Some calculations involving configuration spaces of distinct points

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(Uniqueness theorems)
(Introduction)
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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics<!-- [Gl68, Va92]-->. 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.
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‘The complement of the diagonal’ and ‘the Gauss map’ ideas play a great role in different branches of mathematics<!-- [Gl68, Va92]-->. 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. See \cite{Vassiliev1992}.
In introducing notation and definitions we follow [https://www.mccme.ru/circles/oim/eliminat_talk.pdf slides by A. Skopenkov]
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In introducing notation and definitions we follow \cite{Skopenkov2020a}.
The ''deleted product''
The ''deleted product''
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Denote by $\mathrm{Emb}^{m}K$ the set embeddings of $K$ into $\mathbb R^{m}$ up to isotopy.
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.
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Let $\pi_{\mathrm{eq}}^{m}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}}$ be the set of equivariant maps $\widetilde K \to S^m$ up to equivariant homotopy. Denote by $[f]$ the homotopy class of the map $f$.
<!--Definition of the Haefliger-Wu invariant $\alpha$-->
<!--Definition of the Haefliger-Wu invariant $\alpha$-->
'''The Haefliger-Wu invariant'''
'''The Haefliger-Wu invariant'''
$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$
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$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is defined by formula
is induced by the Gauss map.
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<!--is induced by the Gauss map.-->
I.e. $\alpha([f]) = [\widetilde f]$.
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$\alpha([f]) = [\widetilde f]$.
<!-- 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.-->
{{beginthm|Theorem}}
{{beginthm|Theorem}}
The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one for $2m\ge 3n+4$.
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The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one either
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(a) $K$ is a compact connected $n$-complex and $2m\ge 3n+4$ or
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(b) $K$ is a compact connected $n$-manifold with nonempty boundary and $2m\ge 3n+4$ or
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(c) $K$ is a compact $n$-manifold with nonempty boundary, $(K, \partial K)$ is $k$-connected, $\pi_1(\partial K) = 0$,
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$k + 3 \le n$, $(n, k) \notin \{(5, 2), (4, 1)\}$ and $2m\ge 3n+1-k$.
{{endthm}}
{{endthm}}

Revision as of 13:27, 9 April 2020

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. See [Vassiliev1992].

In introducing notation and definitions we follow [Skopenkov2020a].

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}(\widetilde K) = [\widetilde K;S^{m}]_{\mathrm{eq}} be the set of equivariant maps \widetilde K \to S^m up to equivariant homotopy. Denote by [f] the homotopy class of the map f.


The Haefliger-Wu invariant \alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K}) is defined by formula \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 either

(a) K is a compact connected n-complex and 2m\ge 3n+4 or

(b) K is a compact connected n-manifold with nonempty boundary and 2m\ge 3n+4 or

(c) K is a compact n-manifold with nonempty boundary, (K, \partial K) is k-connected, \pi_1(\partial K) = 0, k + 3 \le n, (n, k) \notin \{(5, 2), (4, 1)\} and 2m\ge 3n+1-k.

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

V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications., Amer. Math. Soc., Providence, RI, (1992).

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