Stable equivalence

Given a topological space X we can construct its suspension \Sigma X. This construction can be lifted to a functor from topological spaces to topological spaces that acts on arrows by sending a continuous map f : X \to Y to its suspension \Sigma f : \Sigma X \to \Sigma Y which is just f in each “level” of the suspensions.

Problem 1: Find two continuous non homotopic maps such that their suspensions are homotopy equivalent.

Problem 2: Find a non nullhomotopic map such that its suspension is nullhomotopic, but now we require the domain and the codomain to be connected!

Stable equivalence