In Chapter 4.2 of Schwarz's book on Morse homology there is a brief discussion of Morse theory on the total space of a smooth vector bundle $E \to M$. In particular, one can take the Morse function $f_E : E \to \mathbb{R}$ given by
\begin{equation*}
f_E(v_m)=f(m) + q(v_m) \ ,
\end{equation*}
where $f: M \to \mathbb{R}$ is a Morse function on the (closed) base and $q$ is the quadratic form associated to a Riemannian metric on $E$. Then one has $f_E|_M = f$ and there is a bijection
\begin{equation*}
\mbox{Crit}_*(f_E) \cong \mbox{Crit}_*(f) \ .
\end{equation*}
Given that the singular homology of $E$ is just that of $M$, I would assume there is some way of relating the moduli space of gradient flow lines of $f_E$ on $E$ with those of $f$ on $M$. However, this is not explained in Schwarz's book (there is a different objective) and I have not seen this done anywhere else. In particular, is it possible to get an identification at the chain level or only at the level of Morse homology?
1 Answer
The function $q$ strictly decreases along the solutions of the gradient flow outside of the zero section. Hence any orbit that starts outside the zero section will not converge to a critical point in backwards time, and does not show up in some moduli space of orbits connecting critical points. The orbits on the zero section correspond bijectively to orbits on $M$ if you choose a metric which respects a horizontal-vertical splitting of $TE$ in the obvious way.
If one chooses orientations of all the unstable manifolds in $M$, this naturally also gives you orientations for all the unstable manifolds of the critical points in $E$, hence even the orientations of the moduli spaces on $M$ and $E$ agree.
Much more interesting is to not take the function you give, but the function
$$ f^2_E(v_m)=f(m)-q(v_m) $$
The critical points on $E$ correspond to the ones on $M$ shifted by the dimension of $E$, as there are this many more unstable directions. The moduli spaces on $M$ and $E$ again correspond bijectively to each other, but the orientations might not agree. However, if $E$ is an oriented vector bundle, we can orient the unstable manifolds of the critical points on $E$ taking into account this orientation on the fibers. Doing this correctly will show that the moduli spaces are isomorphic as oriented manifolds. This will give an isomorphism
$$ HM_*(M)\cong HM_{*+\dim E}(E,f_E^2) $$
which is the Morse theoretic Thom isomorphism. This can be found for example in the appendix of this paper http://arxiv.org/abs/0810.1995 by Abbondandolo and Schwarz
To see that the orientation assumptions are necessary, you can compute this explicitly for the Mobius strip.
