Connecting models of configuration spaces: From double loops to strings

<p>Foundational to the subject of operad theory is the notion of an <em>En</em> operad, that is, an operad that is quasi-isomorphic to the operad of little <em>n-</em>cubes <em>Cn.</em> They are central to the study of iterated loop spaces, and the specific case of <em>n</em> = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two <em>E</em> 2 operads, namely the little 2-cubes operad <em>C</em> 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of <em>C</em>2, which we name the operad of tethered 2-cubes. Much of our analysis involves examining trees labeled by elements of the operad of little intervals, <em>C</em>1. In the final chapter, we generalize this idea of graphs decorated by elements of an operad to the notion of a decorated Feynman category, building off of the work of Kaufmann and Ward. As an immediate application, we will give a simple definition of non-Σ modular operads.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2016

Author dc:creator

• Lucas, Jason M

Contributors dc:contributor

• Ralph Kaufmann
• James McClure
• David B. McReynolds
• Jeremy Miller

• Pure sciences
• Algebraic topology
• Category theory
• Operad theory
• Mathematics

Repository record dc:identifier

https://docs.lib.purdue.edu/open_access_dissertations/802

OAI identifier oai:identifier

oai:docs.lib.purdue.edu:open_access_dissertations-1994