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?



Dirac operator on manifold with periodic end
Not necessarily. The condition from Taubes's paper is stronger than just requiring the vanishing of the kernel of the Dirac operator $D\_W$. Choosing $f:W \to S^1$ that is Poincaré dual to a multiple of [Y], Taubes requires that for all $r \in \mathbb{R}$, the twisted Dirac operators $D\_{W,r}= D\_W - ir f^\*d\theta$ also have vanishing kernel. 


The paper you are referring to [Taubes, [Gauge theory on asymptotically periodic 4-manifolds](https://doi.org/10.4310/jdg/1214440981). JDG 25 (1987), no. 3, 363–430] gives a general criterion for end-periodic operators. The formulation for Dirac operators is discussed in section 2 of [Mrowka-Ruberman-Saveliev, [An index theorem for end-periodic operators](https://arxiv.org/abs/1105.0260). Compos. Math. 152 (2016)].


