Quantum Chern–Simons invariants of differentiable manifolds are analyzed from the point of view of homological algebra. Given a manifold M and a Lie (or, more generally, an L∞) algebra g, the vector space H⁎(M)⊗g has the structure of an L∞ algebra whose homotopy type is a homotopy invariant of M . We formulate necessary and sufficient conditions for this L∞ algebra to have a quantum lift. We also obtain structural results on unimodular L∞ algebras and introduce a doubling construction which links unimodular and cyclic L∞ algebras.