Date of Award
Doctor of Philosophy
Dr. Jeffrey Strom
Dr. Michele Intermont
Dr. John Martino
Dr. Jay Wood
Homotopy theory, rational homotopy theory, algebraic topology, resolving classes, closed classes, Witt group
A class of topological spaces is called a resolving class if it is closed under weak equivalences and homotopy limits. Letting R(A) denote the smallest resolving class containing a space A, we say X is A-resolvable if X is in R(A), which induces a partial order on spaces. These concepts are dual to the well-studied notions of closed class and cellular space, where the induced partial order is known as the Dror Farjoun Cellular Lattice. Progress has been made toward illuminating the structure of the Cellular Lattice. For example: Chachólski, Parent, and Stanley have shown that it is a complete lattice, while Hess and Parent have shown that the sublattice of rational spaces admits an embedding of a quotient of the Witt group.
Using the current theory surrounding closed classes and cellular spaces as a framework, this investigation is an attempt to further develop the theory of resolving classes and resolvable spaces. In particular, topological and algebraic criteria for determining if X is A-resolvable when X and A are rational spaces are found. Beyond a characterization of the rational case, these criteria are used to uncover some of the structure inherent to the resolvability relation on spaces, including the emergence of the same quotient of the Witt group discovered by Hess and Parent in the cellular lattice. Comparisons (and perhaps more interestingly, contrasts) are drawn between the theory of cellular spaces with that of resolvable spaces, and potential directions for future inquiry are discussed.
Clark, Timothy L., "Resolving Classes and Resolvable Spaces in Rational Homotopy Theory" (2016). Dissertations. 1623.