Kathryn HESS
An algebraic model for the loop space homology of a homotopy
fiber
Abstract
In this talk I will present joint work with Ran Levi on a
``neoclassical" approach to computing the homology algebra of the loops
on a homotopy fiber, based on developing a deep understanding of
``strongly homotopy" structures for coalgebras, a notion that goes back
more than 30 years, to work of Gugenheim, Halperin, Munkholm and
Stasheff. We also make extended use of twisting cochains, defined
first by E. Brown in 1959, which we apply in innovative ways.
The model that we propose for the loop homology of a homotopy fiber
offers certain advantages. First, there are no extension problems to
be solved, even over the integers: the homology algebra of the model
is exactly isomorphic to the homology algebra of the loops on the
homotopy fiber. Second, if K is a simplicial set with n
nondegenerate simplices, where n<\infty, then, our model for the
double loop space on K is a subalgebra of a free algebra on 2n
generators. By way of comparison, note that the iterated cobar
construction on the chains on K, which is another model of the double
loop space, is free on an infinite number of generators. Finally our
model is functorial, so that it can be applied to determining the
homomorphism induced on double loop space homology by a simplicial map.
20 janvier 2005