Date of Award
Doctor of Philosophy
Dr. Jeffrey Strom
Dr. Le Minh Hà
Dr. John Martino
Dr. Jay Wood
Phantom maps, decomposability, spaces, finiteness conditions. maps
The purpose of this dissertation is to extend principles for detecting the existence of essential phantom maps into spaces meeting particular finiteness conditions. Zabrodsky shows that a space Y having the homotopy type of a finite CW complex is the target of essential phantom maps if and only if Y has a nontrivial rational homology group. We show this observation holds on the collection of finite generalized CW complexes. Similarly, Iriye shows a finite-type, simply connected suspension space is the target of essential phantom maps if and only if it has a nontrivial rational homology group. We show this observation holds on a large class of simply connected, finite-type co-H-spaces, and begin investigating extensions to the collection of spaces having finite LS category.
To locate phantom maps into finite generalized CW complexes we study the Gray index of phantom maps and make use of a highly natural filtration on the set of phantom maps between two spaces studied by Hà and Strom. To locate phantom maps into co-H-spaces, we develop decomposition methods in phantom map theory and make use of geometric realizations of natural decompositions of tensor algebras discovered by Selick, Grbrić, Theriault, and Wu. We also study the Gray index of phantom maps into co-H-spaces.
Our observations on the Gray index of phantom maps lead to the definition and study of a new homotopy invariant of spaces: the Minimal Inbound Gray Index.
Schwass, James, "Phantom Maps, Decomposability, and Spaces Meeting Particular Finiteness Conditions" (2015). Dissertations. 540.