Talk:Normal bordism - definitions (Ex)
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Revision as of 13:09, 2 April 2012 by Diarmuid Crowley (Talk | contribs)
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
![\displaystyle \mathcal{N}_n(X,k):= \left\{ [\xi,M,i,f,\overline{f}] \right\}/\simeq](/images/math/7/3/d/73dd33522855af53bc4c87afa287054f.png)
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
and such that
:
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
![\displaystyle \mathcal{N}^T_n(X,k):= \left\{ [\xi,M,a,f,\overline{f}] \right\}/\simeq](/images/math/d/6/8/d681026d44615a208f14102b36eed814.png)
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
.