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A standing conjecture in L-cohomology is that every finite CW complex X is of L-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L-acyclic, we also prove that the L-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L-Betti numbers.

$L^2$-determinant class and approximation of $L^2$-Betti numbers