HomTorT

HomTorT is a theoretical framework in mathematics and mathematical physics designed to study spaces with toroidal symmetry by combining ideas from homology theory with torus actions. The name blends homology and torus, and is sometimes read as an acronym for Homology-Torus Theory. In its core formulation, a HomTorT space consists of a topological space X together with a continuous action of a torus T^k, and a set of invariants that encode both the structure of cycles in X and how those cycles behave under torus rotations.

Core concepts in HomTorT center on invariants that arise from a combined chain complex respecting the torus action. These invariants are graded objects with potential product structures intended to capture interactions between nontrivial cycles and the symmetry imposed by the torus. The theory introduces tools such as a HomTorT product and a spectral sequence designed to relate ordinary homology to the torus-equivariant information carried by the action. In this way, HomTorT aims to provide a unified view of topological features and symmetry.

Examples and applications are largely exploratory, with HomTorT used as a thought experiment to model spaces that exhibit toroidal symmetry, such as complex tori or quotient spaces appearing in several areas of geometry and theoretical physics. While the framework is primarily presented as a speculative construct, it serves to illustrate how symmetry can influence homological invariants and inspire new questions in topology and data analysis.

Historically, HomTorT was introduced in speculative literature and has received limited practical adoption, functioning mainly as a conceptual tool rather than an established theory. See also: Homology, Equivariant cohomology, Torus, Group action, Topological data analysis.