The loop cohomology of a space with the polynomial cohomology algebra

Details The loop cohomology of a space with the polynomial cohomology algebra The loop cohomology of a space with the polynomial cohomology algebra

2008-10-24

arxiv.org/abs/0810.4531v5

Mathematics

Algebraic Topology

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

Scientific paper

Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\Omega X;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(\Omega X;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel.

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