Abstract
The goal of this dissertation is to extend the theory of picture groups and picture spaces (as developed in works of Igusa, Todorov, and Weyman) to the class of τ-tilting finite alge- bras. In particular, we aim to generalize the result that the picture space has the structure of a K(π,1) (or Eilenberg MacLane space) for the picture group. In Chapter 1, we further explain this motivation, discuss the fundamentals of representation theory, and introduce the theory on which this thesis lies (namely τ-tilting theory). In Chapter 2, we extend the definitions of the picture group and picture space to our class of interest. We also show that the picture space is a K(π, 1) for the picture group when the algebra’s 2-simple minded collections are characterized by pairwise conditions. In Chapter 3, we show that there exist τ-tilting finite algebras whose 2-simple minded collections are not characterized by pair- wise conditions. In doing so, we develop internal criteria for determining when a set of “bricks” is contained in a 2-simple minded collection and use this criteria to study gentle algebras and preprojective algebras in detail. In Chapter 4, we prove the 2-simple minded collections of Nakayama algebras and Nakayama-like algebras are characterized by pairwise conditions, providing new examples of algebras whose picture spaces are K(π,1)’s for their picture groups.