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
Famous quotes containing the word normal:
“Everyone in the full enjoyment of all the blessings of his life, in his normal condition, feels some individual responsibility for the poverty of others. When the sympathies are not blunted by any false philosophy, one feels reproached by one’s own abundance.”
—Elizabeth Cady Stanton (1815–1902)