Some naturally ocurring examples of A-infinity bialgebras

TL;DR

This paper demonstrates that certain tensor factors in the homology of spheres over Z_p are naturally occurring A-infinity bialgebras with explicitly described structure maps and relations.

Contribution

It provides explicit formulas and quadratic relations for the A-infinity bialgebra structures on tensor factors in homology, illustrating naturally occurring examples.

Findings

Tensor factors E ⊗ Γ in H(Z,n;Z_p) are A-infinity bialgebras for n>2.

Explicit formulas for structure maps are provided.

The internal structure of these bialgebras is well-understood.

Abstract

Let p be an odd prime. When n>2, we show that each tensor factor of form E \otimes \Gamma in H(Z,n;Z_p) is an A-infinity bialgebra with non-trivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E \otimes \Gamma is a naturally occurring example of an A-infinity bialgebra whose internal structure is well-understood.