Two results relating nilpotent spaces and cofibrations

Abstract

We first prove a Blakers-Massey Theorem for nilpotent spaces: If (X, A) is an n-connected, n ⩾ 1 n \geqslant 1 , pair of nilpotent spaces, then under suitable conditions the map π ∗ ( X , A ) → π ∗ X / A {\pi _ \ast }(X,A) \to {\pi _ \ast }X/A is an isomorphism in dimension n + 1 n + 1 and an epimorphism in dimension n + 2 n + 2 . Next, we dualize the well-known fact that if the total space of a fibration is nilpotent, so is the fiber. Our dual theorem can be used to construct new examples of finite nilpotent CW complexes.