Let \(q: \widetilde X \to X\) be a regular cover of a finite CW space \(X\) with the group \(\mathbb Z^r\) of covering transformations, and let \(A: \widetilde X \to \mathbb R^r\) be the Abel–Jacobi map for \(q\). Since \(\mathbb R^r=(\pi_1(X)/{\text{Im}} q_*)\otimes \mathbb R\), any class \(\xi\in H^1(X;\mathbb R)\) with \(q^*\xi=0\) yields a map \(\xi_{\mathbb R}:\mathbb R^r\to \mathbb R\). A neighborhood of infinity with respect to \(\xi\) is a subset of \(\widetilde X\) that contains a set \(\{x\in \widetilde X: \xi_{\mathbb R}(A(x))>c\}\) for some \(c\in \mathbb R\). The authors characterize homology classes that can be moved into an arbitrary small neighborhood of infinity.