Grant: Discovery Grants Program - Individual

The long-term goal of the programme is to study homotopy Gerstenhaber algebras and other structures up to higher homotopies as well as their applications to problems arising in topology and in transformation groups in particular. Starting point of the programme is the recent breakthrough achieved in my preprints "The cohomology rings of homogeneous spaces" (arxiv:1907.04777) and "The cohomology rings of smooth toric varieties and quotients of moment-angle complexes" (arxiv:1907.04791). By determining these cohomology rings, I answer a question that has been open for 50 years (homogeneous spaces) and correct a mistake that has been in the literature for 20 years (partial quotients of moment-angle complexes). A key ingredient in the proofs is the notion of a homotopy Gerstenhaber algebra (hga). So far, the cup product in the cohomology of a homogeneous space is only established under the assumption that 2 is invertible in the given coefficient ring. My first objective is to extend this, ideally to arbitrary coefficients. Another hard problem is to generalize the proof obtained so far from classifying spaces of groups to arbitrary spaces with polynomial cohomology. These two objectives will serve as a guiding problems for the programme. The progress obtained so far rests on a refined study of homotopy Gerstenhaber algebras and also of so-called strongly homotopy commutative (shc) algebras. I have "pedestrian" proofs that require extremely long, but not very illuminating calculations. However, the form of the solutions suggest that there might be a deeper structure lurking behind the formulas. Obtaining a better understanding in this direction is another objective. Another crucial ingredient so far is an hga formality result for classifying spaces of tori. It has already been extended to Davis-Januszkiewicz spaces. I want to study further extensions of hga formality or possibly weaker notions to classifying spaces of other groups or more general classes of spaces. There are also other instances where up-to-homotopy structures appear or are expected to appear in the study of group actions, and addressing them are further objectives. Let us mention the case of real toric spaces, the toral rank conjecture and applications to free loop spaces and string topology. With Prof. J. Minác I want to apply my recent results to problems in Galois theory. Manual calculations with up-to-homotopy structures soon become tedious because the size of the formulas increases very quickly. A secondary objective of the programme is to develop software tools to facilitate these computations (and to identify suitable computer algebra systems to implement the tools).