Given a topological space we can construct its suspension . This construction can be lifted to a functor from topological spaces to topological spaces that acts on arrows by sending a continuous map to its suspension which is just 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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For the first problem, consider the two possible inclusions of a point in the 0-sphere.

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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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Another example. Let be any acyclic (but not simply-connected) simplicial complex. Then is not null-homotopic. But is simply-connected (since is connected) and acyclic, so is contractible and is null-homotopic.

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