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We known that there exists smooth integer homology n-sphere (n>4) with some non-trivial fundamental group by the Kervaire theorem [Michel A. Kervaire, MR 253347 Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67--72.]


(1) Have another examples about that?
(2) Is it possible that some aspherical manifolds are higher-dimensions integer homology spheres?---Asked by Thurston and Kan.


Remark: As far as I know, Andrejz Szczepanski [Aspherical manifolds with the Q-homology of a sphere] proved (2) in the odd dimensions, but it is the rationnal homology sphere. And it is also true in the 3-dimension and 4-dimension, see [John G. Ratcliffe and Steven T. Tschantz, MR 2114711 Some examples of aspherical 4-manifolds that are homology 4-spheres, Topology 44 (2005), no. 2, 341--350.]

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    $\begingroup$ In (1) it is not clear what you mean by "realize". The construction of Kervaire is quite explicit as far as these things go. Regarding (2) I think the existence pf aspherical homology spheres is an open problem in any dimension $>4$. $\endgroup$ Aug 14, 2016 at 13:44
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    $\begingroup$ Regarding the edited version of (1) a general method of building cutom made homology spheres in higher dimensions is developed in [Hausmann, Jean-Claude, Manifolds with a given homology and fundamental group. Comment. Math. Helv. 53 (1978), no. 1, 113–134.] The method is surgery theoretic, and generalises Kevaire's work. $\endgroup$ Aug 14, 2016 at 16:50
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    $\begingroup$ A conjecture of Hopf is that for closed aspherical manifolds $M$ of dimension $2k$, $(-1)^k \chi(M) \geq 0$. This conjecture implies that a manifold of dimension $4k+2$ could not be a homology sphere. en.wikipedia.org/wiki/Hopf_conjecture $\endgroup$
    – Ian Agol
    Aug 14, 2016 at 18:08
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    $\begingroup$ ^ of course, I meant that an aspherical manifold of dimension $4k+2$ could not be a homology sphere if the Hopf conjecture holds. $\endgroup$
    – Ian Agol
    Aug 15, 2016 at 5:10

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