Some calculations involving configuration spaces of distinct points
(→Uniqueness theorems) |
m (→Uniqueness theorems) |
||
Line 54: | Line 54: | ||
''Proof.'' | ''Proof.'' | ||
− | |||
− | |||
Take an arbitrary equivariant triangulation $T$ of $\widetilde N$. | Take an arbitrary equivariant triangulation $T$ of $\widetilde N$. | ||
Revision as of 14:49, 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.
The deleted product is This is the configuration space of ordered pairs of distinct points of .
Suppose that is an embedding of a subset . Then the map is well-defined by the Gauss formula
We have , i.e. this map is equivariant with respect to the `exchanging factors' involution on and the antipodal involution on . Thus the existence of an equivariant map is a necessary condition for the embeddability of in .
Definition 1.1.[of the Haefliger-Wu invariant ]
The Haefliger-Wu invariant is induced by the Gauss map, also denoted by .
Theorem 1.2. The Haefliger-Wu invariant is one-to-one for .
2 Uniqueness theorems
Lemma 2.1. Assume that is a compact -manifold and either
(a) or
(b) is connected and .
Then every two every two equivariant maps are equivariantly homotopic.
Proof.
Take an arbitrary equivariant triangulationTex syntax errorof .
(a) One can easily construct an equivariant homotopy between restrictions of and
on vertices ofTex syntax error.
By general position a homotopy of on the boundary of a -simplex can be extended to a homotopy on the whole -simplex since . We extend equivariant homotopy on symmetric simplices in the symmetric way, so we obtain an equivariant homotopy.
(b) Since has non-empty boundary, there exists an equivariant deformational retraction of to an equivariant -subcomplex ofTex syntax error. A homotopy on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on .
QED
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...