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Jun 13, 2014 at 12:24 vote accept Bruno Stonek
Apr 6, 2014 at 15:54 comment added Ben Wieland An example of where things go bad in your last paragraph is where $Y$ is a point. Then the composite is already constant, but the trivial nullhomotopy is the wrong one.
Apr 6, 2014 at 0:22 history answered Tom Goodwillie CC BY-SA 3.0