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Chain complexes and assembly. (English) Zbl 0616.55016

Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 28, 48 p. (1987).
Let X be a simplicial complex with universal cover \(\tilde X\) and fundamental group \(\pi_ 1(X)\). Let R be a commutative ring, so that the fundamental group ring \(R[\pi_ 1(X)]\) is defined. A local system of R- module chain complexes \(\{C[x]|\) \(x\in X\}\) and chain maps \(C[x]\to C[y]\) \((y\leq x)\) can be assembled into a single global object, an \(R[\pi_ 1(x)]\)-module chain complex \(C[\tilde X]\). An \(R[\pi_ 1(X)]\)- module chain complex can be ”assembled over X” if it is chain equivalent to the assembly \(C[\tilde X]\) of a local system. In this paper we study the following question: Problem. Given a finite f.g. free \(R[\pi_ 1(X)]\)-module chain complex C is it possible to assemble C from a local system of finite f.g. free R-module chain complexes over X ?

MSC:

55U15 Chain complexes in algebraic topology
55N25 Homology with local coefficients, equivariant cohomology
57M10 Covering spaces and low-dimensional topology