# 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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## 3 thoughts on “Stable equivalence”

1. For the first problem, consider the two possible inclusions of a point in the 0-sphere.

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2. Spoiler a-coming: consider the map from the circle to the figure 8 space representing the commutator of the two generators of the fundamental group of the figure 8 space.

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3. 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.

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