Talk:Normal bordism - definitions (Ex)

In both parts let $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ be a connected finite Poincare complex of dimension $n$$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

$\displaystyle \mathcal{N}_n(X,k):= \left\{ (\xi,M,i,f,\overline{f}) \right\}/\sim$ $$\displaystyle \mathcal{N}_n(X,k):= \left\{ (\xi,M,i,f,\overline{f}) \right\}/\sim$$

Here

1. $\xi$$\xi$ is a vector bundle of rank k over $X$$X$,
2. $M$$M$ is a closed manifold of dimension n,
3. $i:\,M\to\mathbb{R}^{n+k}$$i:\,M\to\mathbb{R}^{n+k}$ is an embedding,
4. $(f,\overline{f}):\,\nu(M,i) \to \xi$$(f,\overline{f}):\,\nu(M,i) \to \xi$ is a bundle map,
5. $f:\,M\to X$$f:\,M\to X$ is of degree $1$$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})$$(\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\}$$F(\partial_jW)\subset X\times\{j\}$
   • $F$$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
   • $U_j|_{M_j}$$U_j|_{M_j}$: $M_j\to\partial_jW$$M_j\to\partial_jW$ is a diffeomorphism
   • $F\circ U_j|_{M_j}=f_j$$F\circ U_j|_{M_j}=f_j$
   • the induced bundle map $(U_j,\nu(U_j))$$(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

$\displaystyle \mathcal{N}^T_n(X,k):= \left\{ (\xi,M,a,f,\overline{f}) \right\}/\sim$ $$\displaystyle \mathcal{N}^T_n(X,k):= \left\{ (\xi,M,a,f,\overline{f}) \right\}/\sim$$

Here

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