Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a chain map whose induced maps on homology $H_\bullet f: H_\bullet C \to H_\bullet D$ are isomorphisms. Here's the question:

Is there some (possibly cohomological or $K$-theoretic) obstruction to $f$ forming one-half of a chain homotopy equivalence between $C_\bullet$ and $D_\bullet$?

Note that the question is explicitly not asking whether given two quasi-isomorphic chain complexes, there exists a chain homotopy equivalence. I'm interested in the prescribed quasi-isomorphism $f$ taking part in such an equivalence.

What I know so far is that quasi isomorphism and chain equivalence coincide for bounded complexes of vector spaces, I have gone through this related MO question and hence Weibel's Chapter 10 (particularly Theorem 10.4.8). But neither of those appear to address this question directly.

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a chain map whose induced maps on homology $H_\bullet f: H_\bullet C \to H_\bullet D$ are isomorphisms. Here's the question:

Is there some (possibly cohomological or $K$-theoretic) obstruction to $f$ forming one-half of a chain homotopy equivalence between $C_\bullet$ and $D_\bullet$?

Note that the question is explicitly not asking whether given two quasi-isomorphic chain complexes, there exists a chain homotopy equivalence. I'm interested in the prescribed quasi-isomorphism $f$ taking part in such an equivalence.

What I know so far is that quasi isomorphism and chain equivalence coincide for bounded complexes of vector spaces, I have gone through this related MO question and hence Weibel's Chapter 10 (particularly Theorem 10.4.8). But neither of those appear to address this question directly.