{"title":"Secondary cohomology operations and the loop space cohomology","authors":"Samson Saneblidze","doi":"arxiv-2409.04861","DOIUrl":null,"url":null,"abstract":"Motivated by the loop space cohomology we construct the secondary operations\non the cohomology $H^*(X; \\mathbb{Z}_p)$ to be a Hopf algebra for a simply\nconnected space $X.$ The loop space cohomology ring $H^*(\\Omega X;\n\\mathbb{Z}_p)$ is calculated in terms of generators and relations. This answers\nto A. Borel's decomposition of a Hopf algebra into a tensor product of the\nmonogenic ones in which the heights of generators are determined by means of\nthe action of the primary and secondary cohomology operations on\n$H^*(X;\\mathbb{Z}_p).$ An application for calculating of the loop space\ncohomology of the exceptional group $F_4$ is given.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"46 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04861","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by the loop space cohomology we construct the secondary operations
on the cohomology $H^*(X; \mathbb{Z}_p)$ to be a Hopf algebra for a simply
connected space $X.$ The loop space cohomology ring $H^*(\Omega X;
\mathbb{Z}_p)$ is calculated in terms of generators and relations. This answers
to A. Borel's decomposition of a Hopf algebra into a tensor product of the
monogenic ones in which the heights of generators are determined by means of
the action of the primary and secondary cohomology operations on
$H^*(X;\mathbb{Z}_p).$ An application for calculating of the loop space
cohomology of the exceptional group $F_4$ is given.