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Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in dimension 7.) Yes, the topological category is fine, to avoid the smooth Poincaré conjecture.

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In dimension 4, we have the following: Simply-connectedness implies that $H_1(M)=0$. The condition that $M$ be a rational homology sphere implies that $H_2(M), H_3(M)$ are finitely generated torsion groups. It follows that $H^3(M) = Ext(H_2(M),\mathbb{Z})$, which is noncanonically isomorphic to $H_2(M)$ again (that's true for finitely generated torsion groups).

But Poincare duality tells us that $H^3(M)=H_1(M) =0$, so $H_2(M)=0$. Similarly, we can obtain $H_3(M)=0$. It follows that $M$ is already a homology sphere.

In dimension 5, there's the first counterexample: The so-called Wu manifold $SU(3)/SO(3)$ has homology groups $\mathbb{Z}, 0, \mathbb{Z}/2, 0, 0, \mathbb{Z}$, so rationally, it is a homology sphere.

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  • $\begingroup$ The calculation of the homology groups of the Wu manifold can be found here. $\endgroup$ – Michael Albanese Mar 19 '18 at 14:38
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A complete answer can be found in a paper by D. Ruberman Null-homotopic embedded spheres of codimension one: a simply-connected rational homology $n$-sphere that is not homeomorphic to $S^n$ exists if and only if $n\ge 5$. See the bottom of page 230 and example 7 on p.232.

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Yes, every simply-connected rational homology $4$-sphere is topologically the $4$-sphere. Simply-connected closed topological $4$-manifolds are classified by their intersection form $Q_X:H^2(X;\Bbb Z) \times H^2(X ;\Bbb Z) \to \Bbb Z$ and their Kirby-Siebenmann invariant by a famous theorem of Freedman. If the form is even, the KS invariant automatically vanishes. If $X$ is a rational homology sphere, $Q_X$ clearly vanishes (as $H^2(X;\Bbb Z)=0$), and therefore $X$ must be homeomorphic to the $4$-sphere.

See: Michael H. Freedman & Frank Quinn Topology of 4-Manifolds (PMS-39)

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In dimension 5 and higher, there are simply connected rational homology spheres that are not spheres, e.g. the Wu manifold $SU(3)/SO(3)$, see Theorem 6.7 in [2] and Remark, p. 374 in [1]. See also [3] and Lemma 1.1 in [1] for more examples.

[1] D. Barden, Simply connected five-manifolds. Ann. of Math. 82 (1965), 365-385.

[2] M. Mimura, H. Toda, Topology of Lie groups. I, II. Translated from the 1978 Japanese edition by the authors. Translations of Mathematical Monographs, 91. American Mathematical Society, Providence, RI, 1991.

[3] Ruberman, D. Null-homotopic embedded spheres of codimension one. In: Tight and taut submanifolds (Berkeley, CA, 1994), volume 32 of Math. Sci. Res. Inst. Publ., pp. 229-232. Cambridge Univ. Press, Cambridge, 1997.

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