Some calculations involving configuration spaces of distinct points

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

Contents

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}, \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.

Definition 1.1.[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.

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


2 Construction and examples


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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