We investigate the existence of 4-torsion in the integral cohomology of oriented Grassmannians. We establish bounds on the characteristic rank of oriented Grassmannians and prove some cases of our previous conjecture on the characteristic rank. We also discuss the relation between the characteristic rank and a result of Stong on the height of w1 in the cohomology of Grassmannians. The existence of 4-torsion classes follows from the results on the characteristic rank via Steenrod square considerations. We thus exhibit infinitely many examples of 4-torsion classes for oriented Grassmannians. We also prove bounds on torsion exponents of oriented flag manifolds. The article also discusses consequences of our results for a more general perspective on the relation between the torsion exponent and deficiency for homogeneous spaces.

We show that the Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16 is true.

This paper constructs hyper-homology spectral sequences of $\mathbb{Z}$-graded and $RO_{G}$-graded Mackey functors that compute $\mathrm{Ext}$ and $\mathrm{Tor}$ over $G$-equivariant $S$-algebras ($A_{\infty}$ ring spectra) for finite groups $G$. These specialize to universal coefficient and K\"unneth spectral sequences.

Applying the Sullivan conjecture we construct compacta of certain cohomological and extensional dimensions.