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!

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Stable equivalence

3 thoughts on “Stable equivalence

  1. Iván Sadofschi says:

    Another example. Let X be any acyclic (but not simply-connected) simplicial complex. Then 1_X:X\to X is not null-homotopic. But \Sigma X is simply-connected (since X is connected) and acyclic, so \Sigma X is contractible and \Sigma 1_X : \Sigma X \to \Sigma X is null-homotopic.

    Liked by 1 person

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