Systoles of 2-complexes, Reeb graph, and Grushko decomposition
Systoles of 2-complexes, Reeb graph, and Grushko decomposition
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2-复合体的收缩期、Reeb 图和 Grushko 分解
DOI：
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发表时间：
2006
期刊：
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通讯作者：
S. Sabourau
中科院分区：
文献类型：
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作者：
M. Katz;Yuli B. Rudyak;S. Sabourau
设X是一个具有非自由基群的有限2-复形。我们用X上不可收缩环的最小长度的平方证明了X上度规面积的下界，从而建立了一个适用于所有非自由2-络合物的统一收缩不等式。我们的不等式在这个维度上改进了M. Gromov不等式中的常数。该论证依赖于Reeb图和共面积公式，结合Grushko基本群分解中自由不可分解因子数量的归纳。更具体地说，我们为X构建了一种Reeb空间的“最小模型”，让人想起Gromov在表面语境中使用的“砍掉长手指”结构。因此，我们证明了2-复合体的Lusternik-Schnirelmann范畴与收缩范畴的一致性。
Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model” for X, reminiscent of the “chopping off long fingers” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex.