Talk:Normal bordism - definitions (Ex)
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2) there exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have | 2) there exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have | ||
$I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally | $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally | ||
+ | |||
+ | 3) there exists a vector bundle $\eta$: $E'\to X\times[0,1]$ of rank $k$ | ||
+ | and for $j=0,1$ there exist vector bundle isomorphisms $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$ | ||
+ | |||
+ | 4) there exists a bundle map $(F,\overline{F})$: $\nu(W,I)\to\eta$ such that for $j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$ | ||
+ | and such that $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ | ||
+ | has degree one as a map between Poincare pairs. | ||
+ | |||
+ | 5) for $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that | ||
+ | |||
+ | a) $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism | ||
+ | |||
+ | b) $F\circ U_j|_{M_j}=f_j$ | ||
+ | |||
+ | c) the induced bundle map $(U_j,\nu(U_j))$: $\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$ satisfies $H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$. | ||
</wikitex> | </wikitex> |
Revision as of 11:06, 2 April 2012
Part 1
Let be a connected finite Poincare complex of dimension and let . 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 .