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.**
 <A HREF="http://library.msri.org/books/Book32/files/ruberman.pdf"><FONT FACE="Arial">Null-homotopic embedded spheres of codimension one.</FONT></A><FONT FACE="Arial"> 
In: *Tight and taut submanifolds* (Berkeley, CA, 1994),
  volume 32 of *Math. Sci. Res. Inst. Publ.*, pp. 229-232. Cambridge
  Univ. Press, Cambridge, 1997.