Talk:Normal bordism - definitions (Ex)
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\mathcal{N}_n(X,k):= | \mathcal{N}_n(X,k):= | ||
\left\{ | \left\{ | ||
− | + | (\xi,M,i,f,\overline{f}) \right\}/\simeq$$ | |
Here | Here | ||
# $\xi$ is a vector bundle of rank k over $X$, | # $\xi$ is a vector bundle of rank k over $X$, | ||
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# There exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally. | # There exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally. | ||
# There exists a vector bundle $\eta$: $E'\to X\times[0,1]$ of rank $k$ and for $j=0,1$ there exist vector bundle isomorphisms $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$. | # There exists a vector bundle $\eta$: $E'\to X\times[0,1]$ of rank $k$ and for $j=0,1$ there exist vector bundle isomorphisms $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$. | ||
− | # There exists a bundle map $(F,\overline{F})$: $\nu(W,I)\to\eta$ such that for $j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$ | + | # There exists a bundle map $(F,\overline{F})$: $\nu(W,I)\to\eta$ such that |
+ | #* for $j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$ | ||
+ | #* $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ has degree one as a map between Poincare pairs. | ||
# For $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that | # For $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that | ||
#* $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism | #* $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism | ||
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\mathcal{N}^T_n(X,k):= | \mathcal{N}^T_n(X,k):= | ||
\left\{ | \left\{ | ||
− | + | (\xi,M,a,f,\overline{f}) \right\}/\simeq $$ | |
Here | Here | ||
# $\xi$ is a vector bundle of rank k over $X$, | # $\xi$ is a vector bundle of rank k over $X$, |
Revision as of 08:18, 3 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
Here
- is a vector bundle of rank k over ,
- is a closed manifold of dimension n,
- is an embedding,
- is a bundle map,
- is of degree .
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
- : 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
Here
- is a vector bundle of rank k over ,
- is a closed manifold of dimension n,
- ,
- is a bundle map and
- is degree 1.
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 .
Part 1
The following definition of the set of normal maps is similar to [Lück2001, Definition 3.46]. We define
Here
- is a vector bundle of rank k over ,
- is a closed manifold of dimension n,
- is an embedding,
- is a bundle map,
- is of degree .
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
- : 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
Here
- is a vector bundle of rank k over ,
- is a closed manifold of dimension n,
- ,
- is a bundle map and
- is degree 1.
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 .