Graduate Seminar

Location:  CL 312 and Livestream

Speaker: Shashen Gounden

MSc Student supervised by Don Stanley

Title: Steenrod’s realization problem and graded Stanley-Reisner rings

Zoom:  https://uregina-ca.zoom.us/j/92053635111?pwd=U25NcEp6amlGV1YrRDdnUm5qNlhaUT09

Abstract:

If \(X\) is a topological space the cohomology ring \(H^*(X)\) of \(X\) is a ring formed from the cohomology groups of \(X\) together with the cup product serving as the ring multiplication that is also graded-commutative on \(H^*(X)\). The realization problem posed by Steenrod in 1960 asks, "What kind of ring can be represented as the cohomology ring of a space?" That is, given a unital graded-commutative ring \(R\), does there exist a space \(X\) such that \(H^*(X) \cong R\).

In this talk we will consider this problem when the rings are graded Stanley-Reisner rings, that is a quotient of a polynomial ring by a square-free monomial ideal. We review the result of Takeda (2024) who gives a necessary and sufficient condition for a graded Stanley-Reisner ring to be realizable when the graded Stanley-Reisner ring has no pair of generators \(x,y\) such that \(xy \neq 0\) and \(x\) and \(y\) have the same degree which is a power of two.