Algebraic topology-homotopy and homology
- R. M. Switzer
- Springer-Verlag Berlin- Heidelberg, 1975…
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  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 .