1 Erratum to: Math. Z. DOI 10.1007/s00209-014-1332-4

In the original publication, the theorem number 4 was incorrectly cited as theorem 1, in page 3, below the heading “Theorem 4”. The correct text should read as follows.

The notion of simple equivalence can be found on [14]. The homology of the complex \(C_{*}(\alpha , \partial ^{\xi ,B}_{*}\)), which we denote by \(H_{*}(M, u)\), only depends on the class \(u\) and it is called the Novikov homology of the class \(u\). The historical reason is that the first theorem with the flavour of theorem 4 was given by Novikov in his foundational paper [18], which first gave Morse-type inequalities for \({\mathbb {S}}^1\)-valued functions \(f : M \rightarrow {\mathbb {S}}^1\). These inequalities are related to a homology theory of an abelian cover associated with \(f\). Later, Sikorav proved in [20, Ch. IV] that the homology defined by Novikov is indeed a homology with local coefficients and extended it to non-abelian covers.

Latour also proved that \(H_{*}(M, u)\) coincides with the version of Novikov homology on the universal cover defined in Sikorav’s thesis.

Further versions of theorem 4 can be found on [19, Ch. 14, Th. 2.2 and Th. 2.4] and on [4, Th. 3.1].