The topology of local quaternionic toric actions

TL;DR

This paper explores the topology of manifolds with local quaternionic toric actions, revealing how their global structure is determined by orbit spaces and providing tools to compute their cohomology and K-theory.

Contribution

It extends classical toric topology methods to the quaternionic setting, including spectral sequences and a quaternionic Meyer signature formula.

Findings

Explicit descriptions of cohomology and K-theory for these manifolds

Development of a quaternionic Meyer signature formula

Topological classification based on orbit space data

Abstract

In this paper we examine the topology of manifolds equipped with a local quaternionic toric action modeled on the regular representation of the quaternionic torus $Q^n=(S^3)^n$. Building on our previous work, where the toric, differential and tetraplectic foundations were established, we show that the global topology of such manifolds is determined by the orbit space and its characteristic data. We construct Leray--Serre and Atiyah--Hirzebruch spectral sequences for the orbit projection, yielding explicit descriptions of the cohomology and $K$-theory of manifolds equipped with local quaternionic toric actions. In dimension four, we develop a quaternionic analogue of the Meyer signature formula and we briefly outline an $L$-theoretic interpretation of the resulting signature invariants. These results extend the methods of the classical (complex) toric topology to the quaternionic setting.