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N}) is induced by the Gauss map, also denoted by \alpha. The Gauss map assigns to an individual embedding f:N\to\R^{k} an equivariant map \widetilde{N}\to S^{k-1} defined by the formula (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N. The Haefliger-Wu invariant and the Gauss map are analogously defined for N_0; we will denote them by \alpha_0 in this case. {{definition}} == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]X$, $\widetilde X$$\widetilde X$ denotes the deleted product of $X$$X$, i.e. $X^2$$X^2$ minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution. Template:Theorem[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

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

The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$$N_0$; we will denote them by $\alpha_0$$\alpha_0$ in this case. Template:Definition

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