Talk:Thom spaces (Ex)

Revision as of 08:57, 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>; {\bf Part 1} We define $$ \mathrm{Th}(\xi_1)\wedge\mathrm{Th}(\xi_2)\to\mathrm{Th}(\xi_1\times\xi_2),\quad [v_1,v_2]\mapsto \left\{ \begin{array}{ll}\infty, & \textrm{if }v_1=\infty\textrm{ or }v_2=\infty \ v_1\oplus v_2, & \textrm{else}\end{array} \right. $$ </wikitex>;\mathrm{arg}(z)\in(0,2\pi)$.

Part 2

The map $\Omega_n(\overline{i_k})$: $\Omega_n(\gamma_k)\to\Omega_n(\gamma_{k+1})$ is given by ;