Some calculations involving configuration spaces of distinct points
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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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− | + | 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. | |
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{{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] | {{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] | ||
\label{DefHaef} | \label{DefHaef} | ||
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The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$ is one-to-one for $2m\ge 3n+4$. | The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^m N\to\pi^{m-1}_{\mathrm{eq}}( \widetilde N)$ is one-to-one for $2m\ge 3n+4$. | ||
{{endthm}} | {{endthm}} | ||
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Revision as of 11:56, 5 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.
For a subset , denotes the deleted product of , i.e. minus the diagonal. This is the configuration space of ordered pairs of distinct points of K.
Definition 1.1.[of the Haefliger-Wu invariant ]
The Haefliger-Wu invariant is induced by the Gauss map, also denoted by . The Gauss map assigns to an individual embedding an equivariant map defined by the formula
Theorem 1.2. The Haefliger-Wu invariant is one-to-one for .
2 Construction and examples
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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