Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Heegaard splittings along orientable surfaces are well-known in 3-manifold theory: the manifold is split into a pair of handlebodies, the embedded discs for which can be used combinatorially to obtain information about both the splitting and the manifold. However, when a non-orientable surface is used in an orientable manifold, the associated Heegaard splitting is one-sided and a single handebody is obtained. There are many natural parallels between one- and two-sided Heegaard splittings, however there are striking and far-reaching differences: the presence of singular meridian discs; and, the connection with Z_2 homology. Both properties serve to hamper existing methods, while offering new approaches. Given the direct connection between geometrically incompressible splittings and Z_2 homology classes of the manifold, a finer degree of control of one-sided splitting surfaces can be established over their two-sided counterparts. In particular, the geometrically incompressible one-sided Heegaard splittings of even Dehn fillings of Figure 8 knot space can be explicitly constructed. This involves a result about the behavior of incompressible non-orientable surfaces under Dehn filling, which shows a marked difference from that of either two-sided splittings or incompressible surfaces. When considering the global properties of one-sided splittings, one is motivated by the two-sided precedent, which provides clear fundamental results to pursue. Whilst existence and finite stable equivalence have been known for some time, versions of the key results that progressed the field of two-sided splittings have been lacking. One of the first of these - Waldhausen's classification of splittings of S^3 - finds an analogue in one-sided splittings of RP^3. Whilst there are many alternative proofs for the S^3 result, it is the original that offers a natural generalization to the one-sided case.

Wed seminar this week only, in 3106