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. See [Vassiliev1992].
In introducing notation and definitions we follow [Skopenkov2020a].
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. Denote by the equivalence class of the map .
The Haefliger-Wu invariant is defined by formula .
Theorem 1.1. The Haefliger-Wu invariant is one-to-one either
(a) is a compact connected -complex and or
(b) is a compact -manifold with nonempty boundary, is -connected, , , and .
2 Uniqueness theorems
Lemma 2.1. Assume that is a compact -manifold and either
(b) is connected and .
Then each two equivariant maps from to are equivariantly homotopic.
Proof. Given two equivariant maps 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 between and on the boundary of a simplex can be extended to a homotopy on the whole simplex since the dimension of the simplex does not exceed . 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 between and on the subcomplex can by constructed similarly to case~(a). This homotopy can be extended to a homotopy on . QED
- [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.
- [Skopenkov2020a] https://www.mccme.ru/circles/oim/eliminat_talk.pdf
V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications., Amer. Math. Soc., Providence, RI, (1992).