Reeb图、超曲面系统与到自由群的满同态之间的关系
Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups
AI总结：
本文构造了紧流形基本群到自由群的满同态与带框不可分离超曲面系统间的对应关系，并利用Reeb图给出了闭流形上满同态的表示，为曲面群满同态的等价分类提供了纯几何拓扑证明。
摘要：
我们构造了从紧流形$M$的基本群到秩为$r$的自由群的满同态$φ\colon π_1(M) \to F_r$与$M$中$r$个带框不可分离超曲面系统之间的对应关系，该对应关系诱导了到此类系统的带框配边类上的双射。因此，对于闭流形，任何此类$φ$都可以由Morse函数$f\colon M \to \mathbb{R}$的Reeb满同态表示，即由商映射$M \to \mathcal{R}(f)$到$f$的Reeb图上诱导的满同态表示。应用此构造，我们讨论了到自由群的满同态在（强）等价下的分类问题，为曲面群该问题的解决提供了新的纯几何拓扑证明。
英文摘要：
We construct a correspondence between epimorphisms $φ\colon π_1(M) \to F_r$ from the fundamental group of a compact manifold $M$ onto the free group of rank $r$, and systems of $r$ framed non-separating hypersurfaces in $M$, which induces a bijection onto framed cobordism classes of such systems. In consequence, for closed manifolds any such $φ$ can be represented by the Reeb epimorphism of a Morse function $f\colon M \to \mathbb{R}$, i.e. by the epimorphism induced by the quotient map $M \to \mathcal{R}(f)$ onto the Reeb graph of $f$. Applying this construction we discuss the problem of classification up to (strong) equivalence of epimorphisms onto free groups, providing a new purely geometrical-topological proof of the solution of this problem for surface groups.
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