Fibration Theorems for TQ-Completion of Structured Ring Spectra

Abstract: By considering algebras over an operad $\mathcal{O}$ in one's preferred category of spectra, we can encode various flavors of algebraic structure (e.g. commutative ring spectra). Drawing intuition from singular homology of spaces and Quillen homology of rings, topological Quillen ($\mathbf{TQ}$) homology is a naturally occurring notion of homology for these objects, with analogies to both singular homology and stabilization of spaces.

For a given $\mathcal{O}$algebra $X$, there is a canonical way (following Bousfield-Kan) to "glue together" iterates $\mathbf{TQ}^n(X)$ of the $\mathbf{TQ}$-homology spectrum of $X$ to construct "the part of $X$ that $\mathbf{TQ}$-homology sees," namely its $\mathbf{TQ}$-completion. We then ask, "When can $X$ be 'recovered from' $\mathbf{TQ}(X)$ in this way?" Bousfield-Kan consider the analogous question in spaces and conclude that all nilpotent spaces are weakly equivalent to their homology completion. The key technical maneuver of their proof involves showing that certain fibration sequences are preserved by completion. In this talk, we will discuss certain types of fibration sequences of $\mathcal{O}$-algebras which are preserved by $\mathbf{TQ}$-completion, drawing analogies along the way to the case of pointed spaces.

algebraic topologydifferential geometrygeneral topologygeometric topology

Audience: researchers in the discipline

GOATS

Series comments: Description: A series of online mini-conferences for graduate students in Geo

Conference will be on Zoom and simultaneously livestreamed to Youtube. Zoom link is obtained by registering on the website, the livestream link will be posted closer to conference date.