Fake projective spaces in dim 6 (Ex)
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There is 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. | There is 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. | ||
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Revision as of 14:46, 1 April 2012
We may pull back the canonical line bundle over . By the Poincaré conjecture, the sphere bundle is -homeomorphic to the sphere . is a homotopy complex projective space obtained by coning off the boundary of the disk bundle.
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. Nevertheless there exists a manifold such that
is a homotopy equivalence of manifolds with boundary. Again since the boundary is already -homeomorphic to a sphere we may extend the homotopy equivalence after coning off the boundaries.
Lemma 0.1. , and , 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
- [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
Category:Exercises without solution