Agenda

Abstract: We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean 3-space. Since most humans, including the speaker, cannot visualise Euclidean spaces of higher dimensions, it appears to be impossible to give pictorial representations of higher-dimensional manifolds. However, one can in fact draw 1-dimensional pictures truly representing the topology of surfaces. By analogy, one can draw 2-dimensional pictures of 3- manifolds (Heegaard diagrams), and 3-dimensional pictures of 4-manifolds (Kirby diagrams). With a little trick, one can even draw 2-dimensional (sic!) pictures of at least some 5-manifolds. In this talk I shall explain how to draw such pictures and how to use them for answering topological and geometric questions. The work on 5-manifolds is joint with Fan Ding and Otto van Koert.