Talk:Thom spaces (Ex)

Latest revision as of 12:17, 2 April 2012

Part 1

We define

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

and

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

where $<wikitex>; '''Part 1''' We define $$ \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\to\mathrm{Th}(\xi_1\times\xi_2),\quad [v_1,v_2]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }v_1=\infty\textrm{ or }v_2=\infty \ v_1\oplus v_2, & \textrm{else}\end{array} \right. $$ and $$ S^1\wedge\mathrm{Th}(\xi)\to\mathrm{Th}(\xi\oplus\underline{\mathbb{R}}),\quad [z,v]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }z=1\textrm{ or }v=\infty \ v\oplus\cot(\mathrm{arg}(z)/2), & \textrm{else}\end{array} \right. $$ where $\mathrm{arg}(z)\in(0,2\pi)$. '''Part 2''' If $i$: $M\to\mathbb{R}^{n+k}$ is an embedding, we denote by $j$: $M\to\mathbb{R}^{n+k+1}$ the composition of $i$ with the inclusion $\mathbb{R}^{n+k}=\mathbb{R}^{n+k}\times\{0\}\subset\mathbb{R}^{n+k+1}$. In particular the normal bundles are related by $\nu(M,j)=\nu(M,i)\oplus\underline{\mathbb{R}}$. The bundle map $(i_k,\overline{i_k})$ induces $$ \Omega_n(\overline{i_k}): \Omega_n(\gamma_k)\to\Omega_n(\gamma_{k+1}),\quad [M,i,f,\overline{f}]\mapsto[M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})]. $$ From the definition $$ V_k:\quad\Omega_n(\gamma_k)\to\Omega_n(X),\quad [M,i,f,\overline{f}]\mapsto[M,\mathrm{pr}_X\circ f] $$ we find for all $k\geq0$ $$ (V_{k+1}\circ\Omega_n(\overline{i_k}))([M,i,f,\overline{f}]) =[M,\mathrm{pr}_X\circ i_k\circ f] =[M,\mathrm{pr}_X\circ(\mathrm{id_X\times j_k})\circ f] =[M,\mathrm{pr}_X\circ f] =V_k([M,i,f,\overline{f}]). $$ '''Part 3''' Let the embeddings $i$ and $j$ be as in Part 2. Define the collapse maps $$ c_k:\quad S^{n+k}\to\mathrm{Th}(\nu(M,i)),\quad c_{k+1}:\quad S^{n+k+1}\to\mathrm{Th}(\nu(M,j)) $$ as in {{citeD|Lück2001|page 57}}. Then we have $c_{k+1}=\Sigma c_k$. For all $k\geq0$ we obtain $$ (P_n(\gamma_{k+1})\circ\Omega_n(\overline{i_k}))([M,i,f,\overline{f}]) =P_n(\gamma_{k+1})([M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})]) =[\mathrm{Th}(\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}}))\circ c_{k+1}] =[\mathrm{Th}(\overline{i_k})\circ\mathrm{Th}(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})\circ c_{k+1}] $$ and $$ (s_k\circ P_n(\gamma_k))([M,i,f,\overline{f}]) =s_k([\mathrm{Th}(\overline{f})\circ c_k]) =[\mathrm{Th}(\overline{i_k})\circ\Sigma(\mathrm{Th}(\overline{f})\circ c_k)] =[\mathrm{Th}(\overline{i_k})\circ\mathrm{Th}(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})\circ c_{k+1}]. $$ '''Part 4''' Of course one can do similar things for non oriented manifolds or spin manifolds. One only has to modify the definition of $\Omega_n(X)$ and use the corresponding universal bundle instead of $\xi_k$. </wikitex>\mathrm{arg}(z)\in(0,2\pi)$.

Part 2

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

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

From the definition

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

we find for all $k\geq0$

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

Part 3

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

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

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

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

and

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

Part 4

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