Manifolds which have two projective space bundle structures from the homotopical point of view (Q1174460)

Let \(X\) be a manifold which has the following two bundle structures \(CP^ r\buildrel{i_ 1}\over\longrightarrow X\buildrel {p_ 1}\over \longrightarrow CP^ m\), \(CP^ s\buildrel{ i_ 2}\over\longrightarrow X\buildrel{p_ 2}\over\longrightarrow Y\), where \(r,s,m\geq 1\) and \(Y\) is a manifold. The purpose of this paper is to classify the cohomology ring of \(X\) and describe the cohomology ring of \(Y\) in terms of \(X\). The first result of the paper is when \(Y\) is homotopy equivalent to a complex projective space, then \(Y\) is homotopy equivalent to \(CP^ r\) or \(CP^ m\). If \(Y\) is not homotopy equivalent to a complex projective space, then the fiber of \(X\to Y\) is \(CP^ 1\). The second result of the paper is when \(Y\) is not homotopy equivalent to a complex projective space, then \(\beta=0,\pm 1,\pm 2\). Here \(\beta\) stands for an element of \(\pi_ 2(CP^ m)\simeq Z\) defined by the map \(p_ 1\circ i_ 2: CP^ 1\to CP^ m\), which gives an element of \(\pi_ 2(CP^ m)\simeq H_ 2(CP^ m)\). The proof of the second result requires the cohomology ring of \(Y\) in terms of that of \(X\) which is described in \S 5.