Degrees of Maps and Cohomological Rigidity of Partial Flag Manifolds

Abstract

We study the existence of continuous maps with nonzero Brouwer degree between partial flag manifolds. We prove that every continuous map between distinct complex or quaternionic partial flag manifolds has degree zero if either the domain or the codomain is not a Grassmannian. This establishes, for such partial flag manifolds, the analogue of the results of Ramani-Sankaran and Sankaran-Sarkar for complex and quaternionic Grassmannians, as well as an algebraic-geometric result of Paranjape and Srinivas for complex Grassmannians. As a consequence, we prove that complex and quaternionic partial flag manifolds are cohomologically rigid; that is, their rational cohomology rings determine their homeomorphism types. We conjecture that complex and quaternionic partial flag manifolds are homologically rigid.