Talk:Thom spaces (Ex)
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<wikitex>; | <wikitex>; | ||
− | Part 1 | + | '''Part 1''' |
We define | We define | ||
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where $\mathrm{arg}(z)\in(0,2\pi)$. | where $\mathrm{arg}(z)\in(0,2\pi)$. | ||
− | 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 |
− | </wikitex> | + | 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> |
Latest revision as of 12:17, 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
From the definition
we find for all
Part 3
Let the embeddings and be as in Part 2. Define the collapse maps
as in [Lück2001, page 57]. Then we have . For all we obtain
and
Part 4
Of course one can do similar things for non oriented manifolds or spin manifolds. One only has to modify the definition of and use the corresponding universal bundle instead of .