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. 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
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
(b) is connected and .
Then every two every two equivariant maps are equivariantly homotopic.
Proof. Take an arbitrary equivariant triangulation of .
(a) One can easily construct an equivariant homotopy between restrictions of and on vertices of . 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 of . A homotopy on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on . QED
5 Further discussion