Talk:Normal bordism - definitions (Ex)

Part 1

Let $X$$<wikitex>; Part 1 Let $X$ be a connected finite Poincare complex of dimension $n$ and let $k\geq0$. We define $$ \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\} $$ where we identify $(\xi_0,M_0,i_0,f_0,\overline{f_0})\sim(\xi_1,M_1,i_1,f_1,\overline{f_1})$ iff 1) there exists $W$ compact manifold of dimension $n+1$ such that $\partial W=\partial_0W\amalg\partial_1W$ 2) there exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally </wikitex>X$ be a connected finite Poincare complex of dimension $n$$n$ and let $k\geq0$$k\geq0$. We define

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

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

2) there exists an embedding $I$$I$: $W\to\mathbb{R}^{n+k}\times[0,1]$$W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$$j=0,1$ we have $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$$I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$$W$ meets $\mathbb{R}^{n+k}\times\{j\}$$\mathbb{R}^{n+k}\times\{j\}$ transversally