Talk:Normal bordism - definitions (Ex)
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i:\,M\to\mathbb{R}^{n+k}\textrm{ embedding},\, | i:\,M\to\mathbb{R}^{n+k}\textrm{ embedding},\, | ||
(f,\overline{f}):\,\nu(M,i)\to\xi\textrm{ bundle map},\, | (f,\overline{f}):\,\nu(M,i)\to\xi\textrm{ bundle map},\, | ||
− | f\textrm{ of degree }1 | + | f:\,M\to X\textrm{ of degree }1 |
\end{array} | \end{array} | ||
\right\} | \right\} | ||
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a\in\mathbb{N}_0,\, | a\in\mathbb{N}_0,\, | ||
(f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, | (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, | ||
− | f\textrm{ of degree }1 | + | f:\,M\to X\textrm{ of degree }1 |
\end{array} | \end{array} | ||
\right\} | \right\} | ||
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4) For $j=0,1$ there exist bundle isomorphisms $(\mathrm{id}_X,V_j)$: $\xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}}\to\eta|_{X\times\{j\}}$ | 4) 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 |
+ | $$ | ||
+ | \xymatrix{TM_j\oplus\underline{\mathbb{R}}\oplus\underline{\mathbb{R}^b} | ||
+ | \ar[rr]^{\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}}} | ||
+ | \ar[d]_{TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}}} | ||
+ | & & | ||
+ | \xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}} | ||
+ | \ar[d]^{V_j}\\ | ||
+ | TW|_{\partial_jW}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{F}|_{\partial_jW}} | ||
+ | & & | ||
+ | \eta|_{X\times\{j\}} | ||
+ | } | ||
+ | $$ | ||
+ | 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 $TW|_{\partial_jW}$. | 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 12:11, 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 .