Definition Let p: X˜ → X be a covering map over a topological space X. A deck transformation of the covering space X˜ is a homeomorphism g: X˜ → X˜ of X˜ with the property that p ◦ g = p.

Let p: X˜ → X be a covering map over some topological space X. The deck transformations of the covering space X^~  constitute a group of homeomorphisms of that covering space (where the group operation is the usual operation of composition of homeomorphisms). We shall denote this group by Deck(X˜|X).

Lemma 1.12 Let p: X˜ → X be a covering map, where the covering space X˜ is connected. Let g ∈ Deck(X˜|X) be a deck transformation that is not equal to the identity map. Then g(w) ≠ w for all w ∈ X˜.

Proof The result follows immediately on applying Proposition (Let p: X˜ → X be a covering map, let Z be a connected topological space, and let g:Z → X˜ and h:Z → X˜ be continuous maps. Suppose that p ◦ g = p ◦ h and that g(z) = h(z) for some z ∈ Z. Then g = h.).

Proposition 1.13 Let p: X˜ → X be a covering map, where the covering space X˜ is connected. Then the group Deck(X˜|X) of deck transformations acts freely and properly discontinuously on the covering space X˜.

Proof Let w be a point of the covering space X˜. Then there exists an evenly-covered open set U in X such that p(w) ∈ U. Then the preimage p⁻¹ (U) of U in X˜ is a disjoint union of open subsets, where each of these open subsets is mapped homeomorphically onto U by the covering map.