Simply-connected rational homology spheres

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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)

Answer 4

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.