Talk:Thom spaces (Ex)

Part 1

We define

$\displaystyle \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\to\mathrm{Th}(\xi_1\times\xi_2),\quad [v_1,v_2]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }v_1=\infty\textrm{ or }v_2=\infty \\ v_1\oplus v_2, & \textrm{else}\end{array} \right.$ $$\displaystyle \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\to\mathrm{Th}(\xi_1\times\xi_2),\quad [v_1,v_2]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }v_1=\infty\textrm{ or }v_2=\infty \\ v_1\oplus v_2, & \textrm{else}\end{array} \right.$$

and

$\displaystyle S^1\wedge\mathrm{Th}(\xi)\to\mathrm{Th}(\xi\oplus\underline{\mathbb{R}}),\quad [z,v]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }z=1\textrm{ or }v=\infty \\ v\oplus\cot(\mathrm{arg}(z)/2), & \textrm{else}\end{array} \right.$ $$\displaystyle S^1\wedge\mathrm{Th}(\xi)\to\mathrm{Th}(\xi\oplus\underline{\mathbb{R}}),\quad [z,v]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }z=1\textrm{ or }v=\infty \\ v\oplus\cot(\mathrm{arg}(z)/2), & \textrm{else}\end{array} \right.$$

where $\mathrm{arg}(z)\in(0,2\pi)$$\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}{^}\mathrm{arg}(z)\in(0,2\pi)$.

Part 2

If $i$$i$: $M\to\mathbb{R}^{n+k}$$M\to\mathbb{R}^{n+k}$ is an embedding, we denote by $j$$j$: $M\to\mathbb{R}^{n+k+1}$$M\to\mathbb{R}^{n+k+1}$ the composition of $i$$i$ with the inclusion $\mathbb{R}^{n+k}=\mathbb{R}^{n+k}\times\{0\}\subset\mathbb{R}^{n+k+1}$$\mathbb{R}^{n+k}=\mathbb{R}^{n+k}\times\{0\}\subset\mathbb{R}^{n+k+1}$. In particular the normal bundles are related by $\nu(M,j)=\nu(M,i)\oplus\underline{\mathbb{R}}$$\nu(M,j)=\nu(M,i)\oplus\underline{\mathbb{R}}$. The bundle map $(i_k,\overline{i_k})$$(i_k,\overline{i_k})$ induces

$\displaystyle \Omega_n(\overline{i_k}): \Omega_n(\gamma_k)\to\Omega_n(\gamma_{k+1}),\quad [M,i,f,\overline{f}]\mapsto[M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})].$ $$\displaystyle \Omega_n(\overline{i_k}): \Omega_n(\gamma_k)\to\Omega_n(\gamma_{k+1}),\quad [M,i,f,\overline{f}]\mapsto[M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})].$$

From the definition

$\displaystyle V_k:\quad\Omega_n(\gamma_k)\to\Omega_n(X),\quad [M,i,f,\overline{f}]\mapsto[M,\mathrm{pr}_X\circ f]$ $$\displaystyle V_k:\quad\Omega_n(\gamma_k)\to\Omega_n(X),\quad [M,i,f,\overline{f}]\mapsto[M,\mathrm{pr}_X\circ f]$$

we find for all $k\geq0$$k\geq0$

$\displaystyle (V_{k+1}\circ\Omega_n(\overline{i_k}))([M,i,f,\overline{f}]) =[M,\mathrm{pr}_X\circ i_k\circ f] =[M,\mathrm{pr}_X\circ(\mathrm{id_X\times j_k})\circ f] =[M,\mathrm{pr}_X\circ f] =V_k([M,i,f,\overline{f}]).$ $$\displaystyle (V_{k+1}\circ\Omega_n(\overline{i_k}))([M,i,f,\overline{f}]) =[M,\mathrm{pr}_X\circ i_k\circ f] =[M,\mathrm{pr}_X\circ(\mathrm{id_X\times j_k})\circ f] =[M,\mathrm{pr}_X\circ f] =V_k([M,i,f,\overline{f}]).$$

Part 3

Let the embeddings $i$$i$ and $j$$j$ be as in Part 2. Define the collapse maps

$\displaystyle c_k:\quad S^{n+k}\to\mathrm{Th}(\nu(M,i)),\quad c_{k+1}:\quad S^{n+k+1}\to\mathrm{Th}(\nu(M,j))$ $$\displaystyle c_k:\quad S^{n+k}\to\mathrm{Th}(\nu(M,i)),\quad c_{k+1}:\quad S^{n+k+1}\to\mathrm{Th}(\nu(M,j))$$

as in [Lück2001, page 57]. Then we have $c_{k+1}=\Sigma c_k$$c_{k+1}=\Sigma c_k$. For all $k\geq0$$k\geq0$ we obtain

$\displaystyle (P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k}))([M,i,f,\overline{f}]) =P_n(\gamma_{k+1})([M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})]) =[\mathrm{Th}(\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}}))\circ c_{k+1}] =[\mathrm{Th}(\overline{i_k})\circ\mathrm{Th}(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})\circ c_{k+1}]$ $$\displaystyle (P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k}))([M,i,f,\overline{f}]) =P_n(\gamma_{k+1})([M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})]) =[\mathrm{Th}(\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}}))\circ c_{k+1}] =[\mathrm{Th}(\overline{i_k})\circ\mathrm{Th}(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})\circ c_{k+1}]$$

and

$\displaystyle (s_k\circ P_n(\gamma_k))([M,i,f,\overline{f}]) =s_k([\mathrm{Th}(\overline{f})\circ c_k]) =[\mathrm{Th}(\overline{i_k})\circ\Sigma(\mathrm{Th}(\overline{f})\circ c_k)] =[\mathrm{Th}(\overline{i_k})\circ\mathrm{Th}(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})\circ c_{k+1}].$ $$\displaystyle (s_k\circ P_n(\gamma_k))([M,i,f,\overline{f}]) =s_k([\mathrm{Th}(\overline{f})\circ c_k]) =[\mathrm{Th}(\overline{i_k})\circ\Sigma(\mathrm{Th}(\overline{f})\circ c_k)] =[\mathrm{Th}(\overline{i_k})\circ\mathrm{Th}(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})\circ c_{k+1}].$$

Part 4

Of course one can do similar things for non oriented manifolds or spin manifolds. One only has to modify the definition of $\Omega_n(X)$$\Omega_n(X)$ and use the corresponding universal bundle instead of $\xi_k$$\xi_k$.