Talk:Normal bordism - definitions (Ex)

Revision as of 08:18, 3 April 2012 (view source) Andreas Hermann (Talk | contribs) ← Older edit		Latest revision as of 08:20, 3 April 2012 (view source) Andreas Hermann (Talk | contribs)
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	\mathcal{N}_n(X,k):=		\mathcal{N}_n(X,k):=
	\left\{		\left\{
−	(\xi,M,i,f,\overline{f}) \right\}/\simeq$$	+	(\xi,M,i,f,\overline{f}) \right\}/\sim$$
	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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	\mathcal{N}^T_n(X,k):=		\mathcal{N}^T_n(X,k):=
	\left\{		\left\{
−	(\xi,M,a,f,\overline{f}) \right\}/\simeq $$	+	(\xi,M,a,f,\overline{f}) \right\}/\sim $$
	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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Latest revision as of 08:20, 3 April 2012

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 $$ \mathcal{N}_n(X,k):= \left\{ (\xi,M,i,f,\overline{f}) \right\}/\sim$$ Here # $\xi$ is a vector bundle of rank k over $X$, # $M$ is a closed manifold of dimension n, # $i:\,M\to\mathbb{R}^{n+k}$ is an embedding, # $(f,\overline{f}):\,\nu(M,i) \to \xi$ is a bundle map, # $f:\,M\to X $ is of degree 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

Here

1. $\xi$ is a vector bundle of rank k over $X$,
2. $M$ is a closed manifold of dimension n,
3. $i:\,M\to\mathbb{R}^{n+k}$ is an embedding,
4. $(f,\overline{f}):\,\nu(M,i) \to \xi$ is a bundle map,
5. $f:\,M\to X$ is of degree $1$.

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 $F(\partial_jW)\subset X\times\{j\}$
   • $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ 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
   • $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism
   • $F\circ U_j|_{M_j}=f_j$
   • 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

Here

1. $\xi$ is a vector bundle of rank k over $X$,
2. $M$ is a closed manifold of dimension n,
3. $a\in\mathbb{N}_0$,
4. $(f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi$ is a bundle map and
5. $f:\,M\to X$ is degree 1.

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 $F(\partial_jW)\subset X\times\{j\}$
   • $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ 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
   
   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 $TW|_{\partial_jW}$.

$. 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 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 $I^{-1}(\mathbb{R}^{n+k}\times\{j\})=\partial_jW$ 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 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 #* $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism #* $F\circ U_j|_{M_j}=f_j$ #* 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 $$ \mathcal{N}^T_n(X,k):= \left\{ (\xi,M,a,f,\overline{f}) \right\}/\sim $$ Here # $\xi$ is a vector bundle of rank k over $X$, # $M$ is a closed manifold of dimension n, # $a\in\mathbb{N}_0$, # $ (f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi$ is a bundle map and # $ f:\,M\to X$ is degree 1. 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 $F(\partial_jW)\subset X\times\{j\}$ #* $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ 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 $$ \xymatrix{TM_j\oplus\underline{\mathbb{R}}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{f_j}\oplus\mathrm{id}_{\underline{\mathbb{R}^{b-a_j+1}}}} \ar[d]_{TU_j\oplus n_j\oplus\mathrm{id}_{\underline{\mathbb{R}^b}}} & & \xi_j\oplus\underline{\mathbb{R}^{b-a_j+1}} \ar[d]^{V_j}\ TW|_{\partial_jW}\oplus\underline{\mathbb{R}^b} \ar[rr]^{\overline{F}|_{\partial_jW}} & & \eta|_{X\times\{j\}} } $$ 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 $TW|_{\partial_jW}$. 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

Here

1. $\xi$ is a vector bundle of rank k over $X$,
2. $M$ is a closed manifold of dimension n,
3. $i:\,M\to\mathbb{R}^{n+k}$ is an embedding,
4. $(f,\overline{f}):\,\nu(M,i) \to \xi$ is a bundle map,
5. $f:\,M\to X$ is of degree $1$.

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 $F(\partial_jW)\subset X\times\{j\}$
   • $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ 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
   • $U_j|_{M_j}$: $M_j\to\partial_jW$ is a diffeomorphism
   • $F\circ U_j|_{M_j}=f_j$
   • 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

Here

1. $\xi$ is a vector bundle of rank k over $X$,
2. $M$ is a closed manifold of dimension n,
3. $a\in\mathbb{N}_0$,
4. $(f,\overline{f}):\,TM\oplus\underline{\mathbb{R}^a}\to\xi$ is a bundle map and
5. $f:\,M\to X$ is degree 1.

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 $F(\partial_jW)\subset X\times\{j\}$
   • $F$: $(W,\partial W)\to(X\times[0,1],X\times\partial[0,1])$ 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
   
   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 $TW|_{\partial_jW}$.