Is there some general description of all homology 3-spheres?
[Math] Classification of homology 3-spheres
3-manifoldsgt.geometric-topology
Related Solutions
Yes, there's loads of other contractible 4-manifolds bounding homology spheres with other geometries.
For example $1$-surgery on the Stevedore knot is a hyperbolic manifold with volume $1.3985\cdots$. This bounds a contractible manifold by the same kind of arguments Casson and Harer used to cook up their big list of Mazur manifolds. There are other examples like this that occur in my arXiv preprint: https://arxiv.org/abs/0810.2346 Some are geometric, some have incompressible tori.
I don't think any Nil manifolds bound contractible 4-manifolds. Crisp and Hillman determined all the Nil manifolds that embed smoothly (or topologically) in the $4$-sphere. The only Nil manifolds on that list have non-trivial homology. Crisp & Hillman "Embedding Seifert fibred 3-manifolds and Sol-manifolds in 4-space".
See edit at bottom for further information answering the question in all dimensions.
In all odd dimensions $2k -1 > 3$, there are non-homeomorphic homology spheres with fundamental group G = the binary icosahedral group (fundamental group of the Poincaré homology sphere). This follows from basic surgery theory; the Wall group $L_{2k}(G)$ contains a subgroup of the form $\mathbb{Z}^n$ for some $n$ related to the number of irreducible complex representations of $G$. This subgroup is detected by the so-called multisignature, as described in Wall's book on surgery theory.
Choose one homology sphere $M^{2k-1}$ with $\pi_1(M) = G$. For instance, you can repeatedly spin the Poincaré sphere, where spinning $P$ means doing surgery on the obvious $S^1$ in $S^1\times P$. For any $M$ with $\pi_1(M) = G$, there's an invariant originally described by Atiyah and Singer. A priori, it's a smooth invariant, but is known to be a homeomorphism invariant. Roughly speaking, one knows that for some $d \in \mathbb{N}$, the manifold $d\cdot M$ is the boundary of a $2k$-manifold $X$ with $\pi_1(X) = G$. Then $X$ has a collection of equivariant signatures (associated to the action of $G$ on the universal cover of $X$), known collectively as the multisignature. The multisignature (divided by $d$) is a topological invariant. (Technically, this is only well-defined up to a choice of isomorphism $\pi_1(X) \to G$ but this is readily dealt with.)
Now, the group $L_{2k}(G)$ acts on the structure set of $M$, as described in Wall's book. The effect of the action is to change the multisignature, and hence it changes the homeomorphism type of $M$. In this construction, you not only preserve the fundamental group, but also the (simple) homotopy type of $M$.
It's possible that acting on an even-dimensional homology sphere $M^{2k}$ with fundamental group $G$ by elements of $L_{2k+1}(G)$ could change the homeomorphism type. But odd dimensional $L$-groups are much harder, and you'd need some serious expertise to see what the effect should be. I rather suspect that you could do something simpler, and change the homology with local coefficients or something like that.
Addendum: There are two papers of Alex Suciu that answer this question in dimensions at least 4. In "Homology 4-spheres with distinct k-invariants," Topology and its Applications 25 (1987) 103-110, he gives examples of homology 4-spheres with the same $\pi_1$ and $\pi_2$ that are not homotopy equivalent. In "Iterated spinning and homology spheres", Trans AMS 321 (1990) he constructs homology n-spheres in dimensions $n \geq 5$ with the same property. Combined with the remarks above about dimension 3, this answers your question.
Best Answer
Certainly. There's a general description of all compact 3-manifolds now that geometrization is about.
So for homology 3-sphere's you have the essentially unique connect sum decomposition into primes.
A prime homology 3-sphere has unique splice decomposition (Larry Siebenmann's terminology). The splice decomposition is just a convenient way of encoding the JSJ-decomposition. The tori of the JSJ-decomposition cut the manifold into components that are atoroidal, so you form a graph corresponding to these components (as vertices) and the tori as edges.
The splice decomposition you can think of as tree where the vertices are decorated by pairs (M,L) where M is a homology 3-sphere and L is a link in M such that M \ L is an atoroidal manifold.
By geometrization there's not many candidates for pairs (M,L). The seifert-fibred homology spheres that come up this way are the Brieskorn spheres, in that case L will be a collection of fibres in the Seifert fibering. Or the pair (M,L) could be a hyperbolic link in a homology sphere. That's a pretty big class of manifolds for which there aren't quite as compact a description, compared to, say, Brieskorn spheres.