Generalized Freudenthal Suspension Theorem and De-suspending and De-looping Cellular Inequalities

The following theorem is surprising because it says that if A is a suspension, then for any map A→ X, the homotopy fiber of the cofiber Fib(X → X/A) and A have unstably quite a lot in common (see corollary 8.9): Theorem. Let A→ X → X/A be a cofibration sequence of pointed and connected spaces and Y be a pointed space. If A is weakly equivalent to a suspension of a connected space, then the following statements about Y are equivalent: • map∗(A, Y ) is weakly contractible. • map∗(ΩΣA, Y ) is weakly contractible. • map∗ ( Fib(X → X/A), Y ) is weakly contractible.