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.

The deleted product \widetilde K=K^{\underline2} is \widetilde K:=\{(x,y)\in K\times K\ :\ x\ne y\}. This is the configuration space of ordered pairs of distinct points of K.

Suppose that f:K\to\R^d is an embedding of a subset K\subset \mathbb R^m. Then the map \widetilde f:\widetilde K\to S^{d-1} is well-defined by the Gauss formula

\displaystyle \widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.

We have \widetilde f(y,x)=-\widetilde f(x,y), i.e. this map is equivariant with respect to the `exchanging factors' involution (x,y)\mapsto(y,x) on \widetilde K and the antipodal involution on S^{d-1}. Thus the existence of an equivariant map \widetilde K\to S^{d-1} is a necessary condition for the embeddability of K in \R^d.

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

The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{k}K\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{K}) is induced by the Gauss map, also denoted by \alpha.

Theorem 1.2. The Haefliger-Wu invariant \alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K) 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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