Talk:Normal bordism - definitions (Ex)

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In both parts let $<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 $$ \mathcal{N}_n(X,k):= \left\{ [\xi,M,i,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, i:\,M\to\mathbb{R}^{n+k}\textrm{ embedding},\, (f,\overline{f}):\,\nu(M,i)\to\xi\textrm{ bundle map},\, f\textrm{ of degree }1 \end{array} \right\} $$ 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 $$ \mathcal{N}^T_n(X,k):= \left\{ [\xi,M,a,f,\overline{f}] | \begin{array}{l} \xi\textrm{ vector bundle of rank }k\textrm{ over }X,\, M\textrm{ closed manifold of dimension }n,\, a\in\mathbb{N}_0,\, (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi\textrm{ bundle map},\, f\textrm{ of degree }1 \end{array} \right\} $$ 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 $V_j\circ(\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}})=\overline{F}|_{\partial_jW}\circ(TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}})$. 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 $n$ and let $k\geq0$ .

Part 1

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

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

where we identify $(\xi_0,M_0,i_0,f_0,\overline{f_0})\sim(\xi_1,M_1,i_1,f_1,\overline{f_1})$ 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 [Lück2001, Definition 3.50]. We define

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

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 $V_j\circ(\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}})=\overline{F}|_{\partial_jW}\circ(TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}})$ . 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 $TW|_{\partial_jW}$ .

Talk:Normal bordism - definitions (Ex)

Revision as of 11:44, 2 April 2012

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@@ Line 22: / Line 22: @@
 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
-) there exists $W$ compact manifold of dimension $n+1$ such that $\partial W=\partial_0W\amalg\partial_1W$
+) There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$.
-) there exists an embedding $I$: $W\to\mathbb{R}^{n+k}\times[0,1]$ such that for $j=0,1$ we have
+) 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
+$I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ and $W$ meets $\mathbb{R}^{n+k}\times\{j\}$ transversally.
-) there exists a vector bundle $\eta$: $E'\to X\times[0,1]$ of rank $k$
+) 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 $(\mathrm{id}_X,H_j)$: $\eta|_{X\times\{j\}}\to\xi_j$
+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 $F(\partial_jW)\subset X\times\{j\}$
 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.
-) 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
 a) $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism
@@ Line 59: / Line 59: @@
 \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
+) There exists a compact manifold $W$ of dimension $n+1$ whose boundary can be written as $\partial W=\partial_0W\amalg\partial_1W$.
+) 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.
+) For $j=0,1$ there exist diffeomorphisms $U_j$: $M_j\to\partial_jW$ such that $F\circ U_j=f_j$.
+) 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 $V_j\circ(\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}})=\overline{F}|_{\partial_jW}\circ(TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}})$.
+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>