Dirac operator on manifold with periodic end

Question

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?

Answer 1

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. 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. Compos. Math. 152 (2016)].