Talk:Thom spaces (Ex)
(Difference between revisions)
Line 62: | Line 62: | ||
and | and | ||
$$ | $$ | ||
− | (s_k\circ P_n(\gamma_k))[M,i,f,\overline{f}] | + | (s_k\circ P_n(\gamma_k))([M,i,f,\overline{f}]) |
=s_k([\mathrm{Th}(\overline{f})\circ c_k]) | =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\Sigma(\mathrm{Th}(\overline{f})\circ c_k)] |
Revision as of 10:20, 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 .