发表机构
Universidad de Buenos Aires; CONICET; IMAS (CONICET)（布宜诺斯艾利斯大学; 阿根廷国家科学研究委员会; IMAS（阿根廷国家科学研究委员会））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出Cartan-Hadamard定理的拓扑版本，证明具有特殊路径性质的空间是非球面的，定义非球面复杂度并将其与等变拓扑复杂度关联，为研究空间非球面性提供新视角。
AI 中文摘要
我们研究Cartan-Hadamard定理的一个拓扑版本，该版本可通过特殊路径族研究空间的非球面性。拓扑空间X具有特殊路径性质（distinguished path property，简称dpp），当且仅当存在从路径同伦类空间Π(X)（相对于端点）到路径空间X^I的连续映射，且该映射是典范商映射的右逆。对于赋予局部凸度量的完备度量空间，特殊路径恰好是局部测地线。我们证明，若X具有dpp，则其通用覆盖空间是可缩的，特别地X是非球面的。空间Π(X)是X上的纤维丛，其纤维为X的通用覆盖空间；对于非正曲率的黎曼流形，Π(X)自然同构于切丛。我们还证明，每个局部有限或可数的非球面CW复形都具有dpp，这使我们能通过连续截面的存在性重新诠释CW复形的非球面性。此外，我们借助从Π(X)到X^I的局部截面定义并研究空间的非球面复杂度概念，并将其与Colman和Grant引入的等变拓扑复杂度概念相关联。
英文摘要
We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space $Π(X)$ of homotopy classes of paths (relative endpoints) to the path space $X^I$ that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if $X$ has the dpp, then its universal cover is contractible and, in particular, $X$ is aspherical. The space $Π(X)$ is a fiber bundle over $X$ whose fiber is the universal cover of $X$, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from $Π(X)$ to $X^I$ and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.
Comments15 pages
