The Classification of Cohomology Endomorphisms of Certain Flag Manifolds

Abstract

Let U(n) be the unitary group of rank n, SO(m) the special orthogonal group of rank m, and Sp(n), the symplectic group of rank n. Fix, once and for all, a maximal torus Tn ⊂ U(n), T [m2 ] ⊂ SO(m), Tn ⊂ Sp(n) in each of the groups. For a topological space X the problem of finding all integral cohomology endomorphisms H∗(X;Z) → H∗(X;Z) is a step toward classifying all homotopy classes of self maps X → X. In [3] M. Hoffman solved this problem for the flag manifold F (n) = U(n)/Tn. In this note we settle the problem for the spaces D(m) = SO(m)/T [ m 2 ] and S(n) = Sp(n)/Tn. It is worth to point out that S. Papadima determined all cohomology automorphisms of G/T with coefficients in R or Q, where G is a compact connected Lie Group and T its maximal torus [5].

Cite this paper

@inproceedings{Haibao1999TheCO, title={The Classification of Cohomology Endomorphisms of Certain Flag Manifolds}, author={Duan Haibao and Zhao Xuan}, year={1999} }