Some calculations involving configuration spaces of distinct points

Revision as of 14:07, 2 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.

2 Construction and examples

For a manifold $<!-- 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 </wikitex> == Construction and examples == <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. {{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$. {{endthm}} </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> <!-- == Acknowledgments == ... == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]X$, $\widetilde X$ 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.

Definition 2.1.[of the Haefliger-Wu invariant $\alpha$]

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

$$\displaystyle (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N.$$

Theorem 2.2. The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$ is one-to-one for $2m\ge 3n+4$.

3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References