Homotopical effects of dilatation
Homotopical effects of dilatation
复制标题
DOI：
10.4310/jdg/1214434601
复制
发表时间：
1978
影响因子：
2.5
通讯作者：
M. Gromov
中科院分区：
文献类型：
--
作者：
M. Gromov
1.1.几何和拓扑复杂性。令 V 和 W 为黎曼流形，X 为映射空间 V-^W。例如，X 可能由所有平滑映射组成，或者可能是嵌入或浸没的空间。我们询问如何通过 x 的几何形状来估计 x € X 的“拓扑复杂性”的度量。我们通过正函数 F: X -^ R+ 来测量 x 的几何复杂性，例如通过 x 的膨胀或通过像狄利克雷泛函这样的积分特征。 x 的拓扑复杂性可以通过其度数（当度数有意义时）或另一个数值不变量来测量。莫尔斯理论提出了不同的观点。我们取水平 Xλ C X, Xλ = F~\[0, λ]),,λ e R+ 并将 Xλ 的数值不变量（例如分量的数量或所有 Betti 数的总和）与 λ 进行比较。当 λ —· oo 时，Xλ 的拓扑复杂度的第一个渐近项通常与 V 和 W 中度量的特定选择无关（但当然取决于 F 的特定类型），我们遇到一个纯拓扑问题：如何用通常的不变量来表达 Xλ 的渐近拓扑？当我们研究 F 临界值的渐近分布时，我们首先需要的是贝蒂数 bi(Xλ), /, λ —> oo 的渐近行为。
1.1. Geometrical and topological complexity. Let V and W be Riemannian manifolds, and X a space of mappings V-^W. For instance, X may consist of all smooth maps, or may be the space of imbeddings or immersions. We ask how to estimate a measure of the "topological complexity" of an x € X by geometry of x. We measure geometrical complexity of x by a positive functional F: X -^ R+, say, by the dilatation of x or by an integral characteristic like the Dirichlet functional. The topological complexity of x may be measured by its degree (when the degree makes sense) or another numerical invariant. The Morse theory suggests a different point of view. We take the levels Xλ C X, Xλ = F~\[0, λ]),,λ e R+ and compare the numerical invariants of Xλ (say the number of components or the sum of all Betti numbers) with λ. When λ —• oo, the first asymptotic term of the topological complexity of Xλ is often independent of the particular choice of metrics in V and W (but depends, of course, on the particular type of F), and we come to a pure topological problem: how to express this asymptotic topology of Xλ in terms of usual invariants? When we study the asymptotic distribution of the critical values of F, what we need first is the asymptotic behavior of the Betti numbers bi(Xλ), /, λ —> oo .