Casson-type invariants in dimension four

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The principal investigator applies gauge theoretic methods to thelittle studied class of 4-dimensional manifolds, namely, smoothmanifolds having integral homology of the 3--sphere times thecircle. These manifolds arise naturally as the doubles of homologycobordisms of integral homology spheres. They remain rather elusive,mainly because the conventional approach via Donaldson polynomialsand Seiberg--Witten invariants fails due to the lack of secondhomology. Instead, the investigator studies the flat moduli spaceson such manifolds. A creative count of flat connections leads to aninvariant reminiscent of the Casson invariant for homology 3--spheres.The investigator works on developing a torus surgery theory modeledafter Casson's surgery formula and Fintushel--Stern's knot surgery on4--manifolds, with the view of using it to relate the aboveCasson--type invariant to the classical Rohlin invariant. Thisapproach is expected to lead to solution of some old and difficultproblems concerning homology cobordisms of integral homology3--spheres, smoothing of topological 4--manifolds, and triangulationof topological manifolds in higher dimensions. A part of the aboveprogram is a calculation of the degree zero Donaldson polynomial,which is of great interest in its own right. For mapping tori ofintegral homology spheres, the Donaldson polynomial is identifiedwith the equivariant Casson invariant. For homology 4--tori, it isequal to the Seiberg--Witten invariant, at least modulo 2, thuspartially confirming the Seiberg--Witten conjecture.The proposed research is an investigation of various properties ofmanifolds in dimensions three and four by methods of gauge theory.These methods, rooted in theoretical physics, have revolutionizedlow dimensional topology. In the past twenty years, they have beenactively used to solve many difficult problems. Nevertheless, theirpotential is far from being exhausted: the researcher initiatesapplication of gauge theoretic methods to a special class ofmanifolds which was overlooked until recently but which is crucialfor understanding several fundamental questions of topology andtopological field theory. In addition to advancing knowledge inthese specific areas, the research has a broader educational impact.The principal investigator's teaching experience shows that thegraduate classes based on the above topics are of great interest tostudents in both mathematics and physics, thus promoting closercooperation between the two sciences. The researcher intends todevelop further topics courses, and offer parts of his program asindependent research projects for graduate students. The researchresults will be broadly disseminated at scientific conferences, inprofessional journals, and on the internet.