Talk:Normal bordism - definitions (Ex)
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<wikitex>; | <wikitex>; | ||
+ | |||
+ | In both parts let $X$ be a connected finite Poincare complex of dimension $n$ and let $k\geq0$. | ||
+ | |||
Part 1 | Part 1 | ||
− | + | The following definition of the set of normal maps $\mathcal{N}_n(X,k)$ is similar to {{citeD|Lück2001|Definition 3.46}}. | |
We define | We define | ||
$$ | $$ | ||
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Part 2 | Part 2 | ||
− | The following definition of $\mathcal{N}^T_n(X,k)$ differs from | + | The following definition of the set of tangential normal maps $\mathcal{N}^T_n(X,k)$ differs from {{citeD|Lück2001|Definition 3.50}}. |
+ | We define | ||
+ | $$ | ||
+ | \mathcal{N}^T_n(X,k):= | ||
+ | \left\{ | ||
+ | [\xi,M,a,f,\overline{f}] | | ||
+ | \begin{array}{l} | ||
+ | \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, | ||
+ | M\textrm{ closed manifold of dimension }n,\, | ||
+ | a\in\mathbb{N}_0,\, | ||
+ | (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, | ||
+ | f\textrm{ of degree }1 | ||
+ | \end{array} | ||
+ | \right\} | ||
+ | $$ | ||
</wikitex> | </wikitex> |
Revision as of 11:23, 2 April 2012
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