Deck transformations of the universal cover of $S^1$

A deck transformation of a covering \(p:E\to B\) is a homeomorphism \[ \phi:E\to E \] satisfying \[ p\circ\phi=p. \] Thus it permutes each fiber while preserving the covering projection.

For the universal cover \[ p:\mathbb R\to S^1, \qquad p(t)=e^{2\pi i t}, \] every integer translation \[ T_n(t)=t+n,\qquad n\in\mathbb Z, \] is a deck transformation because \(e^{2\pi i(t+n)}=e^{2\pi it}\). Conversely, if \(T\) is a deck transformation, then \(T(0)\in p^{-1}(1)=\mathbb Z\). Uniqueness of lifts of \(p\) implies \(T=T_{T(0)}\). Hence \[ \operatorname{Deck}(\mathbb R/S^1)\cong\mathbb Z. \]