Steenrod's Problem for Face Rings
Given a graded algebra A, Steenrod's problem asks if A can be realized. That is if there is a space X such that H∗(X)=A. The Hopf invariant one problem is a famous example. Around 15 years ago Anderson-Grodal completed the solution of this problem for polynomial rings. We are interested in the case of the face ring SR(K) of a simplicial complexes K. If all generators have degree 2 or 4 then SR(K) can always be realized using polyhedral products. If there is a single generator of degree 4 then the problem has been mostly solved by Takeda. We consider the case where all generators are of degree 4 and 6. This gives rise to a graph coloring problem, and a kind of coloring which can be thought of as a localization of usual coloring.
This is joint work with Masahiro Takeda.