# 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.

**Lemma 0.2.**We can perform surgery on

**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.**

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.**

**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.

## References

- [Madsen&Milgram1979] I. Madsen and R. J. Milgram,
*The classifying spaces for surgery and cobordism of manifolds*, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002