The principal investigator applies gauge theoretic methods to the little studied class of 4-dimensional manifolds, namely, smooth manifolds having integral homology of the 3--sphere times the circle. These manifolds arise naturally as the doubles of homology cobordisms of integral homology spheres. They remain rather elusive, mainly because the conventional approach via Donaldson polynomials and Seiberg--Witten invariants fails due to the lack of second homology. Instead, the investigator studies the flat moduli spaces on such manifolds. A creative count of flat connections leads to an invariant reminiscent of the Casson invariant for homology 3--spheres. The investigator works on developing a torus surgery theory modeled after Casson's surgery formula and Fintushel--Stern's knot surgery on 4--manifolds, with the view of using it to relate the above Casson--type invariant to the classical Rohlin invariant. This approach is expected to lead to solution of some old and difficult problems concerning homology cobordisms of integral homology 3--spheres, smoothing of topological 4--manifolds, and triangulation of topological manifolds in higher dimensions. A part of the above program is a calculation of the degree zero Donaldson polynomial, which is of great interest in its own right. For mapping tori of integral homology spheres, the Donaldson polynomial is identified with the equivariant Casson invariant. For homology 4--tori, it is equal to the Seiberg--Witten invariant, at least modulo 2, thus partially confirming the Seiberg--Witten conjecture.
The proposed research is an investigation of various properties of manifolds in dimensions three and four by methods of gauge theory. These methods, rooted in theoretical physics, have revolutionized low dimensional topology. In the past twenty years, they have been actively used to solve many difficult problems. Nevertheless, their potential is far from being exhausted: the researcher initiates application of gauge theoretic methods to a special class of manifolds which was overlooked until recently but which is crucial for understanding several fundamental questions of topology and topological field theory. In addition to advancing knowledge in these specific areas, the research has a broader educational impact. The principal investigator's teaching experience shows that the graduate classes based on the above topics are of great interest to students in both mathematics and physics, thus promoting closer cooperation between the two sciences. The researcher intends to develop further topics courses, and offer parts of his program as independent research projects for graduate students. The research results will be broadly disseminated at scientific conferences, in professional journals, and on the internet.
