Fake projective spaces in dim 6 (Ex)
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Revision as of 14:59, 23 March 2012 by Marek Kaluba
We may pull back the canonical line bundle over . By the Poincaré conjecture, the sphere bundle is -homeomorphic to the sphere . is a homotopy complex projective space obtained by coning off the boundary of the disk bundle. Similarly we may form the connected sum , where is the Kervaire manifold, and once again, we take pullback of line bundle via the natural degree 1 normal map. Now to obtain a homotopy equivalence we have to use a little bit of surgery. Nevertheless there exists a manifold such that
is a homotopy equivalence of manifolds with boundary. Again since the boundary is already -homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.
Lemma 0.1. , and , are topologically distinct homotopy projective spaces.
The aim of this exercise is to write full details of proof of Lemma 8.24 form [Madsen&Milgram1979].
A sketch of this proof can be found in the book, on page 170.