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Jun 6, 2014 at 16:49 comment added André Henriques The inclusion of the boundary is just one special example of a smooth map between manifolds (with boundary). I think that it's better to ask that same question in a greater level of generality: Given an arbitrary map of manifolds (with boundary), construct the induced map in Morse cohomology, and prove that it agrees with the standard pullback map in cohomology.
Jun 6, 2014 at 16:44 history asked user36931 CC BY-SA 3.0