Talk:Smoothings of products (Ex)
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Revision as of 19:53, 29 August 2013 by Christoph Winges (Talk | contribs)
Let and be the given smooth structures.
Recall the definition of
Given a smooth structure , consider . In the diagram
the lift (which is unique up to homotopy) results from the fact that the composition is nullhomotopic ( and are lifts of the same PL structure). Then
The PL homeomorphisms and induce a smooth structure on via the map
where and classify the stable tangent bundle of and , respectively.
We now have
where we have used that is homotopy associative and homotopy commutative.
Since is the homotopy fibre of the infinite loop map , a lift of this map is now given by ( denoting now the induced -space structure on ).
Therefore, we get