The loop cohomology of a space with the polynomial cohomology algebra
Samson Saneblidze

TL;DR
This paper computes the loop cohomology algebra of certain simply connected spaces with polynomial cohomology, revealing conditions under which it forms an exterior algebra based on Steenrod operations.
Contribution
It provides a method to determine the structure of loop cohomology algebra using Steenrod operations, extending Borel's theorem with a new converse condition.
Findings
Loop cohomology algebra is exterior if and only if $Sq_1$ is multiplicatively decomposable.
Calculated $H^*(\, ext{loop space}\, X)$ for spaces with polynomial cohomology.
Established a criterion linking Steenrod operations to algebraic structure of loop cohomology.
Abstract
Given a simply connected space with the cohomology to be polynomial, we calculate the loop cohomology algebra by means of the action of the Steenrod cohomology operation on As a consequence we obtain that is the exterior algebra if and only if is multiplicatively decomposable on The last statement in fact contains a converse of a theorem of A. Borel.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
