Virtual homology of surgered torus bundles

Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$. We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$ (in some cases all but finitely many) are virtually $\mathbb{Z}$-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are virtually $\mathbb{Z}$-representable.

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