Operations on Spaces over Operads and Applications to Homotopy Groups

2011/11/30 by Wenbin Zhang, Zhang, Wenbin
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT

paper · pdf · doi:10.48550/arxiv.1111.7134

29 pages, part of the author's Ph.D. thesis

arxiv created 2011/11/30 · arxiv updated 2011/12/01

Abstract

We establish certain smash operations on spaces over operads which are general analogues of the Samelson product on single loop spaces, and obtain a conceptual description of the structure of the homotopy groups of spaces Y over a symmetric K(π,1) operad: π_*Y is a module over the free algebraic symmetric operad generated by operations on homotopy groups induced by these smash operations. In particular the homotopy groups of double loop spaces is a module over the free algebraic symmetric operad generated by the conjugacy classes of Brunnian braids modulo the conjugation action of pure braids.

Related

• Group Operads and Homotopy Theory
• Loop spaces and homotopy operations
• Operads and Operadic Algebras in Homotopy Theory
• Enveloping operads and applications
• Homotopy Algebras for Operads