Talk:Normal bordism - definitions (Ex)
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\mathcal{N}_n(X,k):= | \mathcal{N}_n(X,k):= | ||
\left\{ | \left\{ | ||
− | [\xi,M,i,f,\overline{f}] | + | [\xi,M,i,f,\overline{f}] \right\}/\simeq$$ |
− | \ | + | Here |
− | \xi | + | # $\xi$ is a vector bundle of rank k over $X$, |
− | M | + | # $M$ is a closed manifold of dimension n, |
− | i:\,M\to\mathbb{R}^{n+k} | + | # $i:\,M\to\mathbb{R}^{n+k}$ is an embedding, |
− | (f,\overline{f}):\,\nu(M,i)\to\xi | + | # $(f,\overline{f}):\,\nu(M,i) \to \xi$ is a bundle map, |
− | f:\,M\to X | + | # $f:\,M\to X $ is of degree $1$. |
− | + | We identify $(\xi_0,M_0,i_0,f_0,\overline{f_0})\sim(\xi_1,M_1,i_1,f_1,\overline{f_1})$ iff | |
− | + | ||
− | + | ||
− | + | ||
# There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$. | # There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$. | ||
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\mathcal{N}^T_n(X,k):= | \mathcal{N}^T_n(X,k):= | ||
\left\{ | \left\{ | ||
− | [\xi,M,a,f,\overline{f}] | + | [\xi,M,a,f,\overline{f}] \right\}/\simeq $$ |
− | \ | + | Here |
− | \xi | + | # $\xi$ is a vector bundle of rank k over $X$, |
− | M | + | # $M$ is a closed manifold of dimension n, |
− | a\in\mathbb{N}_0 | + | # $a\in\mathbb{N}_0$, |
− | (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi | + | # $ (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi$ is a bundle map and |
− | f:\,M\to X | + | # $ f:\,M\to X$ is degree 1. |
− | + | ||
− | + | We identify $(\xi_0,M_0,a_0,f_0,\overline{f_0})\sim(\xi_1,M_1,a_1,f_1,\overline{f_1})$ iff | |
− | + | ||
− | + | ||
# There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$. | # There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$. |
Revision as of 13:09, 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
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
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 .