A chain coalgebra model for the James map

Abstract

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.

The Hochschild complex of a twisting cochain

“…Along the same lines, Theorem 4.9 in [12] implies that if L is a pointed simplicial set such that C * L is a cocommutative coalgebra, then there is a natural choice of chain algebra map ω EL : ΩC * EL → Ω(C * EL ⊗ C * EL) such that (C * EL, ω EL ) is a balanced Alexander-Whitney coalgebra, where E denotes the simplicial suspension functor [20]. More precisely, ΩC * EL ∼ = T (C * L), the tensor algebra generated by coaugmentation coideal of C * L, endowed with the strictly linear differential induced by the differential on C * L. Moreover, the comultiplication ψ EL = qω EL satisfies…”

“…the explicit formula for t EL given in [12] just before Theorem 4.11), we conclude that Ωβ EL is a chain Hopf algebra map if and only if C * L is a trivial coalgebra. In particular, if L itself is a simplicial suspension (reduced or unreduced), then Ωβ EL is a chain Hopf algebra map, and Corollary 4.4 therefore implies that…”

“…In these terms, the result from [12] cited above in Example 3.9 says that if K is a symmetric simplicial set, then C * EK ⊗ F 2 is a balanced Alexander-Whitney coalgebra.…”

“…Apply Theorem 4.1 to the twisting cochain t Ω : C → ΩC. Hypotheses (1) and (2) Moreover, if C * L is cocommutative, then C * GEL is a cocommutative chain Hopf algebra, as easily follows from an examination of the formulas for the simplicial suspension functor E and for the Kan loop group functor G (cf., e.g., sections 2.1 (a) and (b) in [12]), which imply the existence of a simplicial map…”

“…In [12] it was shown that q is in fact a chain homotopy equivalence, for all C, C ∈ Coalg R . Dually, for all A, A ∈ Alg R , there is a map of chain coalgebras that is a chain homotopy equivalence…”