# Do trivial homology groups imply contractibility of a compact polyhedron?

Is it true that a compact polyhedron X with trivial homology groups (except $H_{0}(X)$ of course) is necessarily contractible? If yes, what is the approach in proving it? If not, do you see a counter-example?

• I think you at least want to specify that $X$ is connected, or else you could take, say, two points. Mar 30, 2011 at 16:21
• The Poincare homology sphere is a counter-example which is a manifold. It's also called the Poincare Dodecahedral Space. There's a Wikipedia page for it. Mar 30, 2011 at 16:22
• @Ryan: But... Doesn't a homology $3$-sphere have $H_{3} \cong \mathbb{Z}$ by definition?
– t.b.
Mar 30, 2011 at 16:28
• @Ryan: In other words: what was wrong with Mariano's now deleted answer along the lines: If $X$ is connected and simply connected then it follows from Hurewicz and Whitehead that the inclusion of any point is a homotopy equivalence. What am I missing?
– t.b.
Mar 30, 2011 at 16:36
• @Theo: ah, right. I forgot about $H_3$. I'll give a proper example as an answer, if someone else doesn't beat me to it. Mar 30, 2011 at 17:43

The 2-skeleton of the Poincare homology sphere, also describable as the presentation complex of the binary icosahedral group, provides a counterexample to your original question. The fundamental group is of order 120 and is perfect, which implies that that $H_1$ is trivial. You can check from the group presentation $<s,t | s^{-3}(st)^2, t^{-5}(st)^2>$ that the second homology group is trivial as well.

• Very nice. This is the example I should have given! Mar 30, 2011 at 17:51

To sum up the comments: when Poincaré worked on the beginnings of algebraic topology, he originally thought that a space with trivial homology groups must be contractible. (More precisely, he thought that having the homology group of a 3-sphere implies being a 3-sphere.) However, he soon found a counterexample, the Poincaré homology sphere, which led him to the construction of the fundamental group.

When taking the fundamental group into account, the statement is indeed true: if a space has trivial fundamental group and trivial higher homology groups, then it must be contractible. This is a consequence of Whitehead's theorem and the Hurewicz map.

• I don't understand how the first two sentences are related. Dec 31, 2011 at 9:59
• @QiaochuYuan: If I were to conjecture that homology is a complete invariant, then this would imply both sentences. As to Poincaré's thinking, I am only aware of evidence that backs up the sentence in parentheses, however. Dec 31, 2011 at 11:55