Normal Invariants
A degree one normal map consists of the following data: an -dimensional oriented closed manifold, a map which is of degree one (that means, and a bundle map from the stable tangent bundle of to some bundle over . Two such maps are equivalent if there exists a normal bordism between them (that means a bordism of the sources covered by suitable bundle data). The equivalence classes of degree one normal maps are called normal invariants.
When defined like this the normal invariants are just a pointed set, with the base point given by . However the Pontrjagin-Thom construction gives a structure of an abelian group. In fact we have a non-natural bijection
where denotes the homotopy fiber of the map, which is an infinite loop space and hence maps into it define a generalized cohomology theory. There are corresponding identifications of the normal invariants with when working with PL-manifolds and with when working with topological manifolds.
Read more about this topic: Surgery Exact Sequence, The Entries
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