Genus versus double for immersed surfaces, and some concordance invariants of knots
Genus versus double for immersed surfaces, and some concordance invariants of knots

Peter Kronheimer, Harvard University
Fine Hall 314
If X is a simplyconnected closed 4manifold containing an oriented embedded surface S of genus g, is there always an immersed sphere S' which represents the same homology class and has only g transverse doublepoints? This is an open question, though a "relative" version of the question (concerning surfaces in the 4ball bounding a given knot in the 3sphere) is known to have a negative answer. This talk will ask whether there are additive invariants of knots which can be used to detect this distinction between genus and doublepoints. A previous talk in this seminar, by Tom Mrowka, introduced the tools from gauge theory that will be used, but I will try to make this talk largely independent.