Finite domination, Novikov homology and nonsingular closed 1-forms

Let X be a finite connected CW-complex and ρ:[inline-graphic not available: see fulltext] a regular covering space with free abelian covering transformation group. For ξ ∈ H1 (Xℝ) we define the notion of ξ-contractibility of X. This notion is closely related to the vanishing of the Novikov homology of the pair (X,ξ). We show that finite domination of [inline-graphic not available: see fulltext] is equivalent to X being ξ-contractible for every nonzero ξ with ρ*ξ =0  ∈ H1([inline-graphic not available: see fulltext]; ℝ). If M is a closed connected smooth manifold the condition that M is ξ-contractible is necessary for the existence of a nonsingular closed 1-form representing ξ. Also ξ-contractibility guarantees the definition of the Latour obstruction τL(M,ξ) whose vanishing is then sufficient for the existence of a nonsingular closed 1-form provided  dim M≥6. Now if ρ:[inline-graphic not available: see fulltext] is a finitely dominated regular ℤk-covering space we get that τL(M,ξ) is defined for every nonzero ξ with ρ*ξ=0 and the vanishing of one such obstruction implies the vanishing of all such τL(M,ξ).