Universität Augsburg
Institut für Mathematik

Riemannian manifolds M with holonomy equal to the exceptional Lie group G_2 are difficult to construct. Joyce proved both that compact examples exist and that their moduli spaces (meaning we mod out by diffeomorphisms isotopic to the identity) are always locally Euclidean (of dimension the third Betti number of M).

Despite Joyce’s results, very little is known about the global topology of G_2-moduli spaces, and until recently it was not known whether any path component of a G_2-moduli space was not contractible. I will report on joint work with Sebastian Goette and Thorsten Hertl, where we exhibit components of some G_2-moduli spaces, which have non-trivial second homotopy groups.

We also show that related non-trivial homotopy classes appear in \pi_2 BDiff(M), where Diff(M) is the group of diffeomorphisms of M and its classifying space BDiff(M) is another interesting space about whose global topology rather little is known.