University of Chicago
Department of Mathematics

A Pi-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object A, one may ask if it can be realized topologically: Is there a space X such that pi_*(X) is isomorphic to A as a Pi-algebra, and if so, can we classify them? Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a Pi-algebra, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of Pi-algebras. What do these groups look like, and can we compute them? We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated Pi-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types, and compare them to what is known from other approaches.