# Fake projective spaces in dim 6 (Ex)

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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.

## 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