# Topology

March 20, 2024

The realizability question is an old question in algebraic topology. The Hopf invariant 1 problem, solved by Adams in the early 1960s is a special case. Later in the 60s by developing rational homotopy theory Quillen was able to show that all graded algebras over the rationals are realizable. Steenrod started work on the problem of polynomial algrebras over the integers, and a complete solution in this case was given by Anderson-Grodal in 2010. After giving an introduction to the problem, I will talk about some recent progress, concentrating on two special cases: algebras that are polynomial modulo a monomial ideal and algebras that are torsion free and become exterior after tensoring with the rationals. The first case is related to a graph colouring problem and the second involves creating a weighted version of polyhedral products. This is joint work with Larry So, Stephen Theriault and Masahiro Takeda.