I was revising some old postgraduate notes of mine in homological algebra (written during a postgrad course on the topic, I had taken more than ten ;) years ago) and I came accross the following problem: Weibel's book "*An introduction to homological algebra*" (which had been among my textbooks by that time), states the following exercise: ([ex. $1.4.4$](http://assets.cambridge.org/97805214/35000/sample/9780521435000ws.pdf)):



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> Consider the homology $H\_\*(C)$ of $C$ as a chain complex with zero differentials. Show that if the complex $C$ is split, then there is a chain homotopy equivalence between $C$ and $H\_\*(C)$. Give an example in which the converse fails. 
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I've solved the exercise but i've failed to find a counterexample for the converse. In fact, I think the converse also holds, but I haven't yet found the time to clear it up. Can somebody help ? 



On an exercise from Weibel's book on homological algebra
You are correct, the converse also holds. This is a mistake in Weibel, see [this answer](https://math.stackexchange.com/a/1405990/99887 "this answer").


