$A(\infty)$-algebra structure in the cohomology and cohomologies of a free loop space

T. V. Kadeishvili

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 87-96

Résumé

The cohomology algebra of the space $H^*(X)$ defines neither cohomology modules of the loop space $H^*(\Omega X)$ nor cohomologies of the free loop space $H^*(\Lambda X)$. But by the author's minimality theorem, there exists a structure of $A(\infty)$-algebra $(H^*(X),\{m_i\})$ on $H^*(X)$, which determines $H^*(\Omega X)$. We also show that the same $A(\infty)$-algebra $(H^*(X),\{m_i\})$ determines also cohomology modules $H^*(\Lambda X)$.

Détail

Keywords: Hochschild homology, $A(\infty)$-algebra, cohomology algebra, cohomology module, loop space.
Mots-clés : morphism