## HA-spaces and Commutative Homology Rings

- Author(s): Nguyen, Nicholas D.
- et al.

## Abstract

In the study of Lie groups, topological groups, and H- spaces, one common problem involved finding characterizations of spaces whose mod p homology is commutative, and in the case that it is not commutative, one could ask what kinds of commutators occurred in the mod p homology. A common theme in H-space theory is trying to extend results for Lie groups to H-spaces by creating new techniques and finding new proofs. As such, our dissertation stems from recent work in characterizing Lie groups whose mod p homology is commutative using a tool called the adjoint action of the Lie group on its loop space. Our main contribution starts with defining and studying the adjoint action, along with similar maps like the commutator, for a generalization of compact simply- connected Lie groups called finite simply-connected HA- spaces, and the homomorphism it induces on mod p cohomology (p odd). After finding a formula for the induced homomorphism in terms of the coproduct, we proceed to several applications. Given a finite simply-connected HA-space X, we use the adjoint action, along with similar maps, and their induced homomorphisms to characterize HA- spaces whose mod p homology is commutative. This generalizes earlier work by showing that X does not need the full structure of a Lie group; it can be weakened to an HA-space structure. For our second application, given a finite simply-connected HA-space X, we study the free loop space of X and use the adjoint action and related maps to compute products in the mod p homology of the free loop space. Finally, given a finite simply-connected HA-space X, we demonstrate how to use the commutator on X in order to construct a new multiplication map on X which induces a commutative algebra structure in mod p homology