Some calculations involving configuration spaces of distinct points

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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.
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 $\alpha$]
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{{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$]
\label{DefHaef}
\label{DefHaef}
The Haefliger-Wu invariant
The Haefliger-Wu invariant
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The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.
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}}
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{{beginthm}}
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Revision as of 12:55, 2 April 2020


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

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2 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.

Definition 2.1.[of the Haefliger-Wu invariant \alpha]

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

\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; we will denote them by \alpha_0 in this case.

{{{1}}} 2.2.


3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

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