Self fibre homotopy equivalences of fibred products (Q1971125)

A product decomposition is given for the group \({\mathcal E}_B(X\times _BY)\) of based fibre homotopy self-equivalence classes of the fibred product \(X\times _BY\) of two CW fibrations \(X\to B\) and \(Y\to B\). A homotopy self-equivalence \(f:F\times G\to F\times G\) is reducible if the composites \(F\to F\times G\to F\times G\to F\) and \(G\to F\times G\to F\times G\to G\) are homotopy self-equivalences. Given that all the homotopy self-equivalences of the fibre \(F\times G\) of \(X\times _BY\to B\) are reducible, then \({\mathcal E}_B(X\times _BY)\cong {\mathcal E}_X(X\times _BY) . {\mathcal E}_Y(X\times _BY)\) where the two groups have trivial intersection. The result is proved in a more general categorical setting. Criteria for reducibility are given, as well as a number of special cases.