Symmetries in topology and algebra

• Applicant: Professor Dr. Markus Land
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 527329998

This project proposal is about cohomological and homotopical invariants of certain groups of symmetries appearing in topology and algebra, and in particular their interaction. The main focus is on mapping class groups of surfaces and symplectic groups over the integers. These groups are related by a natural homomorphism, and one aspect of this proposal is to study the induced map on (stable) cohomology with finite coefficients. A combination of a number of results says that this is equivalently described by studying the map induced on cohomology of a map $\alpha$ of spaces associated to the spectra MTSO(2) and GW^{-s}(Z) - the Thom spectrum of the negative of the universal bundle over BSO(2) and the antisymmetric (aka symplectic) Grothendieck--Witt spectrum of the integers. These spaces also carry interesting higher homotopy groups (in contrast to the classifying spaces of the above mentioned symmetry groups) and another aspect of this proposal is to compare these homotopy groups via the above mentioned map $\alpha$. The proposal consists of several individual milestones, for instance that $\alpha$ is such that one obtains an infinite family of elements of degree $8i-3$ in the stable mod 2 cohomology of mapping class group, which should be studied in more detail, but also to calculate the effect of $\alpha$ on homotopy groups (modulo torsion) via its relation to the signature homomorphism on MTSO(2). Several variants of such questions arise naturally, like studying the spin mapping class group and its relation to the quadratic symplectic group, or to study higher dimensional analogs where surfaces are replaced by suitable high dimensional manifolds.

• DFG Programme: Research Grants