Some calculations involving configuration spaces of distinct points
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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.--> | <!-- 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 $2m\ge 3n+4$. | ||
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Revision as of 14:02, 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 greatrole in different branches of mathematics [Gl68, Va92]. The Haefliger-Wu invariant is amanifestation of these ideas in the theory of embeddings. The complement to the diagonalidea originated from two celebrated theorems: the Lefschetz Fixed Point Theorem andthe Borsuk-Ulam Antipodes Theorem
2 Construction and examples
For a manifold , denotes the deleted product of , i.e. 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 ]
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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