Some calculations involving configuration spaces of distinct points
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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.
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The Haefliger-Wu invariant
is induced by the Gauss map, also denoted byTex syntax error.
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 Uniqueness theorems
Theorem 2.1. Assume that is a compact -manifold and either
(a) or
(b) is connected and .
Then any two embeddings of into are isotopic.
In cases (a-d) inequality is clearly satisfied. Hence by Haefliger-Weber theorem \ref{thm_Haef-Web} we have . So it is sufficient to show that is trivial, i.e. every two every two equivariant maps are equivariantly homotopic.
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 .
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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