Some calculations involving configuration spaces of distinct points

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{{Stub}}
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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. See \cite{Vassiliev1992}.
<!-- COMMENT:
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In introducing notation and definitions we follow \cite{Skopenkov2020a}.
To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments:
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If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.
- For statements like Theorem, Lemma, Definition etc., use e.g.
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The ''deleted product''
{{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}.
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$$\widetilde K=K^{\underline2}:=\{(x,y)\in K\times K\ :\ x\ne y\}.$$
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This is the configuration space of ordered pairs of distinct points of $K$.
- For references, use e.g. {{cite|Milnor1958b}}.
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Suppose that $f:K\to\R^m$ is an embedding of a subset $K\subset \mathbb R^N$.
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Then the map $\widetilde f:\widetilde K\to S^{m-1}$ is well-defined by the Gauss formula
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$$\widetilde f(x,y)=\frac{f(x)-f(y)}{|f(x)-f(y)|}.$$
END OF COMMENT
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We have $\widetilde f(y,x)=-\widetilde f(x,y)$, i.e. this map is equivariant with respect to the `exchanging factors' involution
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$(x,y)\mapsto(y,x)$ on $\widetilde K$ and the antipodal involution on $S^{m-1}$.
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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 $\R^m$.
-->
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Denote by $\mathrm{Emb}^{m}K$ the set embeddings of $K$ into $\mathbb R^{m}$ up to isotopy.
{{Stub}}
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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. By $[·]$ we denote the isotopy class of an embedding or the homotopy class of a map.
== Introduction ==
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<wikitex>;
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<!--Definition of the Haefliger-Wu invariant $\alpha$-->
...
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'''The Haefliger-Wu invariant'''
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$\alpha:\mathrm{Emb}^{m}K\to \pi_{\mathrm{eq}}^{m-1}(\widetilde{K})$ is defined by formula
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<!--is induced by the Gauss map.-->
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$\alpha([f]) = [\widetilde f]$.
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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.-->
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{{beginthm|Theorem}}
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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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(b) $K$ 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 $2m\ge 3n+2-k$.
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{{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>
== Construction and examples ==
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== Uniqueness theorems ==
<wikitex>;
<wikitex>;
For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution.
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{{beginthm|Lemma}}\label{th::unknotting}
{{theorem|Definition}}[of the Haefliger-Wu invariant $\alpha$]
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Assume that $N$ is a compact $n$-manifold and either
\label{DefHaef}
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The Haefliger-Wu invariant
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(a) $m \ge 2n+2$ or
$\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$
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is induced by the Gauss map, also denoted by $\alpha$.
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(b) $N$ is connected and $m \ge 2n+1 \ge 5$.
The Gauss map assigns to an individual embedding $f:N\to\R^{k}$
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an equivariant map $\widetilde{N}\to S^{k-1}$
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Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic.
defined by the formula
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{{endthm}}
$$
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(x,y)\mapsto
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Hereafter denote by $\widetilde K$ the product $K\times K$ minus tubular neighborhood of the diagonal.
\frac{f(x)-f(y)}
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{\|f(x)-f(y)\|},
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''Proof.''
\quad
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Given two equivariant maps $\phi, \psi\colon\widetilde N \to S^{m-1}$ take an arbitrary equivariant triangulation $T$ of $\widetilde N$.
(x,y)\in\widetilde{N}\subset N\times N.
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$$
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(a) One can easily construct an equivariant homotopy between restrictions of $\phi$ and $\psi$
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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on vertices of $T$.
{{definition}}
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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 $2n+1$.
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We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.
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(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 $\widetilde{N}$.
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QED
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{{beginthm|Lemma}}\label{th::unknotting}
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Assume that $N$ is a closed $k$-connected $n$-manifold and $m-1 \ge 2n-k$.
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Then each two equivariant maps from $\widetilde N$ to $S^{m-1}$ are equivariantly homotopic.
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{{endthm}}
</wikitex>
</wikitex>
== Invariants ==
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<!--== Invariants ==
<wikitex>;
<wikitex>;
...
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</wikitex>
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<!-- == Acknowledgments ==
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== Acknowledgments ==
...
...

Latest revision as of 14:42, 8 January 2021

This page has not been refereed. The information given here might be incomplete or provisional.

[edit] 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].

If the category is omitted, then a result stated below holds in both the smooth and piecewise-linear (PL) category.

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. By [·] we denote the isotopy class of an embedding or the homotopy class of a map.

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 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+2-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.

[edit] 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.

Hereafter denote by \widetilde K the product K\times K minus tubular neighborhood of the diagonal.

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

Lemma 2.2. Assume that N is a closed k-connected n-manifold and m-1 \ge 2n-k.

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


[edit] 3 References

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

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