Talk:Thom spaces (Ex)
(Difference between revisions)
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Part 2 | Part 2 | ||
− | The map $\Omega_n(\overline{i_k}) | + | 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>; | </wikitex>; |
Revision as of 09:14, 2 April 2012
Part 1
We define
and
where .
Part 2
If : is an embedding, we denote by : the composition of with the inclusion . In particular the normal bundles are related by . The bundle map induces
;