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
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