Manchester Geometry Seminar 2001/2002

A self-transverse immersion of a smooth closed manifold M^k+2 in R^2k+2 has a double point set which is the image of an immersion of a smooth surface, the double point surface. It is known that this surface may have odd Euler characteristic if and only if k = 1 (mod 4) or k+1 is a power of 2 [Asadi and Eccles, Geometry and Topology, 4 (2000) 149--170]. In this talk I will review this result and then go on to investigate additional structure on the double point surface, concentrating on the easiest case k=1. In this case the normal bundle of the double point surface in R⁴ has an `unordered projective framing'. I will describe generators for the bordism group of such surfaces and explain why some elements arise as double point surfaces and some do not. [This is joint work with my student Jonathan Burgess.]