On simplicial loops and H-spaces

Abstract

An algebraic loop is a ‘group without associativity’. It holds that a surjective homomorphism of simplicial loops is a Kan fibration, the Moore complex is a loop-valued chain complex with homology the homotopy groups of the simplicial loop, and the simplicial loop is minimal iff the Moore complex is minimal. We show that the minimal simplicial model of a connected H-space X is a simplicial loop. As an application, we give a short proof of a Theorem of Kock, Kristensen and Madsen which represents cohomology theories by the homology of loop-valued cochain functors. Then we show that the Postnikov decomposition of a connected minimal simplicial loop consists of central loop extensions. Using this, we prove a refinement of a Theorem of Iriye and Kono: For a connected 2-antilocal H-space X, there exists a strict commutative loop structure on its minimal model.