Talk:Normal bordism - definitions (Ex)
In both parts let be a connected finite Poincare complex of dimension and let .
Part 1
The following definition of the set of normal maps is similar to [Lück2001, Definition 3.46]. We define
where we identify iff
1) there exists compact manifold of dimension such that
2) there exists an embedding : such that for we have and meets transversally
3) there exists a vector bundle : of rank and for there exist vector bundle isomorphisms :
4) there exists a bundle map : such that for we have and such that : has degree one as a map between Poincare pairs.
5) for there exist diffeomorphisms : such that
a) : is a diffeomorphism
b)
c) the induced bundle map : satisfies .
Part 2
The following definition of the set of tangential normal maps differs from [Lück2001, Definition 3.50]. We define