Cores of s-cobordisms of 4-manifolds

Details Cores of s-cobordisms of 4-manifolds Cores of s-cobordisms of 4-manifolds

2004-03-31

arxiv.org/abs/math/0403554v1

Mathematics

Geometric Topology

12 pages

Scientific paper

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a ``core'' sub s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an invariant, but it shows the topological s-cobordism question reduces to the core case. The simply-connected version of the decomposition (with 1-complex a point) is due to Curtis, Freedman, Hsiang and Stong. Controlled surgery is used to reduce topological triviality of core s-cobordisms to a question about controlled homotopy equivalence of 4-manifolds. There are speculations about further reductions.

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