A chain coalgebra model for the james map

Kathryn Hess; Paul-Eugène Parent; Jonathan Scott

doi:10.4310/HHA.2007.v9.n2.a9

A chain coalgebra model for the james map

Kathryn Hess
, Paul-Eugène Parent
, Jonathan Scott

Mathematics and Statistics

École Polytechnique Fédérale de Lausanne
Department of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, αK: CK → ΩCEK. We compute the cobar diagonal on ΩCEK, not assuming that EK is 1-reduced, and show that αK is comultiplicative. As a result, the natural isomorphism of chain algebras TCK ≅ QCK 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. Copyright © 2007, International Press.

Original language	English
Pages (from-to)	209-231
Number of pages	23
Journal	Homology, Homotopy and Applications
Volume	9
Issue number	2
DOIs	https://doi.org/10.4310/HHA.2007.v9.n2.a9
State	Published - Jan 1 2007

Keywords

Bott-Samelson equivalence
Chain coalgebra
Cobar construction
James map
Simplicial suspension

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10.4310/HHA.2007.v9.n2.a9

Cite this

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AB - Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, αK: CK → ΩCEK. We compute the cobar diagonal on ΩCEK, not assuming that EK is 1-reduced, and show that αK is comultiplicative. As a result, the natural isomorphism of chain algebras TCK ≅ QCK 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. Copyright © 2007, International Press.

KW - Bott-Samelson equivalence

KW - Chain coalgebra

KW - Cobar construction

KW - James map

KW - Simplicial suspension

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A chain coalgebra model for the james map

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Keywords

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