Fake projective spaces in dim 6 (Ex)
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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.''' | ||
+ | |||
+ | However, the construction above 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. |
Latest revision as of 00:15, 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.
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.
However, the construction above gives us a degree 1 normal mapSimilarly 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.
Nevertheless there exists a manifold such thatLemma 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