Double complexes and vanishing of Novikov cohomology

TLDR: In this paper, it was shown that a double complex with exact rows and columns yields an acyclic cochain complex under totalisation using right (resp., left) truncated products.

Abstract: We consider a non-standard totalisation functor to produce a cochain complex from a given double complex D∗,∗: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp., columns) yields an acyclic cochain complex under totalisation using right (resp., left) truncated products. As an application we consider the algebraic mapping torus T (h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T (h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T (h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology. Finiteness conditions for chain complexes of modules play an important role in both algebra and topology. For example, given a group G one might ask whether the trivial G-module Z admits a resolution by finitely generated projective Z[G]-modules; existence of such resolutions is relevant for the study of group cohomology of G, and has applications in the theory of duality groups [B75]. For topologists, finite domination of chain complexes is related, among other things, to questions about finiteness of CW complexes, the topology of ends of manifolds, and obstructions for the existence of nonsingular closed 1-forms [Ran95, S06]. A cochain complex C of R[z, z−1]-modules is called finitely dominated if it is homotopy equivalent, as a complex of R-modules, to a bounded complex of finitely generated projective R-modules. Finite domination of C can be characterised in various ways; Brown considered compatibility of the functors M 7→ H∗(C;M) and M 7→ H∗(C;M) with products and direct limits, respectively [B75, Theorem 1], while Ranicki showed that C is finitely dominated if and only if the Novikov cohomology of C is trivial [Ran95, Theorem 2] (see also Definition 2.3 and Corollary 2.7 below). Our approach to Novikov cohomology involves a non-standard totalisation functor. The key fact is that certain double complexes are converted into acyclic cochain complexes (Proposition 1.2) which is proved by an elementary calculation. As an application we obtain a new result for vanishing of Novikov cohomology of algebraic mapping tori (Theorem 2.5), and recover the “only-if” part of Ranicki’s criterion for finite domination over Date: 08.09.2011. 2000 Mathematics Subject Classification. Primary 18G35; Secondary 55U15. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H018743/1]. 1