# Fake projective spaces in dim 6 (Ex)

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Marek Kaluba (Talk | contribs) (Added beginning of the construction of hCP^5. Expanded exercise range and made it more modular.) |
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− | There | + | Let $M_B^8$ be a closed [[Milnor manifold]]. There exist a natural degree 1 normal map $M_B^8\to S^8$ which induces a degree 1 normal map $$f\colon\mathbb{C}P^4\#_mM_B^8\to \mathbb{C}P^4\#_m S^8\cong \mathbb{C}P^4.$$ |

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

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

+ | '''Prove the following:''' | ||

+ | \begin{lemma} | ||

+ | The map induced above is a degree 1 normal map. | ||

+ | \end{lemma} | ||

+ | |||

+ | Now, since we are working between odd dimensional manifolds we can by surgery below middle dimension assume that $f|_\partial$ bordant to a homotopy equivalence $g$. Thus by Poincare Conjecture it is indeed normally bordant to a PL-homeomorphism. Choose $W'\to S^9\times[0,1]$ to be such bordism. | ||

+ | |||

+ | \begin{lemma} We can perform surgery on $$\bar{g}\colon W'\cup_\partial D(f^*(H_4))\to D(H_4)\cup_\partial S^9\times [0,1]\cong D(H_4)$$ to make it a homotopy equivalence. | ||

+ | \end{lemma} | ||

+ | |||

+ | '''Describe these surgeries.''' | ||

+ | |||

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

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

+ | |||

+ | \begin{lemma} | ||

+ | $\widetilde{\mathbb{C}P^5}$ and $\mathbb{C}P^5$ are not homeomorphic. | ||

+ | \end{lemma} | ||

+ | |||

+ | '''Prove the above lemma.''' | ||

+ | |||

+ | The above construction gives us a degree 1 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 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_sphere|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. | ||

+ | |||

+ | '''Describe these surgeries.''' | ||

+ | |||

+ | 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}} | {{beginthm|Lemma}} | ||

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{{endthm}} | {{endthm}} | ||

− | The aim of this exercise is to write full details of proof of Lemma 8.24 form \cite{Madsen&Milgram1979}. | + | The aim of this exercise is to '''write full details of proof of the above Lemma''' (which is Lemma 8.24 form \cite{Madsen&Milgram1979}). |

A sketch of this proof can be found in the book, on page 170. | A sketch of this proof can be found in the book, on page 170. |

## Revision as of 00:12, 3 April 2012

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