Talk:Normal bordism - definitions (Ex)

Revision as of 11:13, 2 April 2012

Part 1

Let $<wikitex>; Part 1 Let $X$ be a connected finite Poincare complex of dimension $n$ and let $k\geq0$. We define $$ \mathcal{N}_n(X,k):= \left\{ [\xi,M,i,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, i:\,M\to\mathbb{R}^{n+k}\textrm{ embedding},\, (f,\overline{f}):\,\nu(M,i)\to\xi\textrm{ bundle map},\, f\textrm{ of degree }1 \end{array} \right\} $$ 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 1) there exists $W$ compact manifold of dimension $n+1$ such that $\partial W=\partial_0W\amalg\partial_1W$ 2) 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 3) 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 $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$ 4) 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. 5) for $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that a) $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism b) $F\circ U_j|_{M_j}=f_j$ 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 The following definition of $\mathcal{N}^T_n(X,k)$ differs from the definition given in {{citeD|Lück2001|Definition 3.50}}. </wikitex>X$ be a connected finite Poincare complex of dimension $n$ and let $k\geq0$. We define

$$\displaystyle \mathcal{N}_n(X,k):= \left\{ [\xi,M,i,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, i:\,M\to\mathbb{R}^{n+k}\textrm{ embedding},\, (f,\overline{f}):\,\nu(M,i)\to\xi\textrm{ bundle map},\, f\textrm{ of degree }1 \end{array} \right\}$$

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

1) there exists $W$ compact manifold of dimension $n+1$ such that $\partial W=\partial_0W\amalg\partial_1W$

2) 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

3) 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 $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$

4) 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.

5) for $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that

a) $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism

b) $F\circ U_j|_{M_j}=f_j$

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

The following definition of $\mathcal{N}^T_n(X,k)$ differs from the definition given in [Lück2001, Definition 3.50].