On the Betti numbers of a loop space

2009/05/15 by Samson Saneblidze, Saneblidze, Samson · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55P35 #msc:55S30 #msc:55U20

paper · pdf · doi:10.48550/arxiv.0905.2591

11 pages

arxiv created 2009/12/24 · arxiv updated 2010/01/07

Abstract

Let A be a special homotopy G-algebra over a commutative unital ring \Bbbk such that both H(A) and Tor_i^A(\Bbbk,\Bbbk) are finitely generated \Bbbk-modules for all i, and let τ_i(A) be the cardinality of a minimal generating set for the \Bbbk-module Tor_i^A(\Bbbk,\Bbbk). Then the set τ_i(A) is unbounded if and only if H(A) has two or more algebra generators. When A=C^∗(X;\Bbbk) is the simplicial cochain complex of a simply connected finite CW-complex X, there is a similar statement for the "Betti numbers" of the loop space ΩX. This unifies existing proofs over a field \Bbbk of zero or positive characteristic.

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