Talk:Normal bordism - definitions (Ex)
(Difference between revisions)
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where 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 | where 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 vector bundle $\eta$: $E'\to X\times[0,1]$ and there exist $b\in\mathbb{N}_0$ and a bundle map $(F,\overline{F})$: $TW\oplus\underline{\mathbb{R}^b}\to\eta$ 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 $U_j$: $M_j\to\partial_jW$ such that $F\circ U_j=f_j$. | |
− | a bundle map $(F,\overline{F})$: $TW\oplus\underline{\mathbb{R}^b}\to\eta$ | + | # For $j=0,1$ there exist bundle isomorphisms $(\mathrm{id}_X,V_j)$: $\xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}}\to\eta|_{X\times\{j\}}$ such that $$ |
− | such that for $j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$ | + | |
− | and such that $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ | + | |
− | has degree one as a map between Poincare pairs. | + | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | such that | + | |
− | $$ | + | |
\xymatrix{TM_j\oplus\underline{\mathbb{R}}\oplus\underline{\mathbb{R}^b} | \xymatrix{TM_j\oplus\underline{\mathbb{R}}\oplus\underline{\mathbb{R}^b} | ||
\ar[rr]^{\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}}} | \ar[rr]^{\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}}} | ||
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\eta|_{X\times\{j\}} | \eta|_{X\times\{j\}} | ||
} | } | ||
− | $$ | + | $$ commutes. Here $TU_j$: $TM_j\to TW|_{\partial_jW}$ is the differential of $U_j$ and $n_j$: $\underline{\mathbb{R}}\to TW|_{\partial_jW}$ is given by an inward normal field of $TW|_{\partial_jW}$. |
− | commutes. | + | |
− | Here $TU_j$: $TM_j\to TW|_{\partial_jW}$ is the differential of $U_j$ and $n_j$: $\underline{\mathbb{R}}\to TW|_{\partial_jW}$ is given by an inward normal field of $TW|_{\partial_jW}$. | + | |
</wikitex> | </wikitex> |
Revision as of 12:31, 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
- 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 .