The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the rel ∂ case the surgery on a manifold pair is considered with the given fixed manifold structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps on the boundary. Consider a normal map to a manifold pair (Y, ∂Y) ⊂ (X, ∂X) with boundary which is a simple homotopy equivalence on the boundary∂X. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups of obstructions to splitting of such mixed structures along submanifold with boundary (Y, ∂Y). We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric examples.
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Communicated by: R. Diestel
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Cencelj, M., Muranov, Y.V. & Repovš, D. On the splitting problem for manifold pairs with boundaries. Abh.Math.Semin.Univ.Hambg. 76, 35–55 (2006). https://doi.org/10.1007/BF02960853
2000 Mathematics Subject Classification
- Primary 57R67
- Secondary 57R10
Key words and phrases
- Surgery obstruction groups
- surgery on manifold pairs
- splitting obstruction
- splitting obstruction groups
- surgery on closed manifold with boundary
- surgery exact sequence
- relative surgery obstruction groups
- relative splitting obstruction groups