The twisted Eilenberg-Zilber Theorem

TL;DRAbstract

The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F) such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular chains of E over a ring Λ). We work in the context of (semi-simplicial) twisted cartesian products (thus we assume as do the proofs of the theorem given in [Gug60, Shi62, Szc61] the results of [BGM59] on the relation between fibre spaces and twisted cartesian products). In fact we prove a general result on filtered chain complexes; this result applies to give proofs not only of Brown’s theorem but also of a theorem of G. Hirsch, [Hir53]. Our proof is suggested by the formulae (1) of [Shi62, Ch. II, §1]. Let (X,d), (Y,d) be chain complexes over a ring Λ. Let (Y,d) ∇ f − → (X,d) − → (Y,d) be chain maps and let Φ: X → X be a chain homotopy such that Let X,Y have filtrations (1.1) f ∇ = 1; (1.2

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