A chain coalgebra model for the James map

Details A chain coalgebra model for the James map A chain coalgebra model for the James map

2006-09-15

arxiv.org/abs/math/0609444v1

Mathematics

Algebraic Topology

20 pages

Scientific paper

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.

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