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.

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 Uniqueness theorems

Theorem 2.1. 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.

In cases (a-d) inequality 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). 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}.

3 Invariants


4 Classification/Characterization


5 Further discussion


6 References

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