By Liviu I Nicolaescu

This can be a cutting-edge creation to the paintings of Franz Reidemeister, Meng Taubes, Turaev, and the writer at the proposal of torsion and its generalizations. Torsion is the oldest topological (but no longer with appreciate to homotopy) invariant that during its virtually 8 many years of lifestyles has been on the middle of many very important and excellent discoveries. up to now decade, within the paintings of Vladimir Turaev, new issues of view have emerged, which became out to be the "right ones" so far as gauge concept is worried. The e-book gains normally the recent points of this venerable thought. The theoretical foundations of this topic are offered in a mode available to these, who desire to research and comprehend the most principles of the idea. specific emphasis is upon the numerous and fairly varied concrete examples and strategies which seize the subleties of the

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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].