Talk:Thom spaces (Ex)

Revision as of 09:41, 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 $$ \Omega_n(\overline{i_k}): \Omega_n(\gamma_k)\to\Omega_n(\gamma_{k+1}),\quad [M,i,f,\overline{f}]\mapsto[M,j,i_k\circ f,\overline{i_k}\circ(\overline{f}\oplus\mathrm{id}_{\underline{\mathbb{R}}})]. $$ </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 $k\geq0$

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