10.17192/z2019.0528
Zoller, Leopold
Leopold
Zoller
On the Toral Rank Conjecture and Variants of Equivariant Formality
Zur Toral Rank Vermutung und Varianten von äquivarianter Formalität
Philipps-Universität Marburg
2019
formality
Toral Rank Conjecture
group action
cohomology
Kohomology
Torus
Toral Rank Conjecture
Mathematics
Mathematik
Gruppenwirkung
Formalität
torus
Goertsches, Oliver (Prof. Dr.)
Mathematik und Informatik
2020-01-08
2020-01-08
en
DoctoralThesis
https://archiv.ub.uni-marburg.de/diss/z2019/0528
urn:nbn:de:hebis:04-z2019-05288
https://archiv.ub.uni-marburg.de/diss/z2019/0528/cover.png
application/pdf
https://rightsstatements.org/vocab/InC-NC/1.0/
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.