4-torsion classes in the integral cohomology of oriented Grassmannians

Abstract

We investigate the existence of 4-torsion in the integral cohomology of oriented Grassmannians. We prove a general criterion for the appearance of 4-torsion classes based on (twisted) Steenrod squares and show that there are many cases where this criterion is satisfied for minimal-degree anomalous classes, assuming a conjecture on the characteristic rank. We also establish the upper bound in the characteristic rank conjecture for oriented Grassmannians $\tilde{Gr}_k(n)$, and prove the equality in the cases $k=5, n=2^t-1,2^t$ and $k=6, n=2^t$. This provides infinitely many examples of oriented Grassmannians having 4-torsion in their integral cohomology. On the way, we clarify the relation between minimal-degree anomalous classes and results of Stong on the height of the first Stiefel-Whitney class $w_1$ in the mod 2 cohomology of real Grassmannians, for which we give an independent proof. We also establish some bounds on torsion exponents for the integral cohomology of oriented flag manifolds. Based on these findings and further computational evidence, we formulate a conjectural relationship between the torsion exponent in the integral cohomology of homogeneous spaces and their deficiency.