Operations on Spaces over Operads and Applications to Homotopy Groups

Details Operations on Spaces over Operads and Applications to Homotopy Groups Operations on Spaces over Operads and Applications to Homotopy Groups

2011-11-30

arxiv.org/abs/1111.7134v1

Mathematics

Algebraic Topology

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

Scientific paper

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 $K(\pi,1)$ operad: $\pi_*Y$ 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.

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