The loop cohomology of a space with the polynomial cohomology algebra

2008/10/24 by Samson Saneblidze, Saneblidze, Samson
Mathematics · #55P35 (Primary) 55U99 #55S05 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:55P35 #msc:55S05 #msc:55U99

paper · pdf · doi:10.48550/arxiv.0810.4531

6 pages, The mod 2 part of the main theorem is strengthened

arxiv created 2011/11/01 · arxiv updated 2011/11/03

Abstract

Given a simply connected space X with the cohomology H^*(X;\mathbb Z₂) to be polynomial, we calculate the loop cohomology algebra H^*(ΩX;\mathbb Z₂) by means of the action of the Steenrod cohomology operation Sq₁ on H^*(X;\mathbb Z₂). As a consequence we obtain that H^*(ΩX;\mathbb Z₂) is the exterior algebra if and only if Sq₁ is multiplicatively decomposable on H^∗(X;\mathbb Z₂). The last statement in fact contains a converse of a theorem of A. Borel.

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