Some calculations involving configuration spaces of distinct points

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(Introduction)
(Uniqueness theorems)
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(b) $N$ is connected and $m \ge 2n+1 \ge 5$.
(b) $N$ is connected and $m \ge 2n+1 \ge 5$.
Then every two every two equivariant maps $f, g:\widetilde N\to S^{m-1}$ are equivariantly homotopic.
+
Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic.
{{endthm}}
{{endthm}}
''Proof.''
''Proof.''
Take an arbitrary equivariant triangulation $T$ of $\widetilde N$.
+
Given two equivariant maps $\phi, \psi\colon\widetilde N \to S^{m-1}$ take an arbitrary equivariant triangulation $T$ of $\widetilde N$.
(a) One can easily construct an equivariant homotopy between restrictions of $f$ and $g$
+
(a) One can easily construct an equivariant homotopy between restrictions of $\phi$ and $\psi$
on vertices of $T$.
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 $k<2n+1$.
+
By general position a homotopy between $\phi$ and $\psi$ on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed $2n+1$.
We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.
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 $\widetilde{N}$.
+
(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 between $\phi$ and $\psi$ on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on $\widetilde{N}$.
QED
QED
</wikitex>
</wikitex>

Revision as of 11:50, 9 April 2020

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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.

In introducing notation and definitions we follow slides by A. Skopenkov

The deleted product

\displaystyle \widetilde K=K^{\underline2}:=\{(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^m is an embedding of a subset K\subset \mathbb R^n. Then the map \widetilde f:\widetilde K\to S^{m-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^{m-1}. Thus the existence of an equivariant map \widetilde K\to S^{m-1} is a necessary condition for the embeddability of K in \R^m.

Denote by \mathrm{Emb}^{m}K the set embeddings of K into \mathbb R^{m} up to isotopy. Let \pi_{\mathrm{eq}}^{m}(K) = [K;S^{m}]_{\mathrm{eq}} be the set of equivariant maps ̃K\to S^m up to equivariant homotopy.


The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K}) is induced by the Gauss map. I.e. \alpha([f]) = [\widetilde f].

Theorem 1.1. 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.

See [Skopenkov2006, \S 5] and [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1\alpha\partial] for the DIFF case and [Skopenkov2002, Theorem 1.3\alpha\partial] for the PL case.

2 Uniqueness theorems

Lemma 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 each two equivariant maps from \widetilde N to S^{m-1} are equivariantly homotopic.

Proof. Given two equivariant maps \phi, \psi\colon\widetilde N \to S^{m-1} take an arbitrary equivariant triangulation T of \widetilde N.

(a) One can easily construct an equivariant homotopy between restrictions of \phi and \psi on vertices of T. By general position a homotopy between \phi and \psi on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed 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 between \phi and \psi on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on \widetilde{N}. QED

3 Invariants

...

4 Classification/Characterization

...

5 Further discussion

...

6 References

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