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.
In introducing notation and definitions we follow slides by A. Skopenkov
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 .
Denote by the set embeddings of into up to isotopy. Let be the set of equivariant maps ̃ up to equivariant homotopy.
The Haefliger-Wu invariant
is induced by the Gauss map.
I.e. .
Theorem 1.1. The Haefliger-Wu invariant is one-to-one for .
See [Skopenkov2006, 5] and [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1] for the DIFF case and [Skopenkov2002, Theorem 1.3] for the PL case.
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 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
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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6 References
- [Haefliger1963] A. Haefliger, Plongements différentiables dans le domain stable., Comment. Math. Helv.37 (1963), 155-176.
- [Skopenkov2002] A. Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions., Comment. Math. Helv. 77 (2002), no.1, 78-124. MRMR1898394 (2003c:57023) Zbl 1012.57035
- [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.