Year: 1998
Type: preprint
Abstract: The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups π_{*-1} X (which are isomorphic to π_* Y, if X=Ω\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be formulated in the framework of an obstruction theory for realizing Π-algebras, which is simplified here by showing that any (suitable) Δ-simplicial space may be made into a full simplicial space (a result which may be useful in other contexts).
Source: arXiv (Cornell University)
Author: Blanc, David
Institution: University of Haifa
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1553394926