[Math] On an exercise from Weibel’s book on homological algebra

Question

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. $1.4.4$):

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.

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 ?

Answer 1

You are correct, the converse also holds. This is a mistake in Weibel, see this answer.