Aleksandar Milivojevi&cacute, Max Planck Institute for Mathematics in Bonn

Title: Approaching Hirzebruch's prize question via rational surgery

Abstract: In his 1990's book on manifolds and modular forms, Hirzebruch asked whether there exists a 24-dimensional closed spin manifold satisfying certain conditions on its Pontryagin classes, motivated by the observation that one could compute the dimensions of the irreducible representations of the Monster group via certain characteristic numbers of such a manifold. Hopkins and Mahowald showed in the early 2000's that such manifolds exist by understanding the bordism theory of manifolds admitting string structures (a further "lift" of the special orthogonal group beyond spin). I will present an alternative, relatively elementary construction of a manifold as asked for by Hirzebruch, which also offers a large amount of flexibility in constructing other solutions, using an adaptation of rational homotopy theoretic results of Sullivan from the 1970's to spin manifolds.