Talk:Normal bordism - definitions (Ex)

In both parts let $X$$<wikitex>; In both parts let $X$ be a connected finite Poincare complex of dimension $n$ and let $k\geq0$. Part 1 The following definition of the set of normal maps $\mathcal{N}_n(X,k)$ is similar to {{citeD|Lück2001|Definition 3.46}}. We define $$ \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:\,M\to X\textrm{ of degree }1 \end{array} \right\} $$ where we identify $(\xi_0,M_0,i_0,f_0,\overline{f_0})\sim(\xi_1,M_1,i_1,f_1,\overline{f_1})$ iff 1) There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$. 2) There exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally. 3) There exists a vector bundle $\eta$: $E'\to X\times[0,1]$ of rank $k$ and for $j=0,1$ there exist vector bundle isomorphisms $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$. 4) There exists a bundle map $(F,\overline{F})$: $\nu(W,I)\to\eta$ such that for $j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$ and such that $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ has degree one as a map between Poincare pairs. 5) For $j=0,1$ there exist diffeomorphisms $U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that a) $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism b) $F\circ U_j|_{M_j}=f_j$ c) the induced bundle map $(U_j,\nu(U_j))$: $\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$ satisfies $H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$. Part 2 The following definition of the set of tangential normal maps $\mathcal{N}^T_n(X,k)$ differs from {{citeD|Lück2001|Definition 3.50}}. We define $$ \mathcal{N}^T_n(X,k):= \left\{ [\xi,M,a,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, a\in\mathbb{N}_0,\, (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, f:\,M\to X\textrm{ of degree }1 \end{array} \right\} $$ where we identify $(\xi_0,M_0,a_0,f_0,\overline{f_0})\sim(\xi_1,M_1,a_1,f_1,\overline{f_1})$ iff 1) There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$. 2) There exists a vector bundle $\eta$: $E'\to X\times[0,1]$ and there exist $b\in\mathbb{N}_0$ and a bundle map $(F,\overline{F})$: $TW\oplus\underline{\mathbb{R}^b}\to\eta$ such that for $j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$ and such that $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ has degree one as a map between Poincare pairs. 3) For $j=0,1$ there exist diffeomorphisms $U_j$: $M_j\to\partial_jW$ such that $F\circ U_j=f_j$. 4) For $j=0,1$ there exist bundle isomorphisms $(\mathrm{id}_X,V_j)$: $\xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}}\to\eta|_{X\times\{j\}}$ such that $$ \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\}} } $$ commutes. Here $TU_j$: $TM_j\to TW|_{\partial_jW}$ is the differential of $U_j$ and $n_j$: $\underline{\mathbb{R}}\to TW|_{\partial_jW}$ is given by an inward normal field of $TW|_{\partial_jW}$. </wikitex>X$ be a connected finite Poincare complex of dimension $n$$n$ and let $k\geq0$$k\geq0$.

Part 1

The following definition of the set of normal maps $\mathcal{N}_n(X,k)$$\mathcal{N}_n(X,k)$ is similar to [Lück2001, Definition 3.46]. We define

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

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 a compact manifold $W$$W$ of dimension $n+1$$n+1$ whose boundary can be written as $\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.

3) There exists a vector bundle $\eta$$\eta$: $E'\to X\times[0,1]$$E'\to X\times[0,1]$ of rank $k$$k$ and for $j=0,1$$j=0,1$ there exist vector bundle isomorphisms $(\mathrm{id}_X,H_j)$$(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$$\eta|_{X\times\{j\}}\to\xi_j$.

4) There exists a bundle map $(F,\overline{F})$$(F,\overline{F})$: $\nu(W,I)\to\eta$$\nu(W,I)\to\eta$ such that for $j=0,1$$j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$$F(\partial_jW)\subset X\times\{j\}$ and such that $F$$F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$$(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$$j=0,1$ there exist diffeomorphisms $U_j$$U_j$: $\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$$\mathbb{R}^{n+k}\to\mathbb{R}^{n+k}\times\{j\}$ such that

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

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

c) the induced bundle map $(U_j,\nu(U_j))$$(U_j,\nu(U_j))$: $\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$$\nu(M_j,i_j)\to\nu(W,I)|_{\partial_jW}$ satisfies $H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$$H_j\circ\overline{F}\circ\nu(U_j)=\overline{f_j}$.

Part 2

The following definition of the set of tangential normal maps $\mathcal{N}^T_n(X,k)$$\mathcal{N}^T_n(X,k)$ differs from [Lück2001, Definition 3.50]. We define

$\displaystyle \mathcal{N}^T_n(X,k):= \left\{ [\xi,M,a,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, a\in\mathbb{N}_0,\, (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, f:\,M\to X\textrm{ of degree }1 \end{array} \right\}$ $$\displaystyle \mathcal{N}^T_n(X,k):= \left\{ [\xi,M,a,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, a\in\mathbb{N}_0,\, (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, f:\,M\to X\textrm{ of degree }1 \end{array} \right\}$$

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})$$(\xi_0,M_0,a_0,f_0,\overline{f_0})\sim(\xi_1,M_1,a_1,f_1,\overline{f_1})$ iff

1) There exists a compact manifold $W$$W$ of dimension $n+1$$n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$$\partial W=\partial_0W\amalg\partial_1W$.

2) There exists a vector bundle $\eta$$\eta$: $E'\to X\times[0,1]$$E'\to X\times[0,1]$ and there exist $b\in\mathbb{N}_0$$b\in\mathbb{N}_0$ and a bundle map $(F,\overline{F})$$(F,\overline{F})$: $TW\oplus\underline{\mathbb{R}^b}\to\eta$$TW\oplus\underline{\mathbb{R}^b}\to\eta$ such that for $j=0,1$$j=0,1$ we have $F(\partial_jW)\subset X\times\{j\}$$F(\partial_jW)\subset X\times\{j\}$ and such that $F$$F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$$(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ has degree one as a map between Poincare pairs.

3) For $j=0,1$$j=0,1$ there exist diffeomorphisms $U_j$$U_j$: $M_j\to\partial_jW$$M_j\to\partial_jW$ such that $F\circ U_j=f_j$$F\circ U_j=f_j$.

4) For $j=0,1$$j=0,1$ there exist bundle isomorphisms $(\mathrm{id}_X,V_j)$$(\mathrm{id}_X,V_j)$: $\xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}}\to\eta|_{X\times\{j\}}$$\xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}}\to\eta|_{X\times\{j\}}$ such that

$\displaystyle \xymatrix{TM_j\oplus\underline{\mathbb{R}}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}}} \ar[d]_{TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}}} & & \xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}} \ar[d]^{V_j}\\ TW|_{\partial_jW}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{F}|_{\partial_jW}} & & \eta|_{X\times\{j\}} }$

commutes. Here $TU_j$$TU_j$: $TM_j\to TW|_{\partial_jW}$$TM_j\to TW|_{\partial_jW}$ is the differential of $U_j$$U_j$ and $n_j$$n_j$: $\underline{\mathbb{R}}\to TW|_{\partial_jW}$$\underline{\mathbb{R}}\to TW|_{\partial_jW}$ is given by an inward normal field of $TW|_{\partial_jW}$$TW|_{\partial_jW}$.