For any 1-reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that any canonical quasi-isomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the category of chain coalgebras and of strongly homotopy coalgebra maps given in [HPS]. ≃ − → C * ΩX, such that θ X restricts to quasi-isomorphisms (T V ≤n , d) ≃ − → C * ΩX n+1 , where X n+1 denotes the (n + 1)-skeleton of X, T V denotes the free (tensor) algebra on a free, 1991 Mathematics Subject Classification. Primary: 55P35 Secondary: 16W30, 18D50, 18G35, 55U10, 55U35, 57T05, 57T30.
Communicated by C. Kassel
MSC:Primary: 16E40 19D55 secondary: 18G60 55M20 55U10 81T30 a b s t r a c t Generalizing the work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra C is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex H (C) admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C * K is its normalized chain complex, then H (C * K ) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on H (C * K ) when K is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g, h : K → L, where K and L are reduced, the homology of the coHochschild complex of C * L with coefficients in C * K is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of g and h, and this isomorphism respects comultiplicative structure. In particular, there is an isomorphism, respecting comultiplicative structure, from the homology of H (C * K ) to H * L|K |, the homology of the free loops on the geometric realization of K .
Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, α K : CK → ΩCEK. We compute the cobar diagonal on ΩCEK, not assuming that EK is 1-reduced, and show that α K is comultiplicative. As a result, the natural isomorphism of chain algebras T CK ∼ = ΩCK preserves diagonals.In an appendix, we show that the Milgram map, Ω(A ⊗ B) → ΩA ⊗ ΩB, where A and B are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when A and B are not 1-connected.
We give a new upper bound for Farber's topological complexity for rational spaces in terms of Sullivan models. We use it to determine the topological complexity in some new cases, and to prove a Ganea-type formula in these and other cases. 55M30, 55P62
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