Mathematical Sciences: Low-Dimensional Topology and Gauge Theory

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9704204 Saveliev The research is concerned with a topic which is central to the theory of smooth 4-dimensional manifolds, namely the homology cobordism group of integral homology 3-spheres. The long-term goal is to make progress in solving the long-standing problem of R. Kirby about the existence of elements of order two in this group carrying the Rohlin invariant. The investigator concentrates his attention at the elements represented by the links of algebraic singularities, in particular, Seifert fibered homology spheres. His approach is based on application of the methods of modern gauge theory to investigation of the invariant introduced in the 70's by W. Neumann and L. Siebenmann for the so-called graph homology 3-spheres. This application is threefold. First, he uses some special resolutions of singularities to construct a 4-cobordism of an algebraic link with the prescribed intersection form, and he then applies the gauge-theoretical results which prohibit certain integral bilinear forms as intersection forms of such a resolution if the link bounds a homology ball. The result of this investigation is further applied to show that Rohlin invariant one algebraic links cannot have order two in the homology cobordism group. Second, the investigator uses the instanton Floer homology to introduce a new invariant for arbitrary homology 3-spheres, and he proves that it agrees with the invariant of Neumann and Siebenmann when the latter is defined (graph manifolds). This result also gives new insights into Floer homology, in particular, answers positively M. Atiyah's question whether there is a Milnor fiber description of Floer homology of certain algebraic links, and whether it is related to the complex conjugation action on the homology of the Milnor fiber. Finally, the investigator applies his new invariant to investigate Kirby's problem for general homology spheres. The first step in this direction is to prove that his invariant should vanish on all hom ology 3-spheres that bound a homology ball with just one 1-handle (and a 0- and 2-handle). His technique is to use the Floer exact triangle to keep track of the Floer homology along the cobordism. One of the central objects of investigation in both mathematics and theoretical physics is a smooth n-dimensional manifold. Although this object looks locally like an n-dimensional Euclidean space, its global structure may still be very rich and complicated. Most major results about 2-dimensional manifolds were obtained in the 19th century. Manifolds in dimensions greater than or equal to 5 were successfully classified in the 1960's. Though some major questions about 3-manifolds remain unanswered, it is manifolds of dimension 4, the dimension of relativity theory, that occupy the most special place in the manifold hierarchy. On the one hand, they are not "big enough" to apply to them the arguments that proved to be so useful in higher dimensions. On the other hand, their dimension is too big to apply more intuitive methods that work effectively in lower dimensions. Progress has been slow here for a few decades until recent developments that involved the application of ideas from the physics of gauge theories. The main results are due to S. Donaldson, who initiated the whole program in the early 1980's, and most recently to N. Seiberg and E. Witten. The gauge-theoretical approach proved to be very fruitful and led to the solution of many hard problems. The investigator is applying these modern methods to make progress on yet another long-standing problem in topology, the structure of the homology cobordism group of homology 3-spheres, which would provide new insight into manifold structure. ***