Normal Invariant - Different Categories

Different Categories

The above bijection gives a structure of an abelian group since the space is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary cohomology theory defined by that inifinite loop space. Note that similar ideas apply in the other categories of manifolds and one has bijections

, and, and

It is well known that the spaces

, and

are mutually not homotopy equivalent and hence one obtains three different cohomology theories.

Sullivan analyzed the cases and . He showed that these spaces possess alternative inifinite loop space structures which are in fact better from the following point of view: Recall that there is a surgery obstruction map from normal invariants to the L-group. With the above described groups structure on the normal invariants this map is NOT a homomorphism. However, with the group structure from Sullivan's theorem it becomes a homomorphism in the categories, and . His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.

Read more about this topic:  Normal Invariant

Famous quotes containing the word categories:

    all the categories which we employ to describe conscious mental acts, such as ideas, purposes, resolutions, and so on, can be applied to ... these latent states.
    Sigmund Freud (1856–1939)

    The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.
    Gerald M. Edelman (b. 1928)