Smooth Covering Maps
Among the local diffeomorphisms, one class plays an outsized role in smooth manifold theory: the
smooth covering maps. The underlying topological notion is already familiar. A
covering map
is a surjective continuous map \(\pi : E \to M\) between connected, locally path-connected spaces
in which each point of \(M\) has an evenly covered neighborhood \(U\) — one whose preimage
\(\pi^{-1}(U)\) breaks into components, the sheets, each mapped homeomorphically onto
\(U\). The exponential map \(\mathbb{R} \to S^1\) is the model: the line wraps around the circle,
and over any small arc the preimage is a stack of intervals. In the smooth category we strengthen
"homeomorphically" to "diffeomorphically," and the result inherits all the structure of a
submersion.
Definition: Smooth Covering Map
Let \(E\) and \(M\) be connected smooth manifolds, with or without boundary. A map
\(\pi : E \to M\) is a smooth covering map if it is smooth and surjective, and
each point of \(M\) has a neighborhood \(U\) such that each component of \(\pi^{-1}(U)\) is
mapped diffeomorphically onto \(U\) by \(\pi\). Such a \(U\) is called
evenly covered. The manifold \(M\) is the base of the
covering, \(E\) is a covering manifold of \(M\), and if \(E\) is
simply connected
it is a universal covering manifold of \(M\).
When the distinction matters, we call an ordinary (not necessarily smooth) covering map a
topological covering map. A smooth covering map is in particular a topological
covering map, but the converse fails in the now-familiar way: a smooth covering map is more than a
topological covering map that happens to be smooth. The definition demands that the restriction of
\(\pi\) to each sheet be a diffeomorphism — not merely a smooth homeomorphism, whose
inverse might fail to be smooth. This added requirement is exactly the smooth analogue of the
upgrade from continuous to diffeomorphic that distinguished smooth embeddings and submersions from
their topological shadows.
Because a smooth covering map is, in particular, a surjective smooth submersion — a fact contained
in the next proposition — every result on submersions applies to it. In particular, the
characteristic property and the descent of fiber-constant maps furnish a powerful tool for defining
smooth maps out of the base of a covering.
Proposition: Properties of Smooth Coverings
Let \(\pi : E \to M\) be a smooth covering map.
(a) \(\pi\) is a local diffeomorphism, a smooth submersion, an open map, and a quotient map.
(b) If \(\pi\) is injective, it is a diffeomorphism.
(c) A topological covering map is a smooth covering map if and only if it is a local
diffeomorphism.
Proof Sketch.
For (a), restricted to each sheet \(\pi\) is a diffeomorphism onto an evenly covered open set,
so near every point \(\pi\) is a
local diffeomorphism;
a local diffeomorphism is both an immersion and a submersion, in particular a
submersion,
and a submersion is an
open map and, being surjective, a quotient map.
Part (b): an injective local diffeomorphism is a smooth bijection that is a local
diffeomorphism, hence a
diffeomorphism.
Part (c): a topological covering map is smooth precisely when each sheet maps
not just homeomorphically but diffeomorphically, which is exactly the condition that \(\pi\) be
a local diffeomorphism. \(\blacksquare\)
The model examples are smooth. The exponential map
\(\varepsilon : \mathbb{R} \to S^1\), \(\varepsilon(t) = (\cos 2\pi t, \sin 2\pi t)\), encountered
earlier as a local diffeomorphism, is a smooth covering map; so is its product
\(\varepsilon^n : \mathbb{R}^n \to T^n\) onto the torus. For each \(n \geq 1\), the quotient
\(q : S^n \to \mathbb{RP}^n\) identifying antipodal points is a two-sheeted smooth covering map,
once \(\mathbb{RP}^n\) carries its
standard smooth structure.
These are the smooth refinements of the covering maps that compute fundamental groups, and the fact
that \(\varepsilon\) is now seen to be a smooth covering — not merely a topological one — closes a
loop opened when covering maps were first introduced.
Sections lift uniquely
For a general submersion, the local section theorem produces local sections but says nothing about
their uniqueness. For a smooth covering, the rigidity of the sheets pins each section down
completely.
Proposition: Local Sections of Smooth Coverings
Let \(\pi : E \to M\) be a smooth covering map. Given any evenly covered open set
\(U \subseteq M\), any point \(q \in U\), and any \(p\) in the fiber \(\pi^{-1}(q)\), there
exists a unique smooth local section \(\sigma : U \to E\) with \(\sigma(q) = p\).
Proof Sketch.
Let \(\widetilde U_0\) be the sheet — the component of \(\pi^{-1}(U)\) — containing \(p\). Since
\(\pi|_{\widetilde U_0} : \widetilde U_0 \to U\) is a diffeomorphism, its inverse
\(\sigma = (\pi|_{\widetilde U_0})^{-1}\) is a smooth local section with \(\sigma(q) = p\). For
uniqueness, any other smooth local section \(\sigma'\) with \(\sigma'(q) = p\) has connected
image (as \(U\) is connected), so its image lies in the single sheet \(\widetilde U_0\); there
\(\sigma'\) is a right inverse of the bijection \(\pi|_{\widetilde U_0}\), hence equals its
inverse \(\sigma\). \(\blacksquare\)
Covering Spaces of Smooth Manifolds
A topological covering map is a topological condition; it knows nothing of smooth structures. Yet
when the base is a smooth manifold, the covering space is forced to be one too, and in exactly one
way compatible with the projection. This is the result that lets us pass freely between the
topological theory of covering spaces and the smooth category — and, with it, manufacture the
universal covering manifold.
Proposition: Covering Spaces of Smooth Manifolds
Suppose \(M\) is a connected smooth \(n\)-manifold and \(\pi : E \to M\) is a topological
covering map. Then \(E\) is a topological \(n\)-manifold, and it has a unique smooth structure
making \(\pi\) a smooth covering map.
Proof Sketch.
That \(E\) is a topological \(n\)-manifold requires checking three things — locally Euclidean,
Hausdorff, second-countable — and then a smooth atlas must be supplied.
Locally Euclidean. Since \(\pi\) is a local homeomorphism (each sheet maps
homeomorphically), every point of \(E\) has a neighborhood homeomorphic to an open subset of
\(M\), hence to an open subset of \(\mathbb{R}^n\).
Hausdorff. Given distinct \(p_1, p_2 \in E\): if \(\pi(p_1) = \pi(p_2)\), they
lie in distinct sheets of one evenly covered neighborhood, which are disjoint and open; if
\(\pi(p_1) \neq \pi(p_2)\), separate them in \(M\) by Hausdorffness and pull the separating
sets back through \(\pi\).
Second-countable. This is the substantive step, and it turns on two facts
established independently. First, each fiber \(\pi^{-1}(q)\) is countable. To see this, fix
\(p_0 \in \pi^{-1}(q)\) and define a map \(\beta : \pi_1(M, q) \to \pi^{-1}(q)\) by sending the
class of a loop \(f\) to the endpoint \(\widetilde f(1)\) of the
lifted path
starting at \(p_0\) — well defined because monodromy makes the endpoint depend only on the path
class. This \(\beta\) is surjective: given any \(p \in \pi^{-1}(q)\), a path in the
connected manifold \(E\) from \(p_0\) to \(p\) projects to a loop at \(q\) whose lift ends at
\(p\), so \(p\) is hit. Since the
fundamental group of a manifold is countable,
the surjection forces \(\pi^{-1}(q)\) to be countable. Second, the evenly covered open sets
cover \(M\), so by second-countability they admit a countable subcover \(\{U_i\}\); over each
\(U_i\), the preimage has one component per point of a fiber, and the fibers were just shown to be
countable, so \(\pi^{-1}(U_i)\) has countably many components.
The components of all the \(\pi^{-1}(U_i)\) thus form a countable open cover of \(E\) by sets
each homeomorphic to a second-countable space, and a space with a countable cover by
second-countable open sets is second-countable.
Smooth structure. For each evenly covered \(U\) that is also the domain of a
smooth chart \(\varphi : U \to \mathbb{R}^n\), and each sheet \(\widetilde U\) over it, set
\(\widetilde\varphi = \varphi \circ \pi|_{\widetilde U}\). Where two such charts overlap, the
transition map is
\(\widetilde\psi \circ \widetilde\varphi^{-1} = (\psi \circ \pi) \circ (\varphi \circ \pi)^{-1}
= \psi \circ \varphi^{-1}\), a transition map of \(M\) and hence smooth. This atlas makes
\(\pi\) a smooth covering map, since in these charts \(\pi\) is the identity; uniqueness of the
structure follows because any compatible structure must use these same charts. \(\blacksquare\)
What the countability argument actually needs
The second-countability step is where the topological theory of covering spaces feeds back into
the smooth one, and it deserves a closer look because two distinct facts must both hold. The
surjectivity of \(\beta\) is a statement about path lifting: every point of the fiber
is reachable by lifting some loop, which requires the connectedness of \(E\) and the existence
of lifts. The countability of \(\pi_1(M, q)\) is a statement about the base: it holds
because a manifold is second-countable, quite independently of the covering. Neither alone
bounds the fiber; only the surjection from a countable group onto the fiber does. This is the
precise point at which the path-lifting and monodromy properties — established earlier in the
topological theory of the fundamental group — are consumed to produce a smooth-manifold
conclusion.
The same construction works when the base has boundary, with the covering space acquiring a
boundary that sits exactly over the boundary below.
Proposition: Covering Spaces of Manifolds with Boundary
Suppose \(M\) is a connected smooth \(n\)-manifold with boundary and \(\pi : E \to M\) is a
topological covering map. Then \(E\) is a topological \(n\)-manifold with boundary satisfying
\(\partial E = \pi^{-1}(\partial M)\), and it has a unique smooth structure making \(\pi\) a
smooth covering map.
The universal covering manifold
Every connected smooth manifold sits beneath a simply connected one. This is the culmination of the
covering-space arc: the topological theory guarantees a simply connected covering space, and the
proposition above promotes it to the smooth category.
Corollary: Existence of a Universal Covering Manifold
If \(M\) is a connected smooth manifold, there exists a simply connected smooth manifold
\(\widetilde M\), called the universal covering manifold of \(M\), together
with a smooth covering map \(\pi : \widetilde M \to M\). It is unique up to diffeomorphism over
\(M\): if \(\widetilde M'\) is any other simply connected smooth manifold admitting a smooth
covering map \(\pi' : \widetilde M' \to M\), then there is a diffeomorphism
\(\Phi : \widetilde M \to \widetilde M'\) with \(\pi' \circ \Phi = \pi\).
Proof Sketch.
The topological theory of covering spaces provides a simply connected topological covering
space \(\widetilde M \to M\) — its construction, which requires \(M\) to be locally simply
connected (automatic for manifolds, as they are locally Euclidean), is the substantial
topological input and we take it here on the strength of that theory. The proposition above then
endows \(\widetilde M\) with the unique smooth structure making the covering smooth. Uniqueness
up to diffeomorphism descends from the corresponding topological uniqueness of universal covers,
upgraded to a diffeomorphism by the smooth structure. \(\blacksquare\)
This closes a thread left open when covering maps were first introduced, where the universal cover
was named but its construction deferred. It also supplies the smooth setting for the covering
\(\mathrm{SU}(2) \to \mathrm{SO}(3)\) and its kin, where a simply connected group covers a
non-simply-connected one — the geometric origin of the distinction between a rotation and the
spinor that double-covers it.
Criteria for Smooth Coverings
Recognizing a smooth covering map in the wild is not always easy. Being a surjective local
diffeomorphism is necessary but not sufficient — the sheets must stack up evenly, with no point of
the base having more preimage components than its neighbors. There are few simple criteria that
guarantee this, but one is genuinely useful: properness. A proper local diffeomorphism cannot have
sheets appearing or disappearing, because properness controls the preimages of compact sets.
Proposition: Proper Local Diffeomorphisms Are Smooth Coverings
Suppose \(E\) and \(M\) are nonempty connected smooth manifolds, with or without boundary. If
\(\pi : E \to M\) is a
proper
local diffeomorphism, then \(\pi\) is a smooth covering map.
Proof Sketch.
Surjectivity. A local diffeomorphism is an open map, so \(\pi(E)\) is open;
and because \(\pi\) is proper with \(M\) locally compact and Hausdorff, \(\pi\) is also a
closed map,
so \(\pi(E)\) is closed. Being nonempty, open, and closed in the connected manifold \(M\), the
image \(\pi(E)\) is all of \(M\); thus \(\pi\) is surjective.
Fibers are finite. Fix \(q \in M\). Each point of \(\pi^{-1}(q)\) has a
neighborhood on which \(\pi\) is injective, so \(\pi^{-1}(q)\) is discrete; and it is compact,
because \(\pi\) is proper and \(\{q\}\) is compact. A discrete compact set is finite, say
\(\pi^{-1}(q) = \{p_1, \dots, p_k\}\).
Building an evenly covered neighborhood. Choose disjoint neighborhoods \(V_i\)
of the \(p_i\), each mapped diffeomorphically by \(\pi\) onto an open set \(U_i \subseteq M\),
and set \(U' = U_1 \cap \cdots \cap U_k\), a neighborhood of \(q\). The leftover set
\(K = E \setminus (V_1 \cup \cdots \cup V_k)\) is closed, so its image \(\pi(K)\) is closed
(again by the closed-map property), and it misses \(q\). Replacing \(U'\) by
\(U = U' \setminus \pi(K)\), and then by its connected component containing \(q\), yields a
connected neighborhood \(U\) of \(q\) with \(\pi^{-1}(U) \subseteq V_1 \cup \cdots \cup V_k\).
Setting \(\widetilde V_i = \pi^{-1}(U) \cap V_i\), the sets \(\widetilde V_i\) are disjoint and
each mapped diffeomorphically onto \(U\) by \(\pi\). It is precisely here that taking \(U\)
connected pays off: each \(\widetilde V_i\), being diffeomorphic to the connected set \(U\), is
itself connected, so the disjoint union \(\pi^{-1}(U) = \widetilde V_1 \sqcup \cdots \sqcup
\widetilde V_k\) exhibits the \(\widetilde V_i\) as exactly the connected components of
\(\pi^{-1}(U)\) — which is what the definition of an evenly covered neighborhood demands of its
sheets. Hence \(U\) is evenly covered, and \(\pi\) is a smooth covering map. \(\blacksquare\)
The condition is sufficient but not necessary. A covering map with finitely many sheets is always
proper: the preimage of a compact set is covered by finitely many compact sheet-pieces and so is
compact. The gap opens only for infinite-sheeted coverings — the exponential map
\(\varepsilon : \mathbb{R} \to S^1\) is the standard witness, with \(\varepsilon^{-1}(\{1\}) =
\mathbb{Z}\) non-compact over the compact point \(\{1\}\), so \(\varepsilon\) is a covering that is
not proper. Properness thus acts as a filter that picks out the finite-sheeted coverings, detecting
coverings without exhausting them. Its value lies in being checkable — one verifies a
local-diffeomorphism condition pointwise and a properness condition on compact sets, neither of
which requires foreknowledge of how the sheets are arranged.
Properness, closed maps, and the recurring role of one lemma
It is worth noting how much of this proof rests on a single topological fact: that a proper
continuous map into a locally compact Hausdorff space is closed. That lemma appeared first in
the criterion for an injective immersion to be an embedding, where properness forced the image
to be closed and so upgraded the immersion to an embedding; it returns here to force the image
of a proper local diffeomorphism to be closed, and to keep the leftover set \(K\) from
contaminating the neighborhood of \(q\). The same closed-map property is the workhorse behind
both the embedding theory of the previous page and the covering theory of this one — a
reminder that the topological scaffolding erected earlier is what makes the smooth conclusions
possible. With this criterion, the arc that began with the bare topological notion of a
covering map reaches its smooth completion: coverings are recognized, their total spaces are
smooth manifolds, and the universal cover exists in the smooth category.