Some calculations involving configuration spaces of distinct points

Revision as of 15:31, 5 April 2020

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 $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <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. For a subset $X\subset \mathbb R^{m}$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $\widetilde X:= \{(x, y)\in X\times X, x\neq y\} = X^2$ minus the diagonal. <!-- It is a manifold with boundary and has the standard free involution.--> This is the configuration space of ordered pairs of distinct points of K. {{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] \label{DefHaef} The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$ is induced by the Gauss map, also denoted by $\alpha$. The Gauss map assigns to an individual embedding $f:N\to\R^{k}$ an equivariant map $\widetilde{N}\to S^{k-1}$ defined by the formula $$ (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N. $$ <!-- 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}} The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$ is one-to-one for m\ge 3n+4$. {{endthm}} </wikitex> == Uniqueness theorems == <wikitex>; {{beginthm|Theorem}}\label{th::unknotting} 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. {{endthm}} In cases (a-d) inequality m\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 $\widetilde{N}$. </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> <!-- == Acknowledgments == ... == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]\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 $\R^d$.

2 Uniqueness theorems

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

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4 Classification/Characterization

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5 Further discussion

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6 References