Talk:Structure set (Ex)
From Manifold Atlas
Revision as of 00:59, 28 August 2013 by Marek Kaluba (Talk | contribs)
We begin with a map from the set of manifolds simply homotopy equivalent to to and then we show two things:
- if manifolds and simply homotopy equivalent to are diffeomorphic then their images by the map belong to the same orbit of -action on .
- if two elements of belong to the same orbit of -action, then we will show that they are diffeomorphic.
Let be a smooth manifold and a simple homotopy equivalence. Consider a map which takes to . Suppose now that is a manifold diffeomorphic to , and a simple homotopy equivalence (possibly and ). Then there exists such that the following diagram commutes.
Map is given by composition (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in we have the following equalities.
where denotes the -action. Therefore and belong to the same orbit.Suppose now, that belong to the same orbit of -action. It means, that there exist a simple homotopy equivalence such that
But equality in the simple structure set is just the existence of a diffeomorphism making the following diagram commute.