Carlsson conjectured that if a finite CW complex admits a free action by a $p$-torus of rank $n$, then the sum of its mod-$p$ Betti numbers is at least $2^n$. An algebraic version predicted the same bound for the total dimension of the homology of nonacyclic, finite, free $(\mathbb{Z}/p)^n$-equivariant chain complexes. In 2017, Iyengar and Walker constructed counterexamples to this algebraic conjecture and raised the question if they can be realized topologically in order to produce counterexamples to Carlsson’s conjecture.

In this talk, I will explain that these counterexamples can not be realized topologically based on multiplicative properties of the spectral sequence obtained by filtration with powers of the augmentation ideal. This is joint work with Henrik Rüping.