Date of Award

6-2026

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Jeffrey Strom, Ph.D.

Second Advisor

John Martino, Ph.D.

Third Advisor

Nicholas Scoville, Ph.D.

Fourth Advisor

Timothy Clark, Ph.D.

Keywords

Conic space, extension, module, monoid

Abstract

In 1955, I. M. James introduced the James Construction, a free topological monoid that models the loops on the suspension of a given space. In 1969, S. Y. Husseini generalized this idea to RPT monoids: topological monoids with a free-like monoid structure that can be used to model a broader class of loop spaces. In order to prove that these topological monoids are models of loop spaces, both I. M. James and S. Y. Husseini constructed contractible spaces on which these topological monoids act. We define a topological module as a space equipped with an action by a topological monoid. For convenience, we define a monoid-module as a tuple containing a module and its acting monoids.

While the original setting introduced by I. M. James and S. Y. Husseini utilized CW-complexes, spaces built by sequentially attaching disks, in this dissertation we employ the generalized framework of conic spaces introduced by J. Strom. Conic spaces are spaces built by sequentially attaching cones, and conic maps are maps that respect this structure by sending cones to cones. In this context, an RPT monoid is a conic topological monoid where the induced monoid structure on the set of cones is a free monoid, and an RPT module is a conic topological module where the induced action upon the cone set is a free module action.

S. Y. Husseini’s construction of RPT monoid models relied on the ability to extend them, attaching a disk and then minimally completing to an RPT monoid. As the primary objective of this dissertation, we formalize extending RPT monoid-modules within this conic setting. Because we must transport monoids and modules between categories, we employ monoidal categories and monoidal functors. In a broad categorical context, we provide a sufficient condition for monoid-module extensions to exist and provide an explicit construction. Additionally, we prove that such monoid-module extensions are preserved under a special class of monoidal functors.

Applying this general result, we demonstrate that an RPT monoid-module with new cones attached can always be minimally completed to an RPT monoid-module. The underlying topological structure and set structure of the completed RPT monoid-module must match the corresponding extension in the category of topological spaces and the category of sets, respectively. Furthermore, we explore two consequences of extensions: first a characterization of RPT monoid-modules in terms of extensions, and second a sufficient condition for RPT monoid-module pushouts to exist.

Access Setting

Dissertation-Open Access

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