On Suspension Equivalent Homeomorphisms

Russian Journal of Nonlinear Dynamics. 2026. Vol. 22. No. 2. P. 427–435.

Pochinka O., Shmukler V., Yakovlev E.

Every discrete dynamical system (cascade) generated by a homeomorphism induces a continuous dynamic system (flow) — a suspension. However, not every flow is equivalent to a suspension over a cascade, a necessary and sufficient condition for this is the existence of a global section for the flow. In the case of the existence, the flow is equivalent to a suspension over a Poincaré map on the global section. The basis of the topological dynamics is the topological classification of cascades (flows) up to a conjugacy (equivalence) realized by a homeomorphism that sends the trajectories of one system into the trajectories of another while preserving the direction of the motion. The paper explores the deep relationship between a homeomorphism and its suspension. The core question is: if two of such suspensions are topologically equivalent, does it mean the original homeomorphisms were topologically conjugate? Usually, the answer is “no”, and here a natural question arises about the relationship between the invariants of the topological conjugacy and the topological equivalence of suspensions for homologically reducible homeomorphisms. In this paper we identify the exact boundary where the answer becomes “yes”. We find conditions under which the topological conjugacy of the homeomorphisms on manifolds is tantamount to the equivalence of the suspensions over them. The found condition, called homological irreducibility, consists of the absence of the eigenvalue 1 in the first homology group action. It allows us to distinguish some classes of homologically irreducible homeomorphisms. In particular, to give an exhaustive description of them in the class of homeomorphisms of surfaces with nonnegative Euler characteristic.

Keywords: suspension, homology group, Topological equivalence