[Paper Review] Orientability and fundamental classes of Alexandrov spaces with applications

Ayato Mitsuishi|arXiv (Cornell University)|Oct 25, 2016

Geometric Analysis and Curvature Flows33 references3 citations

TL;DR

This paper establishes the equivalence of multiple notions of orientability for Alexandrov spaces—topological, geometric, and analytic—and proves a Poincaré-type duality theorem for orientable open subsets. The key contribution is a duality isomorphism in cohomology and homology for rational coefficients in degrees 0, 1, n-1, and n, with applications to curvature bounds and filling radius inequalities.

ABSTRACT

In the present paper, we consider several valid notions of orientability of Alexandov spaces and prove that all such conditions are equivalent. Further, we give topological and geometric applications of the orientability. In particular, a Poincaré-type duality theorem is proved. As a corollary to the duality theorem, we also prove that if a closed Alexandrov space admits a positive curvature bound in a synthetic sense, then its codimension one homology vanishes. Further, we obtain a filling radius inequality for closed orientable Alexandrov spaces.

Motivation & Objective

• To unify and clarify various definitions of orientability in Alexandrov spaces, which are not necessarily triangulable or homology manifolds.
• To establish that topological, geometric, and analytic notions of orientability are equivalent in the context of Alexandrov spaces.
• To prove a Poincaré-type duality theorem for orientable open subsets of Alexandrov spaces with rational coefficients.
• To derive topological obstructions, such as the vanishing of codimension-one rational homology for positively curved closed Alexandrov spaces.
• To establish a filling radius inequality for closed orientable Alexandrov spaces and explore duality in spaces with boundary.

Proposed method

• Define orientability via local homology isomorphisms and compatibility of orientation classes in the singular cohomology with rational coefficients.
• Use the natural isomorphism between compactly supported cohomology and relative cohomology of the interior to transfer duality results.
• Apply the duality map $ D_M: H^k_c(M;\mathbb{Q}) \to H_{n-k}(M;\mathbb{Q}) $ for open, boundaryless subsets of $ n $-dimensional Alexandrov spaces.
• Prove that the double of a space $ D(X) $ is orientable if and only if the original space $ X $ is orientable, using conical neighborhoods and orientation transfer.
• Establish a Lefschetz-type duality for compact Alexandrov spaces with boundary via the isomorphism $ H^*(M,\partial M) \cong H^*_{\mathrm{c}}(M\setminus\partial M) $.
• Use the structure of NB-spaces (normal bandbodies) and their gluing to define and analyze orientability and boundary behavior.

Experimental results

Research questions

• RQ1 Are the various definitions of orientability in Alexandrov spaces—topological, geometric, and analytic—equivalent?
• RQ2 Does a Poincaré-type duality hold for orientable open subsets of Alexandrov spaces with rational coefficients?
• RQ3 What topological constraints arise for closed Alexandrov spaces with positive curvature in the synthetic sense?
• RQ4 Can a filling radius inequality be established for closed orientable Alexandrov spaces?
• RQ5 Under what conditions is the boundary of an orientable Alexandrov space with boundary itself orientable?

Key findings

• All notions of orientability for Alexandrov spaces are equivalent, including topological, geometric, and analytic definitions.
• A Poincaré-type duality isomorphism $ D_M: H^k_c(M;\mathbb{Q}) \to H_{n-k}(M;\mathbb{Q}) $ holds for orientable open subsets of $ n $-dimensional Alexandrov spaces in degrees $ k = n $ and $ k = n-1 $.
• For a closed orientable Alexandrov $ n $-space, the duality map $ D_M: H^k(M;\mathbb{Q}) \to H_{n-k}(M;\mathbb{Q}) $ is an isomorphism when $ k = 0,1,n-1,n $.
• If a closed $ n $-dimensional Alexandrov space has curvature $ \geq \kappa > 0 $, then its codimension-one rational homology $ H_{n-1}(\Sigma;\mathbb{Q}) $ vanishes.
• For closed orientable Alexandrov spaces with positive curvature, the integral codimension-one homology $ H_{n-1}(\Sigma;\mathbb{Z}) $ also vanishes.
• A filling radius inequality holds for closed orientable Alexandrov spaces, providing a geometric-topological constraint.