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关于Reeb图的流形子复形表示

On Representation of the Reeb Graph as a Sub-Complex of Manifold

Marek Kaluba, Wacław Marzantowicz, Nelson Silva

备注：
18 pages
发表信息：
Topological Methods in Nonlinear Analysis, Vol. 45, No. 1, March 2015
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摘要：

Reeb图$\mathcal{R}(f) $是具有孤立临界点的光滑函数$f\colon M\to \mathbb{R} $的基本不变量之一。它被定义为闭流形$M$的商空间$M/_{\!\sim}$，通过依赖于$f$的关系。本文构造了一个嵌入$M$的$1$维复形$Γ(f)$，它同伦等价于$\mathcal{R}(f)$ .结果表明，对于有限基本群流形上的每个函数$f$，$f$的Reeb图是一棵树.如果$π_1(M)$是一个阿贝尔群，或者更一般的，一个离散顺从群，那么$\mathcal{R}(f)$最多包含一个循环。最后证明了曲面$M_g$上每个函数的Reeb图中的圈数都可以由曲面$M_g$的亏格$g$来估计.

英文摘要：

The Reeb graph $\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\colon M\to \mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\!\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $Γ(f)$ embedded into $M$ which is homotopy equivalent to $\mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $π_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.

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。

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