Talk:Normal bordism - definitions (Ex)
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In both parts let $X$ be a connected finite Poincare complex of dimension $n$ and let $k\geq0$. | 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}}. | The following definition of the set of normal maps $\mathcal{N}_n(X,k)$ is similar to {{citeD|Lück2001|Definition 3.46}}. | ||
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c) the induced bundle map $(U_j,\nu(U_j))$: $\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$ satisfies $H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$. | c) the induced bundle map $(U_j,\nu(U_j))$: $\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$ satisfies $H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$. | ||
− | Part 2 | + | '''Part 2''' |
The following definition of the set of tangential normal maps $\mathcal{N}^T_n(X,k)$ differs from {{citeD|Lück2001|Definition 3.50}}. | The following definition of the set of tangential normal maps $\mathcal{N}^T_n(X,k)$ differs from {{citeD|Lück2001|Definition 3.50}}. |
Revision as of 12:16, 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 a compact manifold of dimension whose boundary can be written as .
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
where we identify iff
1) There exists a compact manifold of dimension whose boundary can be written as .
2) 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.
3) For there exist diffeomorphisms : such that .
4) For there exist bundle isomorphisms : such that
commutes. Here : is the differential of and : is given by an inward normal field of .