M 392C Four-Manifolds (54485)
Dr. Bob Gompf
MWF 11am – 12pm
While work of the past four decades has revolutionized our understanding of 4-manifolds, much of the subject now seems more mysterious than ever. Virtually nothing was known about 4-manifolds until 1981, when Freedman's revolutionary work led to a complete classification of simply connected topological 4-manifolds. Shortly thereafter, Donaldson led a counter-revolution that showed that smooth 4-manifolds were much more complicated than their topological counterparts. That is, many topological 4-manifolds admit infinitely many diffeomorphism types of smooth structures, and others cannot be smoothed at all, frequently in defiance of the predictions of high-dimensional smoothing theory. Currently there isn't even a good guess about how to organize the manifolds that can be distinguished by the gauge-theoretic techniques of Donaldson and others. Many smooth 4-manifolds admit complex structures, and a much larger class admits symplectic structures, but there seems to be a sense in which "most" smooth 4-manifolds do not admit symplectic structures. While much of current research is focused on how to organize such examples, other basic problems remain completely open - for example, does the 4-sphere admit exotic smooth structures? In contrast, while Rn admits a unique smooth structure for n not equal 4, R4 admits uncountably many smooth structures.
We will attempt an overview of the current theory of 4-manifolds, with an emphasis on constructing various types of examples. To visualize 4-manifolds, we will use handle decompositions in the form of Kirby calculus on framed links. This technique is also useful for studying 3-manifolds, since it reduces to surgery theory when we restrict to the 3-dimensional boundaries of 4-manifolds. Various constructions of exotic smooth structures, complex, symplectic, nonsymplectic and Stein manifolds will be discussed as time permits.
Prerequisites: M 382C, M 382D
Gompf and Stipsicz, 4-Manifolds and Kirby Calculus, Grad. Studies in Math. #20, Amer. Math. Soc. 1999.