ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

6. Topology

1991 MS Classification: 55P35 Saneblidze Samson, and Kadeishvili, Tornike, A. Razmadze Mathematical Institute, Tbilisi, Georgia Adams Cobar Construction as a Cubical Complex In this talk we are going to show that the cobar construction ΩC̄_{∗}(X) of the singular chain coalgebra C_{∗}(X) of a 1-connected space X, as it stands, coincides with the alternated chain complex of certain cubical monoidal set; moreower, Adams’s DG algebra weak equivalence

ΩC̄_{∗}(X) → CU_{∗}^{N}(ΩX)

from the cobar construction to the normalized singular cubical chain complex of the loop space ΩX is a map of cubical abelian groups. Consequently, the cobar construction carries all the structures, caused by this fact. Namely, on the cobar construction all Steenrod chain cooperations {▽_{i}, i = 0, 1, 2, ...} canonicaly appear and the Adams weak equivalence preserves them. Note that the cooperations ▽_{0} and ▽_{1} coincide with those constructed by Baues.