Novikov complex is an extension of Morse theory to (closed) Morse 1-form $\omega$, which is not necessarily exact. Suppose for simplicity, $\omega$ is in the integer cohomology class and the universal covering is infinite cyclic. Then $\omega$ can be lifted to the universal cover $\hat M$, i.e., $\pi^\star \omega=d\hat f$, for some Morse function $\hat f$ on $\hat M$. The Morse 1-form $\omega$ generalizes the Morse function and the zeros of $\omega$ take the place of the critical points of a Morse function. Indices of the zeros (or critical points) of $\omega$ can be defined similarly and we have the stable and unstable manifolds of a critical point. The incidence number between a pair of critical points respectively of index $k$ and $k-1$ of $\omega$ can be defined for a Morse-Smale pair $(\omega, g)$, where $g$ is a Riemann metric on $M$. One can lift the stable and unstable manifolds to $\hat M$ and define the corresponding incidence number in $\hat M$ similarly. One can also define the Novikov complex.

My question is: $\hat M$ is not compact and the unstable manifold (or the descending disc) of a critical point in $\hat M$ in general goes infinitely downwards. Also, for fixed $x$ in $\hat M$, it is possible that the incidence numbers between the critical points $x$ of index $k$ and $y$ of index $k-1$ in $\hat M$ are non-zero for infinitely many $y$. These are different from Morse theory on compact manifolds. Can someone give a (simple) example of an unstable manifold in $\hat M$ going infinitely downwards for some Morse 1-form? Maybe also an example of infinite number of non-zero incidence numbers? I am sorry that I pose the question rather imprecisely.


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