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.

20 pages

References in corpus (1)

• A cubical approach to homotopy orbits of circle actions