Fiber-homotopy self-equivalences and a classification of fibrations in rational homotopy

Abstract

For a fibration ξ : X i → E p → B we consider the image of the rationalized homotopy group homomorphism π * (res ξ) Q : π * (aut 1 p) Q → π * (aut 1 X) Q obtained from the fibre-restricting map res ξ : aut 1 p → aut 1 X. Then we consider a finite type classification of fibrations ξ with fibre X and base B. In particular, we measure the size of it by a rational homotopical invariant "depth B X" when X are certain homogeneous spaces and B are spheres.