Some calculations involving configuration spaces of distinct points
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− | For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus | + | For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus the diagonal. It is a manifold with boundary and has the standard free involution. |
{{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] | {{beginthm|Definition}}[of the Haefliger-Wu invariant $\alpha$] | ||
\label{DefHaef} | \label{DefHaef} |
Revision as of 13:11, 2 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.
2 Construction and examples
For a manifold , denotes the deleted product of , i.e. minus the diagonal. It is a manifold with boundary and has the standard free involution.
Definition 2.1.[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 2.2. The Haefliger-Wu invariant is one-to-one for .
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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