Casson-type invariants in dimension four

主要研究者将规范理论方法应用于研究较少的一类四维流形，即具有3球乘以圆的积分同调的光滑流形。这些流形自然地产生为积分同调球的同调配合的对偶。它们仍然相当难以捉摸，主要是因为通过Donaldson多项式和Seiberg—Witten不变量的传统方法由于缺乏二次同调而失败。相反，研究者研究了这种流形上的平模空间。平面连接的创造性计数导致不变量，让人想起同源3-球体的卡森不变量。研究者致力于开发一种环面手术理论，该理论以Casson的手术公式和Fintushel- Stern在4-流形上的结手术为模型，目的是使用它将上述Casson-型不变量与经典的Rohlin不变量联系起来。该方法有望解决积分同调球的同调协、拓扑流形的光滑化、高维拓扑流形的三角化等老难题。上述程序的一部分是零度唐纳森多项式的计算，它本身就很有趣。对于整同调球的映射环面，用等变Casson不变量识别了Donaldson多项式。对于同调4-环面，它等于Seiberg- Witten不变量，至少模为2，从而部分地证实了Seiberg- Witten猜想。本文提出的研究是用规范理论的方法研究三维和四维流形的各种性质。这些根植于理论物理学的方法彻底改变了低维拓扑学。在过去的二十年里，它们被积极地用于解决许多难题。然而，他们的潜力还远远没有被耗尽：研究人员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的学生都很感兴趣，从而促进了两门科学之间的更密切合作。该研究人员打算进一步开发主题课程，并将其部分课程作为独立研究项目提供给研究生。研究成果将在科学会议、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.