Talk:Normal bordism - definitions (Ex)
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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
Here
- is a vector bundle of rank k over ,
- is a closed manifold of dimension n,
- is an embedding,
- is a bundle map,
- is of degree .
We identify iff
- There exists a compact manifold of dimension whose boundary can be written as .
- There exists an embedding : such that for we have and meets transversally.
- There exists a vector bundle : of rank and for there exist vector bundle isomorphisms : .
- There exists a bundle map : such that
- for we have
- : has degree one as a map between Poincare pairs.
- For there exist diffeomorphisms : such that
- : is a diffeomorphism
- 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
Here
- is a vector bundle of rank k over ,
- is a closed manifold of dimension n,
- ,
- is a bundle map and
- is degree 1.
We identify iff
- There exists a compact manifold of dimension whose boundary can be written as .
- There exists a vector bundle : and there exist and a bundle map : such that for we have and such that : has degree one as a map between Poincare pairs.
- For there exist diffeomorphisms : such that .
- For there exist bundle isomorphisms : such that commutes. Here : is the differential of and : is given by an inward normal field of .