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, $\alpha_K : CK \to \Omega CEK$. We compute the cobar diagonal on $\Omega CEK$, not assuming that $EK $is 1-reduced, and show that $\alpha_K$ is comultiplicative. As a result, the natural isomorphism of chain algebras $TCK \cong \Omega CK$ preserves diagonals. In an appendix, we show that the Milgram map, $\Omega (A \otimes B) \to \Omega A \otimes \Omega B$, 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.