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By Allan R. Keith, Stephen A. Spongberg

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VI]). In dealing with the gluing properties of the torsion it is perhaps more convenient to adopt this new point of view because, as explained in [17, ibid], the homology with local coefficients satisfies the same set of defining axioms the ordinary homology. 4 (The circle). Suppose X is the circle S 1 with the natural CW-decomposition consisting of a single 0-cell α0 , and a single 1-cell α1 . Then H = H1 (S 1 , Z) = 0. We write it multiplicatively, and we denote its natural generator by t. The group Z acts on Xˆ = R by translations t n · x = x + n.

The boundary operator ∂ : C k (R × X) → C k−1 (R × X) acts according to the prescriptions ˆ 1 ⊗ σk−1 ) = (1 − t)e0 ⊗ σk−1 − e1 ⊗ ∂σk , ∂(e ˆ 0 ⊗ σk ) = e0 ⊗ ∂σk−1 . ∂(e Define the morphism of Z[t]-module η : C k−1 (R × X) → C k (R × X), e0 ⊗ σk−1 → 1 e1 ⊗ σk−1 , t −1 e1 ⊗ σk−2 → 0. e. η is an algebraic contraction. Then Observe that η2 = 0 and ∂η TSπ1 ×X = det ∂ˆ + η : C even (R × X) → C odd (R ⊗ X) . With respect to the bases e0 ⊗ S0 (X) ∪ e1 ⊗ S1 (X) ∪ e0 ⊗ S2 (X) ∪ · · · of C even , and e1 ⊗ S0 (X) ∪ e0 ⊗ S1 (X) ∪ e1 ⊗ S2 (X) ∪ e0 ⊗ S3 (X) ∪ · · · of C odd , the operator ∂ˆ + η has the description n0 columns n1 columns n2 columns · · · n0 rows (t − 1)−1 −∂ 0 ··· n1 rows 0 (t − 1) ∂ ··· n2 rows ..

It can be non-canonically decomposed as H = FH ⊕ Tors(H ), where FH denotes the free part of H , FH ∼ = H / Tors(H ). Denote by Q(H ) the ring of fractions of the group algebra Z[H ]. 23. Z[H ] is a quasi-integral domain of characteristic zero. 22 1 Algebraic preliminaries Proof. Let us first consider the two extremes, Tors(H ) = 0, or FH = 0. • If Tors(H ) = 0 then H = FH , and if rank H = n, then Q(H ) is the field of rational functions in n variables with rational coefficients. • If FH = 0, so that H = Tors(H ), then Q[H ] is a semisimple, commutative algebra, and thus decomposes as a sum of fields; see [55].

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