Fake projective spaces in dim 6 (Ex)

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There is a degree 1 normal map
\displaystyle h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.
We may pull back the canonical line bundle H_5\to \mathbb{C}P^5 over \widetilde{\mathbb{C}P^5}. By the Poincaré conjecture, the sphere bundle S(h^*(H_5))=\partial D(h^*(H_5)) is PL-homeomorphic to the sphere S^{11}. \widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12} is a homotopy complex projective space obtained by coning off the boundary of the disk bundle. Similarly we may form the connected sum \widetilde{\mathbb{C}P^5}\# M^{10}_A, where M^{10}_A is the Kervaire manifold, and once again, we take pullback of H_5 line bundle via the natural degree 1 normal map. Now to obtain a homotopy equivalence \partial D(f^*(H_5))\to \partial D(H_5) we have to use a little bit of surgery. Nevertheless there exists a manifold W^{12} such that
\displaystyle D(f^*(H_5))\cup W^{12}\to D(H_5)
is a homotopy equivalence of manifolds with boundary. Again since the boundary is already PL-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.

Lemma 0.1. \mathbb{C}P^6, \widetilde{\mathbb{C}P^6} and f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}, 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.


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