【时间】
第一讲时间：12月29日（星期一）下午2:00-4:00
第二讲时间：12月30日（星期二）下午2:00-4:00
第三讲时间：12月31日（星期三）下午2:00-4:00
【地点】
智华楼丁石孙教室
【主讲人】
叶帆（Harvard University）
【摘要】
Problem 3.105(A) in Kirby's list asks whether every compact 3-manifold M other than S³ has a nontrivial representation π₁(M)→SU(2), and if π₁(M) is non-abelian, when such a representation can have non-abelian image—such manifold is called SU(2) non-abelian. A positive answer to the first question would provide an alternative proof of 3D Poincaré conjecture. Kronheimer–Mrowka showed that the Dehn surgery manifold S³_r(K) for a nontrivial knot and a rational slope r∈[0,2] is always SU(2) non-abelian, and Zentner showed that the splicing of two nontrivial knots in S³ is SU(2) non-abelian. Both results rely on instanton Floer homology and also on the study of submanifolds in the SU(2) character variety of the boundary torus of the knot complement, which is homeomorphic to a pillowcase. Later, through work of many people on instanton Floer homology, the SU(2) non-abelian slope for nontrivial knots (except the right-handed trefoil) was extended to all rational r=p/q∈(2,6) with p=x^e or 2x^e for a prime x and a natural number e. For the right-handed trefoil, the only exceptions to SU(2) non-abelian slopes are of the form 6−1/n for a positive integer n, which can be read directly from the pillowcase.
In the first lecture, I will review the background of the above results and introduce the pillowcase. In the second lecture, I will discuss the strategy for proving SU(2) non-abelian slopes via instanton Floer homology and introduce instanton L-space knots. In the third lecture, I will introduce a diagrammatic way on the pillowcase to understand the surgery exact triangle for instanton Floer homology and provide a dimension formula for the instanton Floer homology of surgery manifolds for knots in S³. Throughout the lecture series, I will treat instanton Floer homology as a black box and focus on its properties, so the minicourse is friendly to those familiar only with fundamental groups, ordinary homology theories, and knot surgeries.
