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# Homology, Homotopy and Applications

## Volume 9 (2007)

### Number 2

### A chain coalgebra model for the James map

Pages: 209 – 231

DOI: http://dx.doi.org/10.4310/HHA.2007.v9.n2.a9

#### Authors

#### Abstract

Let $EK$ be the simplicial suspension of a pointed simplicial set $K$. We construct a chain model of the James map, $\alpha_{K}\colon CK \rightarrow \Omega CEK$. 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.

#### Keywords

cobar construction; chain coalgebra; simplicial suspension; James map; Bott-Samelson equivalence

#### 2010 Mathematics Subject Classification

55P35, 55P40, 55U10, 57T30