Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to itself obtained from cutting $\tilde W$ open along $Y$. ($<[Y]>\cong H^1(\tilde W;\mathbb Z)$)

Let $X$ be a oriented manifold with periodic-end modelling in $W$ in the sense of Taubes.

Assume $\tilde W$ admits a metric $g_0$ such that the associated Dirac operator has trivial kernel.

Let $g_1$ be a metric on $X$ such that $g_1$ restricts on the periodic end part equals to the pull-back of the $g_0$.

Q Let $D_1$ be the Dirac operator on $W$ associated with $g_1$, is $D_1$ Fredholm?

PS: I think it is true, and I remember I saw/heard such a result on somewhere? Could anyone give a reference?