Some calculations involving configuration spaces of distinct points

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(Introduction)
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<wikitex>;
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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.
‘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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In introducing notation and definitions we follow [https://www.mccme.ru/circles/oim/eliminat_talk.pdf slides by A. Skopenkov]
The ''deleted product''
The ''deleted product''
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This is the configuration space of ordered pairs of distinct points of $K$.
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 $K\subset \mathbb R^m$.
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Suppose that $f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^n$.
Then the map $\widetilde f:\widetilde K\to S^{d-1}$ is well-defined by the Gauss formula
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Then the map $\widetilde f:\widetilde K\to S^{m-1}$ is well-defined by the Gauss formula
$$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$
$$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$
We have $\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the `exchanging factors' involution
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 $S^{d-1}$.
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$(x,y)\mapsto(y,x)$ on $\widetilde K$ and the antipodal involution on $S^{m-1}$.
Thus the existence of an equivariant map $\widetilde K\to S^{d-1}$ is a necessary condition for the embeddability of $K$ in $\R^d$.
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Thus the existence of an equivariant map $\widetilde K\to S^{m-1}$ is a necessary condition for the embeddability of $K$ in $\R^m$.
{{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$]
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Denote by $\mathrm{Emb}^{m}K$ the set embeddings of $K$ into $\mathbb R^{m}$ up to isotopy.
\label{DefHaef}
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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'''
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$\alpha:\mathrm{Emb}^{k}K\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{K})$
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<!--Definition of the Haefliger-Wu invariant $\alpha$-->
is induced by the Gauss map.
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<!-- The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.-->
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'''The Haefliger-Wu invariant'''
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$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$
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is induced by the Gauss map.
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I.e. $\alpha([f]) = [\widetilde f]$.
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<!-- 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 $2m\ge 3n+4$.
The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m K\to\pi^{m-1}_{\mathrm{eq}}( \widetilde K)$ is one-to-one for $2m\ge 3n+4$.
{{endthm}}
{{endthm}}
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See \cite[$\S$ 5]{Skopenkov2006} and \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} for the PL case.
</wikitex>
</wikitex>

Revision as of 11:44, 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 every two every two equivariant maps f, g:\widetilde N\to S^{m-1} are equivariantly homotopic.

Proof. 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}. QED

3 Invariants

...

4 Classification/Characterization

...

5 Further discussion

...

6 References

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