Given a compact Lie group \(G\) and two closed subgroups \(K\) and \(H\) of \(G\) such that \(K \times H\) operates freely on \(G\) by \(g \cdot (k,h)=k^{-1} gh\), we can form the quotient \(K \backslash G/H\), which is a compact manifold called a double coset manifold. The objective of the present paper is to show that the topology of double coset manifolds is in many respects as easy to handle as that of homogeneous spaces.
In particular, there is an explicit description of the tangent bundle. Since, by the known collapsing results for the Eilenberg-Moore spectral sequence, the cohomology of \(K \backslash G/H\) is readily computable, formulas for the characteristic classes can be deduced, which generalize those of Borel-Hirzebruch for homogeneous spaces. For example, rational characteristic classes vanish in dimensions larger than \(\dim (K \backslash G/H)-(rk G-rk K-rk H)\). Finally, there is a formula for the Euler characteristics of \(K \backslash G/H\).