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Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map

$$ H^*(M) \to H^*(\partial M) $$

I'm wondering if there is a reference that:

1) constructs this map in Morse cohomology and

2) proves that it agrees with the standard pullback map in cohomology.

I would like to consider Morse functions $f$ such that along a collar neighborhood $\partial M \times [0,1)_r$, $f$ is pulled back from the second factor and $f'(r)< 0$.

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    $\begingroup$ 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. $\endgroup$
    – André Henriques
    Commented Jun 6, 2014 at 16:49

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