$A_{\infty }$-actions and Recognition of Relative Loop Spaces

Abstract

We show that relative loop spaces are recognized by $A_{\infty }$-actions. A certain version of the 2-sided bar construction is used to prove such recognition theorem. The operad ${\mathsf{Act}}_{\infty }$ of $A_{\infty }$-actions is presented in terms of the Boardman-Vogt resolution of the operad $\mathsf{Act}$. We exhibit an operad homotopy equivalence between such resolution and the $1$-dimensional Swiss-cheese operad ${\mathcal{SC}}_1$.

21 pages, 1 figure

References in corpus (3)

• Homotopy units in A-infinity algebras
• Finiteness of $A_n$-equivalence types of gauge groups
• Associahedra, Multiplihedra and units in $A_{\infty }$ form