Fake projective spaces in dim 6 (Ex)

Let $M_B^8$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M_B^8$ be a closed Milnor manifold. There exist a natural degree 1 normal map $M_B^8\to S^8$$M_B^8\to S^8$ which induces a degree 1 normal map

$\displaystyle f\colon\mathbb{C}P^4\#_mM_B^8\to \mathbb{C}P^4\#_m S^8\cong \mathbb{C}P^4.$ $$\displaystyle f\colon\mathbb{C}P^4\#_mM_B^8\to \mathbb{C}P^4\#_m S^8\cong \mathbb{C}P^4.$$

We may pull back the Hopf bundle $H_4\to \mathbb{C}P^4$$H_4\to \mathbb{C}P^4$ to $\mathbb{C}P^4\#_mM_B^8$$\mathbb{C}P^4\#_mM_B^8$. The sphere bundle of the Hopf bundle over sphere is the true sphere, hence we get a map:

$\displaystyle f|_\partial\colon S(f^*(H_4))\to S(H_4)=S^9.$ $$\displaystyle f|_\partial\colon S(f^*(H_4))\to S(H_4)=S^9.$$

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 $f|_\partial$$f|_\partial$ bordant to a homotopy equivalence $g$$g$. Thus by Poincare Conjecture it is indeed normally bordant to a PL-homeomorphism. Choose $W'\to S^9\times[0,1]$$W'\to S^9\times[0,1]$ to be such bordism.

Lemma 0.2. We can perform surgery on

$\displaystyle \bar{g}\colon W'\cup_\partial D(f^*(H_4))\to D(H_4)\cup_\partial S^9\times [0,1]\cong D(H_4)$ $$\displaystyle \bar{g}\colon W'\cup_\partial D(f^*(H_4))\to D(H_4)\cup_\partial S^9\times [0,1]\cong D(H_4)$$

to make it a homotopy equivalence.

Describe these surgeries.

We obtain a manifold with boundary $(W^10,\partial W)$$(W^10,\partial W)$ homotopy homotopy equivalent to $D(H_4)$$D(H_4)$ with boundary PL-homeomorphic to $S^9$$S^9$. Thus we may cone off common boundaries extending the homotopy equivalence at the same time. We define

$\displaystyle \widetilde{\mathbb{C}P^5}=W^{10}\cup_{S^9}D^{10}.$ $$\displaystyle \widetilde{\mathbb{C}P^5}=W^{10}\cup_{S^9}D^{10}.$$

Lemma 0.3. $\widetilde{\mathbb{C}P^5}$$\widetilde{\mathbb{C}P^5}$ and $\mathbb{C}P^5$$\mathbb{C}P^5$ are not homeomorphic.

Prove the above lemma.

However, the construction above gives us a degree 1 normal map

$\displaystyle h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$ $$\displaystyle h\colon\widetilde{\mathbb{C}P^5}\to \mathbb{C}P^5.$$

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 Poincaré 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$$H_5$ line bundle via the natural degree 1 normal map. Now to obtain a homotopy equivalence $\partial D(f^*(H_5))\to \partial D(H_5)$$\partial D(f^*(H_5))\to \partial D(H_5)$ we have to use a little bit of surgery.

Describe these surgeries.

Nevertheless there exists a manifold $W^{12}$$W^{12}$ such that

$\displaystyle D(f^*(H_5))\cup W^{12}\to D(H_5)$ $$\displaystyle D(f^*(H_5))\cup W^{12}\to D(H_5)$$

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 0.4. $\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 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