jats:titleAbstract</jats:title> jats:pWhen a multistage high-speed compressor is operated away from its design point, extreme incidence is caused in some blade rows. This results in large, localized separations that are three dimensional in nature. In this paper, topological reasoning is used to describe the behavior of these three-dimensional separations. It is shown that two classes of separation exist: one in which the flow progresses from attached to separate in a smooth way and another where there is a discontinuity in the response of the flow topology. It is shown that the global structure of the flow depends on the type of topological response that occurs. When the response is discontinuous, nonaxisymmetric cells of separated blades are formed. When the response is smooth, the resultant separated flow is axisymmetric. The paper is split into two broad sections: The first section presents examples of the two different classes of topological response that can occur in a single blade row, and it also shows how an engineer can achieve a different response by altering the blade design. The second section covers the analysis of a multistage high-speed compressor. The compressor initially presents the discontinuous behavior with rotating cells of separations. It is then redesigned to reduce the severity of the cell behavior or remove it entirely.</jats:p>