Fake projective spaces in dim 6 (Ex)
We may pull back the Hopf bundle to . The sphere bundle of the Hopf bundle over sphere is the true sphere, hence we get a map:
Prove the following:
Lemma 0.1. The map induced above is a degree 1 normal map.
Now, since we are working between odd dimensional manifolds we can by surgery below middle dimension assume that bordant to a homotopy equivalence . Thus by Poincare Conjecture it is indeed normally bordant to a PL-homeomorphism. Choose to be such bordism.
Describe these surgeries.
We obtain a manifold with boundary homotopy homotopy equivalent to with boundary PL-homeomorphic to . Thus we may cone off common boundaries extending the homotopy equivalence at the same time. We define
Lemma 0.3. and are not homeomorphic.
Prove the above lemma.However, the construction above gives us a degree 1 normal map
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.
Describe these surgeries.Nevertheless there exists a manifold such that
Lemma 0.4. , and , are topologically distinct homotopy projective spaces.
The aim of this exercise is to write full details of proof of the above Lemma (which is Lemma 8.24 form [Madsen&Milgram1979]).
A sketch of this proof can be found in the book, on page 170.