# Fake projective spaces in dim 6 (Ex)

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

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

$normal map $$h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$ We may pull back the canonical line bundle$H_5\to \mathbb{C}P^5$over$\widetilde{\mathbb{C}P^5}$. By the Poincar\'{e} 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_sphere|Kervaire manifold]], and once again, we take pullback of$H_5$line bundle via the natural 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$$H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$$\widetilde{\mathbb{C}P^5}$. By the Poincar\'{e} conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$$S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$$PL$-homeomorphic to the sphere $S^{11}$$S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$$\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$$\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$$M^{10}_A$ is the Kervaire manifold, and once again, we take pullback of $H_5$$H_5$ line bundle via the natural degree $1$$1$ normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$$\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. Nevertheless there exists a manifold $W^{12}$$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$$PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries. Lemma 0.1. $\mathbb{C}P^6$$\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$$\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$$f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces. The aim of this exercise is to write full details of proof of Lemma 8.24 form [Madsen&Milgram1979]. A sketch of this proof can be found in the book, on page 170. ## References$ 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 $$D(f^*(H_5))\cup W^{12}\to D(H_5)$$ 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. {{beginthm|Lemma}} $\mathbb{C}P^6$, $\widetilde{\mathbb{C}P^6}$ and $f^*(H_5)\cup W^{12}\cup_{S^{11}}D^{12}$, are topologically distinct homotopy projective spaces. {{endthm}} The aim of this exercise is to write full details of proof of Lemma 8.24 form \cite{Madsen&Milgram1979}. A sketch of this proof can be found in the book, on page 170. == References == {{#RefList:}} [[Category:Exercises]]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$$H_5\to \mathbb{C}P^5$ over $\widetilde{\mathbb{C}P^5}$$\widetilde{\mathbb{C}P^5}$. By the Poincar\'{e} conjecture, the sphere bundle $S(h^*(H_5))=\partial D(h^*(H_5))$$S(h^*(H_5))=\partial D(h^*(H_5))$ is $PL$$PL$-homeomorphic to the sphere $S^{11}$$S^{11}$. $\widetilde{\mathbb{C}P^6}=D(h^*(H_5))\cup_{S^{11}}D^{12}$$\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$$\widetilde{\mathbb{C}P^5}\# M^{10}_A$, where $M^{10}_A$$M^{10}_A$ is the Kervaire manifold, and once again, we take pullback of $H_5$$H_5$ line bundle via the natural degree $1$$1$ normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$$\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery. Nevertheless there exists a manifold $W^{12}$$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$$PL$-homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.

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