Casson-type invariants in dimension four

主要研究者应用规范理论的方法，以thelittle研究类的4维流形，即光滑流形具有积分同源的3-球倍thecircle。这些流形自然地产生为整同调球面的同调协边的加倍。他们仍然相当难以捉摸，主要是因为传统的方法通过唐纳森多项式和Seiberg-维滕不变量失败，由于缺乏第二同调。相反，研究人员研究平坦模空间上这样的流形。创造性地计算平面连通导致了一个类似于同调3-球面的Casson不变量的不变量.研究者在4-流形上发展了一个仿照Casson的外科手术公式和Fintushel-Stern的结外科手术的环面外科手术理论，以期用它把上述Casson-型不变量与经典的Rohlin不变量联系起来.这一方法有望解决一些古老而困难的问题，如积分同调3-球面的同调配边、拓扑4-流形的光滑化以及高维拓扑流形的三角剖分等。上述程序的一部分是计算零度唐纳森多项式，这本身就很有意义。对于整同调球面的环面映射，用等变Casson不变量来确定唐纳森多项式.对于同调4-环面，它等于Seiberg-维滕不变量，至少模2，从而部分地证实了Seiberg-维滕猜想.本文用规范理论的方法研究了3维和4维流形的各种性质.这些方法源于理论物理，是低维拓扑学的一个革命.在过去的二十年里，它们被积极地用来解决许多难题。然而，它们的潜力远未耗尽：研究人员开始将规范理论方法应用于一类特殊的流形，这类流形直到最近才被忽视，但对于理解拓扑学和拓扑场论的几个基本问题至关重要。除了推进这些特定领域的知识之外，这项研究还具有更广泛的教育影响。首席研究员的教学经验表明，基于上述主题的研究生课程对数学和物理学的学生都有很大的兴趣，从而促进了这两门科学之间的密切合作。研究者打算进一步发展专题课程，并将其部分计划作为独立研究项目提供给研究生。研究结果将在科学会议、专业期刊和互联网上广泛传播。

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.