Talk:Normal bordism - definitions (Ex)
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where 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 | where 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 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. | |
− | + | # 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 $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$. | |
− | $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally. | + | # 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. |
− | + | # For $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that | |
− | + | #* $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism | |
− | and for $j=0,1$ there exist vector bundle isomorphisms $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$. | + | #* $F\circ U_j|_{M_j}=f_j$ |
− | + | #* the induced bundle map $(U_j,\nu(U_j))$: $\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$ satisfies $H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$. | |
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− | 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. | + | |
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'''Part 2''' | '''Part 2''' |
Revision as of 12:29, 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
- 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
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 .