The proposed work on manifolds has been performed in collaboration with Randal-Williams. For manifolds of dimension 6, 8, 10, ... essentially all of the goals outlined in the proposal have been achieved, and have been published in three long papers in the highest-ranking journals. The understanding in high dimension is now almost on par with the understanding in dimension 2. Building in part on this work, other important results have been obtained by Kupers, Krannich, and others. More recently, joint work with Kupers and Randal-Williams has introduced a new method (cellular E_2 algebras) to the study of moduli spaces. In dimension 2 we discovered a new pattern (secondary stability) in the cohomology.
For the number theoretic aspects, much of what was outlined in the proposal has been achieved in joint work with Venkatesh. In a joint paper which is now published, we use methods from topology (homotopy theory) to study deformations of so-called Galois representations. Several further projects of algebraic or number theoretic flavor have been carried out by the PI and collaborators during the support of the grant. In a paper joint with Feng and Venkatesh, we defined an action of the absolute Galois group of Q on a certain kind of higher Grothendieck--Witt theory, using the moduli space of principally polarized abelian varieties and its connection to integral symplectic groups. We then proceeded to determine the induced action on homotopy groups as a certain universal extension.
Another major finding, not explicitly foreseen in the application but very much in the spirit of this project, was joint with Chan and Payne and has been published in the Journal of the American Mathematical Society. We studied the moduli of curves, and in particular the weight filtration on its rational cohomology. Our main result is a new connection between the top weight cohomology and the cohomology of a graph complex defined by Kontsevich in the 1990s. Combining with a result of Willwacher, we deduce an exponentially growing lower bound on the rank of homology of M_g in degree 4g-5. It was known beforehand that the rational homology vanishes in degrees 4g-4 and higher, so our classes live in the highest possible degree.
Work of Kupers and Randal-Williams, supported by this grant and now published in Forum of Math Pi, led to an understanding of the rational cohomology of Torelli groups for high-dimensional manifolds.
Work of Mikala Jansen under this grant, recently accepted for publication in IMRN, determines the stratified homotopy type of reductive Borel--Serre compactifications.
