格上同调的约化定理
Reduction theorem for lattice cohomology
摘要：
连通负定铅垂图的铅垂3-流形$M$的格上同调是研究$M$的拓扑性质以及将$M$实现为复解析奇性链时与解析奇性链拓扑性质比较的重要工具.根据定义，其计算基于$\Z^s$的格点的（Riemann-Roch）权重，其中$s$是管道图的顶点数。本文将这个格的秩减少到图的“坏”顶点的数量。（通常，$M$的几何/拓扑通过手术或其他构造由这些“坏”顶点精确编码。它们的数量衡量了管道图离理性管道图的距离。）减少的效果也出现在某些多变量（拓扑庞加莱）系列的水平。由于从这些系列中也可以读取Seiberg-Witten不变量，约化定理也为这些不变量提供了新的公式。该约化还意味着$q\geq ν$的格上同调的$\bH^q=0$消失，其中$ν$是“坏”顶点的数量。(This边界是尖锐的）。
英文摘要：
The lattice cohomology of a plumbed 3--manifold $M$ associated with a connected negative definite plumbing graph is an important tool in the study of topological properties of $M$, and in the comparison of the topological properties with analytic ones when $M$ is realized as complex analytic singularity link. By definition, its computation is based on the (Riemann--Roch) weights of the lattice points of $\Z^s$, where $s$ is the number of vertices of the plumbing graph. The present article reduces the rank of this lattice to the number of `bad' vertices of the graph. (Usually the geometry/topology of $M$ is codified exactly by these `bad' vertices via surgery or other constructions. Their number measures how far is the plumbing graph from a rational one.) The effect of the reduction appears also at the level of certain multivariable (topological Poincaré) series as well. Since from these series one can also read the Seiberg--Witten invariants, the reduction theorem provides new formulae for these invariants too. The reduction also implies the vanishing $\bH^q=0$ of the lattice cohomology for $q\geq ν$, where $ν$ is the number of `bad' vertices. (This bound is sharp.)
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
