On the Toral Rank Conjecture and Variants of Equivariant Formality

We investigate the topological consequences of actions of compact connected Lie groups. Our focus lies on the \emph{Toral Rank Conjecture}, which states that a suitable space $X$ with an almost free $T^r$-action has to satisfy $\dim H^*(X;\mathbb{Q})\geq 2^r$. We investigate various refinements of formality in an equivariant setting and show that they imply the TRC in several cases. Furthermore, we study the properties of the newly developed terminology with regards to possible implications, inheritance under elementary topological constructions, and characterizations in terms of higher operations on the equivariant cohomology. We also attack the problem of finding bounds for $\dim H^*(X;\mathbb{Q})$ in the spirit of the TRC outside of the formal context. Different lower bounds are constructed and applied in particular to the case of cohomologically symplectic spaces

GKM manifolds are not rigid

We construct effective GKM $T^3$-actions with connected stabilizers on the total spaces of the two $S^2$-bundles over $S^6$ with identical GKM graphs. This shows that the GKM graph of a simply-connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.Comment: v2: simplified construction of the minimal example. Comments are welcome

Poincar\'e dualization and formal domination

We consider the question of whether formality of the domain of a non-zero degree map of closed manifolds implies formality of the target. Though there are various situations where this is indeed the case, we show the answer is negative in general, with a counterexample given by a non-zero degree map from a formal manifold to one that carries a non-vanishing quadruple Massey product. This violates a general heuristic that the domain of a non-zero degree map should be more complicated than its target. For the construction of the counterexample we introduce a method to turn a cdga into one that satisfies Poincar\'e duality, which is natural in certain situations.Comment: 15 pages, comments very welcom