Some completion theorems in algebraic topology.

This thesis explores several different notions of completion. In chapter 2, the representation of a finite category is defined as a generalization of the representation of a group, and it is shown that when X is a simplicial complex, and ${\cal C}(X)$ is the category whose objects are simplices of X and morphisms are face inclusions of X, that the homotopy classes of maps $\lbrack X,\ BGL\sb{n}(\doubc)$) are in bijective correspondence with natural isomorphism classes of functors from ${\cal C}(X)$ to the category of n-dimensional vector spaces and linear isomorphisms. Chapter 3 looks at maps from the classifying space $B{\doubz}/p$ into various $E\sb{\infty}$-spaces Z. It is conjectured that the spectrum associated to the $E\sb{\infty}$-space $Map(B{\doubz}/p, Z)$ has the same $\pi\sb0$ as the function spectrum $F(B{\doubz}/p, E\sb{Z})$ where $E\sb{Z}$ is the function spectrum associated with the $E\sb{\infty}$-space Z, up to some completion. This is shown to be true for the case of stable homotopy, topological complex K-theory, and the algebraic K-theory of ${\rm I\!F}\sb{q},$ where q is a prime different than p. This chapter also extends a theorem of Mislin that $\lbrack B{\doubz}/p, BG\rbrack\simeq\lbrack B{\doubz}/p, BG\sbsp{p}{\wedge}\rbrack$ where the $({-})\sbsp{p}{\wedge}$ denotes Bousfield-Kan p-completion from the case of G compact Lie to the case G an arithmetic group. In chapter 4, Bousfield-Kan's definition of R-completion of a space and Bousfield's R-completion of a group are used along with Lie algebra techniques to show that the free group on p generators is not ${\doubz}\sb{(p)}$-good in the sense of Bousfield-Kan.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/131250/2/9840578.pd