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August 27

When Are Induced Homomorphisms of Homotopy Groups Nonzero?

Continuous maps between spaces induce Homomorphisms between the homotopy groups of those spaces. Is their some general condition that, if true, will imply that the induced map is non-trivial? I'm not a topologist, but I've always been curious if there was any way of guaranteeing this - and, if not, what the value of this fact would be (aside from the homotopy groups being functorial - or is that the value?) Thanks for any help:-)24.3.61.185 (talk) 13:21, 27 August 2018 (UTC)[reply]

There are certainly specific cases where you know an induced map isn't trivial. If a subspace A of a space X is a retract, then the map of homotopy groups induced by inclusion is injective. So if the fundamental group (or higher groups, or homology groups too) aren't trivial, then the map won't be either. This fact can be used to prove the no-retraction theorem, which is a key step in a proof of the Brouwer fixed point theorem. –Deacon Vorbis (carbon • videos) 13:33, 27 August 2018 (UTC)[reply]