Homotopy and Path Homotopy
The invariants studied so far answer coarse questions about a space, among them dimension,
compactness, connectedness and the number of components. They do not distinguish a disk from an
annulus. Both are connected and two-dimensional, yet the annulus has a hole that the disk lacks.
The fundamental group is the
first invariant fine enough to detect such holes. It records, in algebraic form, the loops one can draw in a space
and which of them can be continuously deformed into one another. The whole construction rests on
a single geometric idea, continuous deformation, made precise as
homotopy.
Throughout this page \(I\) denotes the closed unit interval \([0,1]\), and the spaces are arbitrary topological
spaces, with no metric assumed. Where a notion was previously introduced for
metric spaces,
we restate it in the topological generality the theory requires. The metric version is the special
case in which the space happens to carry a metric.
Homotopy of Maps
Two continuous maps with the same domain and codomain are homotopic when one can be continuously deformed into the
other through a one-parameter family of continuous maps. The deformation parameter ranges over \(I\), and the
requirement that the family vary continuously is captured by asking the whole family to assemble into a single
continuous map on \(X \times I\).
Definition: Homotopy and Homotopic Maps
Let \(X\) and \(Y\) be topological spaces and let \(F_0, F_1 : X \to Y\) be continuous maps. A
homotopy from \(F_0\) to \(F_1\) is a continuous map \(H : X \times I \to Y\) such that
\[
H(x, 0) = F_0(x) \quad\text{and}\quad H(x, 1) = F_1(x) \quad \text{for all } x \in X.
\]
If such a homotopy exists, \(F_0\) and \(F_1\) are said to be homotopic, written
\(F_0 \simeq F_1\). For a fixed subset \(A \subseteq X\), the maps are homotopic relative to \(A\)
if in addition \(H(x, t) = F_0(x) = F_1(x)\) for every \(x \in A\) and every \(t \in I\). The
deformation then holds the points of \(A\) fixed throughout.
Writing \(H_t(x) = H(x, t)\), a homotopy is exactly a family \(\{H_t\}_{t \in I}\) of maps that
interpolates continuously from \(H_0 = F_0\) to \(H_1 = F_1\). Continuity of the single map \(H\)
on the product \(X \times I\) is the precise sense in which the family varies continuously in the
parameter \(t\). Both relations, "homotopic" and "homotopic relative to \(A\)", are equivalence
relations on the set of all continuous maps \(X \to Y\). Reflexivity is witnessed by the constant
homotopy \(H(x, t) = F(x)\), symmetry by reversing the parameter, \(H(x, t) \mapsto H(x, 1 - t)\),
and transitivity by concatenating two homotopies in the parameter, running the first on \([0,
\tfrac12]\) and the second on \([\tfrac12, 1]\). The join is continuous because the two pieces
agree at \(t = \tfrac12\) and the interval is covered by two closed sets on which the map is
separately continuous.
Paths and Path Homotopy
The most important application of homotopy is to paths. Recall that a
path in \(X\)
is a continuous map \(f : I \to X\). Its endpoints are the points \(f(0)\) and
\(f(1)\), and \(f\) is a path from \(f(0)\) to \(f(1)\). A homotopy between two
paths that holds the endpoints fixed is the deformation relevant to the fundamental group. It
slides one path to the other without ever detaching either end. This is precisely homotopy relative to the two-point set
\(\{0, 1\} \subseteq I\).
Definition: Path Homotopy and Path Class
Two paths \(f_0, f_1 : I \to X\) with the same endpoints, \(f_0(0) = f_1(0) = p\) and \(f_0(1) = f_1(1) = q\),
are path-homotopic, written \(f_0 \sim f_1\), if they are homotopic relative
to \(\{0, 1\}\), that is, if there is a continuous map \(H : I \times I \to X\) with
\[
\begin{align*}
H(s, 0) &= f_0(s), & H(s, 1) &= f_1(s), && s \in I; \\\\
H(0, t) &= p, & H(1, t) &= q, && t \in I.
\end{align*}
\]
For fixed endpoints \(p, q\), path homotopy is an equivalence relation on the set of all paths from \(p\) to
\(q\). The equivalence class of a path \(f\) is its path class, denoted \([f]\).
The two pairs of conditions play distinct roles. The first row, \(H(s,0) = f_0(s)\) and \(H(s,1) = f_1(s)\), says
that \(H\) is a homotopy of paths interpolating from \(f_0\) to \(f_1\) as the parameter \(t\) runs from \(0\) to
\(1\). The second row, \(H(0,t) = p\) and \(H(1,t) = q\), is the "relative to \(\{0,1\}\)" clause. At
every stage of the deformation the two endpoints stay pinned at \(p\) and \(q\). Without this
constraint any two paths in a path-connected space would be homotopic and the notion would carry
no information. With it, path homotopy becomes sensitive to the holes a loop may enclose, which is
exactly the sensitivity the fundamental group exploits. That
the relation is reflexive, symmetric, and transitive follows from the same constant, reversed, and concatenated
homotopies as for maps, each of which respects the fixed endpoints.
The Fundamental Group
Path classes can be multiplied. When one path ends where another begins, the two can be traversed in
succession, and this operation descends to path classes in a way that makes the loops at a fixed point into a
group. That group is the fundamental group.
The Product of Paths
Given two paths in \(X\) such that the first ends where the second starts, their product is the path that runs
along the first at double speed, then along the second at double speed.
Definition: Product of Paths
Let \(f, g : I \to X\) be paths with \(f(1) = g(0)\). Their product is the path
\(f \cdot g : I \to X\) defined by
\[
(f \cdot g)(s) =
\begin{cases}
f(2s), & 0 \le s \le \tfrac12, \\\\
g(2s - 1), & \tfrac12 \le s \le 1.
\end{cases}
\]
The two formulas agree at \(s = \tfrac12\), where both give the common point \(f(1) = g(0)\), so \(f \cdot g\)
is well-defined and, by the
gluing lemma,
continuous. It is continuous on each of the closed halves \([0, \tfrac12]\) and \([\tfrac12, 1]\),
and these agree on their overlap, the single point \(\tfrac12\). The reparametrization by the
factor of two compresses each path into half the parameter interval, so that the product traverses both within
the single interval \(I\).
This product respects path homotopy. Suppose \(f \sim f'\) and \(g \sim g'\), with matching
endpoints throughout, so that the products are defined. Then \(f \cdot g \sim f' \cdot g'\), since
a path homotopy from \(f\) to \(f'\) and one from \(g\) to \(g'\) can be run side by side under
the same reparametrization, giving a path homotopy from \(f \cdot g\) to \(f' \cdot g'\).
Consequently the product descends to path classes. For path classes \([f]\) and \([g]\) with
\(f(1) = g(0)\), the product
\[
[f] \cdot [g] = [f \cdot g]
\]
is independent of the chosen representatives.
Multiplication of actual paths is not associative on the nose. The paths \((f \cdot g) \cdot h\) and
\(f \cdot (g \cdot h)\) traverse the same three paths but allot them different portions of the parameter
interval, so in general they are different paths. They are, however, path-homotopic, related by a homotopy that
reparametrizes the interval. Multiplication of path classes is therefore associative,
\[
([f] \cdot [g]) \cdot [h] = [f] \cdot ([g] \cdot [h]).
\]
When we form products of three or more actual paths, we adopt the convention that they are evaluated from left
to right, \(f \cdot g \cdot h = (f \cdot g) \cdot h\).
Loops and the Group Structure
Fixing a single point as both the start and the end of every path turns the product into a genuine group
operation, because every product is then defined and the result is again based at the same point.
Definition: Loop Based at a Point
Let \(X\) be a topological space and \(q \in X\). A loop in \(X\) based at \(q\) is a path
\(f : I \to X\) with \(f(0) = f(1) = q\).
The set of path classes of loops based at \(q\), under the product of path classes, has the structure of a group.
Definition: The Fundamental Group
Let \(X\) be a topological space and \(q \in X\). The set of path classes of loops in \(X\) based at \(q\),
equipped with the product \([f] \cdot [g] = [f \cdot g]\), is a
group,
called the fundamental group of \(X\) based at \(q\) and denoted \(\pi_1(X, q)\). Its
identity element is the path class of the constant loop \(c_q(s) \equiv q\).
The inverse of \([f]\) is the path class of the reverse loop \(\bar{f}(s) = f(1 - s)\).
The three group axioms hold at the level of path classes. Associativity is the reparametrization fact established
above. The constant loop acts as identity because \(c_q \cdot f\) and \(f \cdot c_q\) are each
path-homotopic to \(f\). Traversing the constant loop occupies half the interval doing nothing,
and it is deformed away by a homotopy that gradually shrinks the constant portion. The reverse
loop inverts because both \(f \cdot \bar{f}\) and \(\bar{f} \cdot f\) are path-homotopic to the
constant loop. One retraces one's steps, and the resulting loop contracts to \(q\) by the homotopy
that traverses less and less of \(f\) before turning back. Each of these is a statement about path
classes, not about paths themselves, and it is exactly the passage to path classes that converts the
near-misses of path multiplication into the exact identities of a group.
Dependence on the Base Point
The fundamental group is defined with reference to a chosen base point, and a priori different base points could
give different groups. For path-connected spaces they do not. The group is the same, up to isomorphism, wherever it is based.
The reason is that a path from one base point to another conjugates loops at the first into loops at the second.
Let \(\alpha\) be a path from \(q\) to \(q'\), and write \(\bar{\alpha}(s) = \alpha(1 - s)\) for
its reverse, which is a path from \(q'\) to \(q\). For a loop \(f\) based at \(q\) the product
\(\bar{\alpha} \cdot f \cdot \alpha\) is defined, since \(\bar{\alpha}\) ends at \(q\), where
\(f\) begins, and \(f\) ends at \(q\), where \(\alpha\) begins. It runs from \(q'\) to \(q\) along
\(\bar{\alpha}\), traverses \(f\), and returns to \(q'\) along \(\alpha\), so it is a loop based
at \(q'\). Because the product respects path homotopy, the assignment \([f] \mapsto [\bar{\alpha} \cdot f \cdot
\alpha]\) is well defined on path classes. It is also a group homomorphism. The same retracing homotopy that makes the reverse loop an
inverse shows \(\alpha \cdot \bar{\alpha} \sim c_q\), so the middle pair cancels in
\([\bar{\alpha} \cdot f \cdot \alpha] \cdot [\bar{\alpha}
\cdot g \cdot \alpha] = [\bar{\alpha} \cdot (f \cdot g) \cdot \alpha]\). Applying the same
construction to \(\bar{\alpha}\), a path from \(q'\) to \(q\), gives \([h] \mapsto [\alpha \cdot h
\cdot \bar{\alpha}]\), and the two assignments are mutually inverse because \(\alpha \cdot \bar{\alpha} \sim c_q\) and
\(\bar{\alpha} \cdot \alpha \sim c_{q'}\). The result is a group
isomorphism \(\pi_1(X, q) \to \pi_1(X, q')\). In a path-connected space such a path \(\alpha\)
exists between any two points, so all the groups \(\pi_1(X, q)\) are isomorphic. The
isomorphism depends on the choice of \(\alpha\) and so is not canonical, but its existence justifies the common
abuse of writing \(\pi_1(X)\) without a base point when \(X\) is path-connected and only the isomorphism class is at issue.
Simply Connected Spaces and Induced Homomorphisms
Two themes complete the basic theory. The first is the class of spaces whose fundamental group is as small as possible, the simply
connected spaces, in which every loop contracts. The second is the way a continuous
map between spaces produces a homomorphism between their fundamental groups, making \(\pi_1\) into an invariant
that respects continuous maps and, in particular, cannot tell apart spaces that are homeomorphic.
Simply Connected Spaces
The fundamental group is smallest when it is trivial. A path-connected space whose fundamental group vanishes is
one in which every loop can be shrunk to its base point.
Definition: Simply Connected
A topological space \(X\) is simply connected if it is
path-connected
and, for some (hence
every) point \(q \in X\), the fundamental group \(\pi_1(X, q)\) is the trivial group, consisting of the
identity alone. Equivalently, \(X\) is simply connected if it is path-connected and every loop in \(X\) is
path-homotopic to a constant loop.
The parenthetical "some (hence every)" is licensed by the base-point independence established
above. In a path-connected space the groups at different base points are isomorphic, so triviality
at one point is triviality at all. An equivalent and frequently used reformulation is that \(X\)
is simply connected precisely when it is path-connected and any two paths with the same endpoints
are path-homotopic. The two forms are linked by the observation that two paths \(f, g\) with
common endpoints are path-homotopic if and only if the loop
\(f \cdot \bar{g}\) is null-homotopic, so all loops being trivial is the same as all same-endpoint path pairs being homotopic.
A large and convenient supply of simply connected spaces comes from a purely geometric condition. A subset of
Euclidean space is star-shaped if it contains a point from which every other point is visible
along a straight segment lying in the set.
Proposition: Star-Shaped Sets Are Simply Connected
Let \(U \subseteq \mathbb{R}^n\) be star-shaped, so that there is a point \(c
\in U\) for which the line segment from \(c\) to \(x\) lies in \(U\) for every \(x \in U\). Then \(U\) is simply connected.
Proof:
First, \(U\) is path-connected. Any two points are joined by going from one to \(c\) along a
segment and from \(c\) to the other along a segment, both contained in \(U\) by the
star-shaped hypothesis. Now let \(f\) be a
loop in \(U\) based at \(c\). Define \(H : I \times I \to U\) by
\[
H(s, t) = (1 - t)\, f(s) + t\, c.
\]
For each fixed \(s\), the value \(H(s, t)\) traces the segment from \(f(s)\) to \(c\), which lies in \(U\)
because \(U\) is star-shaped with respect to \(c\). Thus \(H\) takes values in \(U\). It is continuous as a
composition of continuous operations, and
\[
H(s, 0) = f(s), \quad H(s, 1) = c, \quad H(0, t) = H(1, t) = (1-t)\, c + t\, c = c,
\]
so \(H\) is a path homotopy from \(f\) to the constant loop \(c_c\), holding the base point fixed. Every loop
based at \(c\) is therefore null-homotopic, \(\pi_1(U, c)\) is trivial, and \(U\) is simply connected.
In particular every nonempty convex subset of \(\mathbb{R}^n\), being star-shaped with respect to each of its
points, is simply connected. This includes \(\mathbb{R}^n\) itself, all open and closed balls, and all the
coordinate balls and half-balls that model manifolds locally. The straight-line homotopy in the proof is the
precise form of the informal phrase, used when first meeting rotation groups, that a loop "can be continuously
shrunk to a point". In a star-shaped set the shrinking is literally the contraction toward the star center.
The Induced Homomorphism
A continuous map carries loops to loops and respects path homotopy, so it descends to a map between fundamental
groups. This is the functoriality that makes \(\pi_1\) useful as an invariant.
Let \(F : X \to Y\) be continuous and \(q \in X\). If \(f\) is a loop in \(X\) based at \(q\), then \(F \circ f\)
is a loop in \(Y\) based at \(F(q)\). If \(f \sim f'\) by a path homotopy \(H\), then \(F \circ
H\) is a path homotopy from \(F \circ f\) to \(F \circ f'\). The assignment \([f] \mapsto [F
\circ f]\) is therefore a well-defined map
\[
F_* : \pi_1(X, q) \to \pi_1\bigl(Y, F(q)\bigr), \quad F_*[f] = [F \circ f].
\]
Proposition: The Induced Homomorphism
If \(F : X \to Y\) is continuous and \(q \in X\), then
\(F_* : \pi_1(X, q) \to \pi_1(Y, F(q))\) defined by \(F_*[f] = [F \circ f]\) is a
group homomorphism,
called the homomorphism induced by \(F\). The induced homomorphisms satisfy:
-
Functoriality:
for continuous \(F : X \to Y\) and \(G : Y \to Z\), one has
\((G \circ F)_* = G_* \circ F_*\).
-
Identity:
the map induced by the identity \(\operatorname{id}_X : X \to X\) is the identity of \(\pi_1(X, q)\).
-
Homeomorphism invariance:
if \(F\) is a homeomorphism, then \(F_*\) is a
group isomorphism.
Consequently
homeomorphic
spaces have isomorphic fundamental groups.
Proof:
That \(F_*\) is a homomorphism is the compatibility of composition with the path product:
\(F \circ (f \cdot g) = (F \circ f) \cdot (F \circ g)\), because composing with \(F\) does not disturb the
reparametrization that defines the product, so
\(F_*([f] \cdot [g]) = F_*[f] \cdot F_*[g]\). Functoriality is the associativity of composition,
\((G \circ F) \circ f = G \circ (F \circ f)\), read at the level of path classes. The identity statement is
immediate, since \(\operatorname{id}_X \circ f = f\). For the last statement, if \(F\) is a homeomorphism with
inverse \(F^{-1}\), then functoriality and the identity statement give
\(F^{-1}_* \circ F_* = (F^{-1} \circ F)_* = (\operatorname{id}_X)_* = \operatorname{id}\) and likewise
\(F_* \circ F^{-1}_* = \operatorname{id}\), so \(F_*\) is invertible as a homomorphism, hence an isomorphism.
Homotopy Invariance
Homeomorphism invariance is in fact a special case of a stronger and more flexible statement. The
fundamental group does not distinguish spaces that are merely homotopy equivalent,
that is, related by maps that are mutually inverse only up to homotopy, a far coarser relation
than homeomorphism.
Definition: Homotopy Equivalence
A continuous map \(F : X \to Y\) is a homotopy equivalence if there is a continuous map
\(G : Y \to X\) with \(F \circ G \simeq \operatorname{id}_Y\) and \(G \circ F \simeq
\operatorname{id}_X\). Such a \(G\) is a homotopy inverse for \(F\). When a homotopy equivalence between \(X\) and
\(Y\) exists, the spaces are homotopy equivalent.
Theorem: Homotopy Invariance of the Fundamental Group
If \(F : X \to Y\) is a homotopy equivalence, then for each \(p \in X\) the induced homomorphism
\(F_* : \pi_1(X, p) \to \pi_1(Y, F(p))\) is an isomorphism. In particular, homotopy equivalent spaces have
isomorphic fundamental groups.
Proof Sketch:
The key step compares the maps induced by two homotopic maps. Let \(H : X \times I \to Y\) be a
homotopy from \(F_0\) to \(F_1\), fix
\(p \in X\), and let \(\alpha(t) = H(p, t)\), a path from \(F_0(p)\) to \(F_1(p)\). For a loop \(f\) at
\(p\), the map \(K(s, t) = H(f(s), t)\) on the square \(I \times I\) takes the values \(F_0 \circ f\) on
the bottom edge, \(F_1 \circ f\) on the top edge, and \(\alpha\) on both vertical edges, since
\(f(0) = f(1) = p\). Let \(a\) be the path in the square that runs along the bottom edge and then up the
right edge, and \(b\) the path that runs up the left edge and then along the top edge. Both run from
\((0, 0)\) to \((1, 1)\), and since the square is convex, the straight-line homotopy
\((s, u) \mapsto (1 - u)\,a(s) + u\,b(s)\) stays in it and fixes these endpoints, so \(a \sim b\).
Composing this path homotopy with \(K\) gives
\((F_0 \circ f) \cdot \alpha \sim \alpha \cdot (F_1 \circ f)\), hence
\(F_1 \circ f \sim \bar{\alpha} \cdot (F_0 \circ f) \cdot \alpha\). That is,
\((F_1)_* = \Psi_\alpha \circ (F_0)_*\), where \(\Psi_\alpha[h] = [\bar{\alpha} \cdot h \cdot \alpha]\)
is the change-of-base-point isomorphism \(\pi_1(Y, F_0(p)) \to \pi_1(Y, F_1(p))\) constructed earlier.
Now let \(G\) be a homotopy inverse of \(F\). Applied to a homotopy from \(G \circ F\) to
\(\operatorname{id}_X\), the key step shows that
\((G \circ F)_* = G_* \circ F_* : \pi_1(X, p) \to \pi_1(X, G(F(p)))\) is an isomorphism, being the
inverse of a change-of-base-point isomorphism, so \(G_* : \pi_1(Y, F(p)) \to \pi_1(X, G(F(p)))\) is
surjective. Applied to a homotopy from \(F \circ G\) to \(\operatorname{id}_Y\), at the base point
\(F(p)\), it shows that \((F \circ G)_* = F'_* \circ G_* : \pi_1(Y, F(p)) \to \pi_1(Y, F(G(F(p))))\) is
an isomorphism, where \(F'_*\) denotes the map induced by \(F\) at the base point \(G(F(p))\), so the
same \(G_*\) is injective. Hence \(G_*\) is an isomorphism, and \(F_* = G_*^{-1} \circ (G \circ F)_*\),
at the base point \(p\), is one as well.
The force of the theorem is that the fundamental group is blind to deformations far more
drastic than homeomorphisms. A solid ball deformation retracts to its center and is thus homotopy
equivalent to a point, so its fundamental group is trivial. The punctured plane retracts onto a
circle and shares the circle's fundamental group. A Euclidean space with a point removed,
\(\mathbb{R}^n \setminus \{0\}\), is homotopy equivalent to the sphere \(\mathbb{S}^{n-1}\) by the
radial retraction \(x \mapsto x / |x|\), so the two have the same fundamental group.
Fundamental Groups in Geometry
The machinery now pays off on the spaces that matter for geometry. We record the fundamental groups of spheres
and products, and then turn to the rotation groups, where the fundamental group explains a fact already
encountered in the study of Lie groups: the existence of loops of rotations that can be shrunk to a point only
after being traversed twice.
Spheres and Products
The circle is the basic source of a nontrivial fundamental group. A loop on the circle can wind around any
integer number of times, and the winding number is a complete invariant of its path class.
Proposition: Fundamental Groups of Spheres
For the spheres
\(\mathbb{S}^n\):
-
\(\pi_1(\mathbb{S}^1)\) is the infinite
cyclic group,
generated by the path class of the loop \(\omega(s) = (\cos 2\pi s, \sin 2\pi s)\). Thus
\(\pi_1(\mathbb{S}^1) \cong \mathbb{Z}\).
-
for \(n \gt 1\), the sphere \(\mathbb{S}^n\) is simply connected.
Proof Sketch:
(1) The map \(\varepsilon : \mathbb{R} \to \mathbb{S}^1\),
\(\varepsilon(t) = (\cos 2\pi t, \sin 2\pi t)\), is a
covering map, in the sense of the final
subsection of this page, whose lifting results the argument uses. Both spaces are connected and locally
path-connected, the circle being the continuous image \(\varepsilon(\mathbb{R})\) and a manifold. If
\(\varepsilon(a) = z\) and \(z^\perp = (-z_2, z_1)\) denotes \(z\) rotated counterclockwise by
\(\pi / 2\), the complement \(U\) of \(-z\) is evenly covered: \(\varepsilon^{-1}(U)\) is the disjoint
union of the intervals \((a + m - \tfrac{1}{2}, a + m + \tfrac{1}{2})\), \(m \in \mathbb{Z}\), and
\(\varepsilon\) maps each of them bijectively onto \(U\) with continuous inverse
\(w \mapsto a + m + \tfrac{1}{\pi}\arctan\bigl((w \cdot z^\perp) / (1 + w \cdot z)\bigr)\), the
denominator being positive on \(U\). The total space \(\mathbb{R}\) is convex, hence simply connected,
so the subgroup \(H\) in the
theorem on sheets and cosets is
trivial, and the map \(\Phi[f] = \widetilde{f}(1)\), where \(\widetilde{f}\) is the lift of \(f\)
starting at \(0\), is a bijection from \(\pi_1(\mathbb{S}^1, (1, 0))\) onto the fiber
\(\varepsilon^{-1}(1, 0) = \mathbb{Z}\). It is a homomorphism into \((\mathbb{Z}, +)\). Given loops
\(f, g\) with \(N = \widetilde{f}(1)\), the translate \(\widetilde{g} + N\) is again a lift of \(g\),
because \(\varepsilon(t + N) = \varepsilon(t)\), and it starts at \(N\). So
\(\widetilde{f} \cdot (\widetilde{g} + N)\) is a lift of \(f \cdot g\) starting at \(0\), hence the lift
by uniqueness, and it ends at
\(N + \widetilde{g}(1)\). Thus \(\Phi\) is an isomorphism onto \(\mathbb{Z}\), and since the lift of
\(\omega\) is \(s \mapsto s\), it sends \([\omega]\) to the generator \(1\).
(2) Let \(n \gt 1\). The sphere is path-connected, since two points \(x, x'\) that are not antipodal are
joined by the normalized segment \(t \mapsto ((1 - t)x + tx') / |(1 - t)x + tx'|\), and antipodal points
are joined through a third point. Let \(f\) be a loop in \(\mathbb{S}^n\) based at \(q\). The open
hemispheres \(\{x \in \mathbb{S}^n : x \cdot v \gt 0\}\), \(v \in \mathbb{S}^n\), cover
\(\mathbb{S}^n\), so by the
Lebesgue number lemma
applied to their preimages in \(I\) there is a subdivision \(0 = s_0 \lt s_1 \lt \cdots \lt s_k = 1\) of
mesh less than a Lebesgue number, so that \(f\) maps each \([s_{i-1}, s_i]\) into a single hemisphere.
On that subinterval, let \(g\) be the normalized segment from \(f(s_{i-1})\) to \(f(s_i)\), parametrized
linearly. It stays in the same hemisphere, and so does the normalized straight-line homotopy
\(((1 - t)f + tg) / |(1 - t)f + tg|\), whose denominator cannot vanish because its dot product with the
hemisphere's center is positive. The pieces of \(g\) agree at the subdivision points, and the homotopies
fix those points, so by the
gluing lemma they
fit together into a loop \(g\) at \(q\) and a path homotopy from \(f\) to \(g\).
The image of \(g\) lies in the union of at most \(k\) linear subspaces of \(\mathbb{R}^{n+1}\) of
dimension at most \(2\), each spanned by the endpoints of one piece. Because \(n + 1 \ge 3\), each of
them has empty interior, since a subspace containing a ball contains a ball about \(0\) and so spans
\(\mathbb{R}^{n+1}\). A finite union of closed sets with empty interior has empty interior: if an open
set \(W\) lies in \(A \cup B\) with \(A\) closed, then \(W \setminus A\) is an open subset of \(B\),
hence empty, so \(W \subseteq A\) and \(W\) is empty, and induction on the number of sets gives the
general case. Linear subspaces are also closed, so this applies. Hence some point of
\(\mathbb{R}^{n+1} \setminus \{0\}\) lies outside the union, and since the union is invariant under
scaling, its normalization \(y \in \mathbb{S}^n\) is not in the image of \(g\). In particular
\(y \ne q\). Let \(K(s, t) = ((1 - t)g(s) - ty) / |(1 - t)g(s) - ty|\). The denominator vanishes only if
\((1 - t)g(s) = ty\), which forces \(t = \tfrac{1}{2}\) and \(g(s) = y\), so \(K\) is continuous on the
square. It equals \(g\) on the bottom edge, the constant \(-y\) on the top edge, and the same path
\(\beta(t) = K(0, t) = K(1, t)\) from \(q\) to \(-y\) on both vertical edges. The square argument in the
proof of homotopy invariance above, applied to \(K\), gives \(g \cdot \beta \sim \beta \cdot c_{-y}\),
hence
\(f \sim g \sim g \cdot \beta \cdot \bar{\beta} \sim \beta \cdot c_{-y} \cdot \bar{\beta} \sim \beta \cdot \bar{\beta} \sim c_q\),
where a constant path is absorbed by the same shrinking homotopy that makes the constant loop an
identity. Every loop is therefore null-homotopic, and \(\mathbb{S}^n\) is simply connected.
The computation \(\pi_1(\mathbb{S}^1) \cong \mathbb{Z}\) is the foundational calculation of the subject. The
integer attached to a loop is its winding number, and
the isomorphism sends a path class to the net number of times it circles the origin. The simple connectivity of the higher spheres
reflects the fact that a loop on \(\mathbb{S}^n\) for \(n \gt 1\) has room to be slid off any point it might wind
around, leaving enough of the sphere to contract it. The one-dimensional circle is too tight for
this, which is why it alone among the spheres has a nontrivial fundamental group.
Fundamental groups of products are computed factor by factor.
Proposition: Fundamental Group of a Product
Let \(X_1, \ldots, X_k\) be topological spaces and \(q_i \in X_i\). The projections induce an isomorphism
\[
\pi_1\bigl(X_1 \times \cdots \times X_k,\, (q_1, \ldots, q_k)\bigr) \cong
\pi_1(X_1, q_1) \times \cdots \times \pi_1(X_k, q_k).
\]
Proof:
Write \(X = X_1 \times \cdots \times X_k\), \(q = (q_1, \ldots, q_k)\), and \(p_i : X \to X_i\) for the
projections. Define \(\Theta\) by sending a path class \([f]\) to the tuple of path classes of its
projections,
\[
\Theta[f] = \bigl([p_1 \circ f], \ldots, [p_k \circ f]\bigr),
\]
with values in the direct product of the groups \(\pi_1(X_i, q_i)\) under componentwise multiplication.
The \(i\)-th component of \(\Theta\) is the
induced homomorphism
\((p_i)_*\), so \(\Theta\) is well defined, and it is a homomorphism because multiplication in the direct
product is computed componentwise.
A map \(g\) into the
product
is continuous if and only if each of its components is.
The linked statement is for two factors, and its argument applies verbatim to \(k\) factors: the preimage under \(g\) of a basic open set
\(U_1 \times \cdots \times U_k\) is \(\bigcap_i (p_i \circ g)^{-1}(U_i)\), and each \(p_i^{-1}(U_i)\) is
itself a basic open set, so in particular the projections are continuous. For surjectivity, let \(f_i\) be a loop in
\(X_i\) based at \(q_i\) for each \(i\). Then \(f(s) = (f_1(s), \ldots, f_k(s))\) is continuous, it is a
loop based at \(q\), and \(\Theta[f] = ([f_1], \ldots, [f_k])\). For injectivity, suppose \(\Theta[f]\) is
the identity, so that for each \(i\) some path homotopy \(H_i\) runs from \(p_i \circ f\) to the constant
loop at \(q_i\). The map \(H(s, t) = (H_1(s, t), \ldots, H_k(s, t))\) is continuous on \(I \times I\) by
the same property, and it satisfies the conditions of a path homotopy from \(f\) to the constant loop at
\(q\) because it satisfies them in each coordinate. Hence the kernel of \(\Theta\) is trivial, so
\(\Theta\) is injective and therefore an isomorphism.
As an immediate consequence, the torus
\(\mathbb{T}^n = (\mathbb{S}^1)^n\) has fundamental group \(\mathbb{Z}^n\). Each circle factor
contributes a copy of \(\mathbb{Z}\), and a loop on the torus is classified by the tuple of
winding numbers about its \(n\) independent directions.
The Rotation Groups
The rotation group of three-dimensional space is the first place where a nontrivial fundamental group carries
direct physical meaning. Its fundamental group is as small as it can be without vanishing.
The rotation group and its double both have fundamental groups computed from the sphere \(\mathbb{S}^3\),
directly for \(\mathrm{SU}(2)\) and through a two-sheeted covering for \(\mathrm{SO}(3)\).
Proposition: Fundamental Groups of \(\mathrm{SU}(2)\) and \(\mathrm{SO}(3)\)
The special unitary group
\(\mathrm{SU}(2)\)
is simply connected, and the rotation group
\(\mathrm{SO}(3)\)
has fundamental group
\[
\pi_1(\mathrm{SU}(2)) \cong 1, \quad
\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2\mathbb{Z}.
\]
Proof Sketch:
The group \(\mathrm{SU}(2)\) is homeomorphic to the sphere \(\mathbb{S}^3\). Writing a special
unitary \(2 \times 2\) matrix in terms of two complex parameters subject to a single
normalization shows that \(\mathrm{SU}(2)\) is exactly the unit sphere in \(\mathbb{C}^2 \cong
\mathbb{R}^4\). Since \(\mathbb{S}^3\) is
simply connected and the
fundamental group is a homeomorphism invariant, \(\mathrm{SU}(2)\)
is simply connected. This part is complete with the tools of this page.
For \(\mathrm{SO}(3)\), the conjugation action \(\rho : \mathrm{SU}(2) \to \mathrm{SO}(3)\), written
\(\Phi\) where it is constructed, is a
surjective homomorphism onto \(\mathrm{SO}(3)\) that is two-to-one with kernel \(\{I, -I\}\).
Each fiber is thus a pair \(\{U, -U\}\). That result is algebraic, and the covering structure needs
topology on top of it. The map \(\rho\) is continuous, as shown with its
construction. Under the
identification of \(\mathrm{SU}(2)\) with \(\mathbb{S}^3 \subseteq \mathbb{R}^4\) above,
\(\tfrac{1}{2}\operatorname{tr}(UV^*)\) is the inner product of \(\mathbb{R}^4\), and \(-U\) is the
antipodal point of \(U\). Fix \(U_0\) and let \(W = \{U : \operatorname{tr}(UU_0^*) \gt 0\}\), an open
hemisphere containing \(U_0\), so that every \(R \in \mathrm{SO}(3)\) lies in \(\rho(W)\) for \(U_0\)
chosen in the fiber over \(R\). The set \(W\) is path-connected (normalize the straight segment between
two of its points), it is disjoint from \(-W\), and \(\rho\) is injective on \(W\) because \(W\) never
contains both \(U\) and \(-U\). Since \(\mathrm{SU}(2)\) is
compact, being closed and
bounded in \(\mathbb{R}^4\), and \(\mathrm{SO}(3) \subseteq \mathbb{R}^9\) is Hausdorff, the
closed map lemma
makes \(\rho\) a quotient map. For \(V \subseteq W\) open, \(\rho^{-1}(\rho(V)) = V \cup (-V)\) is open,
so \(\rho(V)\) is open. Hence \(\rho\) maps \(W\), and likewise \(-W\), homeomorphically onto the open
set \(\rho(W)\), whose preimage has exactly the two components \(W\) and \(-W\), these being disjoint,
open, and connected. The total space \(\mathrm{SU}(2) \cong \mathbb{S}^3\) is connected, being simply
connected, and it is
locally path-connected
as a manifold. The base \(\mathrm{SO}(3)\) is
connected
as a continuous image of \(\mathrm{SU}(2)\), and locally path-connected because it is covered by the
open sets \(\rho(W)\), each homeomorphic to an open subset of \(\mathbb{S}^3\). So \(\rho\) is a
covering map with two sheets, whose
total space is simply connected. The subgroup that the fundamental group of the total space induces in
\(\pi_1(\mathrm{SO}(3))\) is therefore trivial, and the
correspondence between sheets and cosets
established below, built on the
lifting properties of covering maps,
puts \(\pi_1(\mathrm{SO}(3))\) in bijection with the fiber \(\{I, -I\}\). A
group with two elements is cyclic of order two, so this yields
\(\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2\mathbb{Z}\). The lifting machinery used here is supplied in
the final subsection. Finally, the fibers of \(\rho\) are the antipodal pairs, so through the
identification \(\mathbb{S}^3 \cong \mathrm{SU}(2)\) it induces a bijection from
\(\mathbb{RP}^3 = \mathbb{S}^3 / \{x \sim -x\}\) onto \(\mathrm{SO}(3)\). The bijection is continuous by
the
universal property of the quotient topology,
and it is a homeomorphism by the closed map lemma, since \(\mathbb{RP}^3\) is
compact
as a continuous image of \(\mathbb{S}^3\). This model of \(\mathbb{RP}^3\) agrees with the quotient of
\(\mathbb{R}^4 \setminus \{0\}\) by nonzero scalars: the inclusion of \(\mathbb{S}^3\) followed by that
quotient map is a continuous surjection onto a
manifold,
hence a Hausdorff space, whose fibers are the antipodal pairs, and the closed map lemma again turns the
induced bijection, continuous by the universal property, into a homeomorphism.
The nontrivial element of \(\pi_1(\mathrm{SO}(3))\) is represented by a loop that rotates by \(2\pi\) about a fixed
axis. Such a loop returns every rotation to its start, so it is genuinely a loop in
\(\mathrm{SO}(3)\). Yet it cannot be contracted to a point. Take the loop \(t \mapsto \exp(tE_3)\),
\(0 \le t \le 2\pi\) (reparametrized over \([0, 1]\)), where \(E_3 \in \mathfrak{so}(3)\) is the
generator of rotations about the third axis, so
that \(\exp(tE_3)\) is the rotation by the angle \(t\). The path \(t \mapsto \exp(t\tilde{E}_3)\) in
\(\mathrm{SU}(2)\), with \(\tilde{E}_3 = \operatorname{diag}(-i/2, i/2)\), is its lift starting at \(I\),
because the double cover \(\rho\)
carries \(\exp(t\tilde{E}_3)\) to \(\exp(tE_3)\). It ends at \(\exp(2\pi\tilde{E}_3) = -I \neq I\), so by
the correspondence between sheets and cosets the loop is not the identity class. The same holds about every
axis. For a unit vector \(\mathbf{n} = (n_1, n_2, n_3)\), with \(\tilde{E}_1, \tilde{E}_2, \tilde{E}_3\) the
basis of \(\mathfrak{su}(2)\) that
\(\rho\) matches with the basis \(E_1, E_2, E_3\) of \(\mathfrak{so}(3)\), the element
\(\tilde{A} = n_1\tilde{E}_1 + n_2\tilde{E}_2 + n_3\tilde{E}_3\) satisfies \(\tilde{A}^2 = -\tfrac{1}{4}I\),
so \(\exp(t\tilde{A}) = \cos(t/2)\,I + 2\sin(t/2)\,\tilde{A}\), which ends at \(-I\) at \(t = 2\pi\) and at
\(I\) at \(t = 4\pi\), while \(\rho\) carries it to the rotation \(\exp(t(n_1E_1 + n_2E_2 + n_3E_3))\) about
\(\mathbf{n}\). Traversing the \(2\pi\) loop twice, a rotation by
\(4\pi\), produces a loop that
can be contracted, which is the group-theoretic content of the relation that the nontrivial element
squares to the identity in \(\mathbb{Z}/2\mathbb{Z}\). The lift of the doubled loop, \(\exp(t\tilde{A})\)
for \(0 \le t \le 4\pi\), ends at \(I\), so the same correspondence makes the doubled loop the identity
class. This is the topological origin of the physical phenomenon,
observable with a belt or a cup of water held in the hand, that a \(2\pi\) rotation is "tangled" relative to the
surroundings while a \(4\pi\) rotation can be undone without releasing the object.
The simply connected \(\mathrm{SU}(2)\) is thus the "untangled" double of \(\mathrm{SO}(3)\). It
maps onto \(\mathrm{SO}(3)\) by a two-to-one continuous surjection, and the loop in
\(\mathrm{SO}(3)\) that fails to contract
lifts to a path in \(\mathrm{SU}(2)\) running between the two preimages of the identity, so only the doubled loop
closes up to a contractible loop. This relationship is the prototype of a structure that organizes much of the
representation theory underlying quantum mechanics and is taken up in the study of covering groups.
Covering Maps
The two-to-one map \(\mathrm{SU}(2) \to \mathrm{SO}(3)\) is an instance of a covering map, the device by
which the fundamental groups of the circle and of \(\mathrm{SO}(3)\) are actually computed, and the bridge
from the fundamental
group to the deeper invariants of a space.
Definition: Covering Map
Let \(E\) and \(X\) be topological spaces. A continuous surjection \(\pi : E \to X\) is a
covering map if \(E\) and \(X\) are
connected
and
locally path-connected
and every point
\(p \in X\) has an open neighborhood \(U\) that is evenly covered, meaning
that each connected component of \(\pi^{-1}(U)\) is mapped homeomorphically onto \(U\) by
\(\pi\). The space \(X\) is the base
of the covering, \(E\) is a covering space, and the components of \(\pi^{-1}(U)\) over an
evenly covered \(U\) are the sheets of the covering over \(U\).
The exponential map \(\mathbb{R} \to \mathbb{S}^1\), \(t \mapsto (\cos 2\pi t, \sin 2\pi t)\), is the model
example, as shown in the proof sketch for spheres. Each point of the circle has an arc neighborhood whose
preimage is a disjoint union of intervals, each
wrapped homeomorphically onto the arc, and the integer fundamental group of the circle is exactly the count of how
a loop's lift fails to close up in the line above it. The two-to-one map of \(\mathrm{SU}(2)\) onto
\(\mathrm{SO}(3)\) is a covering with two sheets, and the lift of the noncontractible \(2\pi\)-loop to a
non-closed path in \(\mathrm{SU}(2)\) is the same mechanism producing the group \(\mathbb{Z}/2\mathbb{Z}\).
The word lift used informally above has a precise meaning, and the device that makes covering maps
compute fundamental groups is the ability to lift paths and to control the ambiguity in doing so.
Definition: Lift
Let \(\pi : E \to X\) be a covering map and \(F : B \to X\) a continuous map from a topological space \(B\).
A lift of \(F\) is a continuous map \(\widetilde{F} : B \to E\) with
\(\pi \circ \widetilde{F} = F\). In other words, \(\widetilde{F}\) makes the diagram over
\(\pi\) commute, sending
each point of \(B\) to a point in the fiber above its image under \(F\).
Theorem (Lifting Properties of Covering Maps)
Let \(\pi : E \to X\) be a covering map.
- Unique Lifting.
If \(B\) is
connected
and \(F : B \to X\) is continuous, then any two lifts of \(F\) that agree at a single point of \(B\) agree
everywhere.
- Path Lifting.
If \(f : I \to X\) is a path and \(e \in E\) satisfies
\(\pi(e) = f(0)\), then there is a lift \(\widetilde{f} : I \to E\) of \(f\) with
\(\widetilde{f}(0) = e\). It is unique by (1).
- Monodromy.
If \(f, g : I \to X\) are
path-homotopic
and \(\widetilde{f}, \widetilde{g}\) are their lifts starting at the same point \(e \in E\), then
\(\widetilde{f}\) and \(\widetilde{g}\) are path-homotopic. In particular
\(\widetilde{f}(1) = \widetilde{g}(1)\).
Proof Sketch:
(1) The set of points of \(B\) at which two given lifts agree is both open and closed. Near any point, each
lift lands in a single sheet over an evenly covered neighborhood, on which \(\pi\) is a homeomorphism, so
agreement at a point forces agreement on a whole neighborhood, while disagreement likewise persists on a
neighborhood. Connectedness of \(B\) then makes this set all of \(B\) once it is nonempty.
(2) Pull back the evenly covered neighborhoods of \(X\) to an open cover of the
compact
interval \(I\). By the
Lebesgue number lemma
applied to \(I\), there is a subdivision fine enough that each subinterval maps into a single
evenly covered neighborhood. Lift successively across the subintervals, at each stage
choosing the unique sheet that continues from the endpoint already lifted. The choice is forced at every step, which
yields existence and, by (1), uniqueness.
(3) Let \(H : I \times I \to X\) be a path homotopy from \(f\) to \(g\). By the Lebesgue number lemma
applied to the compact square, a fine enough grid divides \(I \times I\) into small squares each mapped
by \(H\) into a single evenly covered set. Lift the small squares one at a time, row by row starting
from the corner \((0, 0)\), whose image is lifted to \(e\). Each new square after the first meets the
part already lifted in its left edge, its bottom edge, or both, a connected set containing its lower
left corner. Compose \(H\) on the new square with the inverse of \(\pi\) on the sheet that contains the
lift already assigned to that corner. By (1), applied on the connected shared edges, the new lift agrees
with the old one there, so the pieces glue to a continuous lift \(\widetilde{H}\) of \(H\). Its left and
right edges lift constant paths, so they are constant by (1). Its bottom and top edges then lift \(f\)
and \(g\) starting at \(e\), so they are \(\widetilde{f}\) and \(\widetilde{g}\). Hence
\(\widetilde{H}\) is a path homotopy from \(\widetilde{f}\) to \(\widetilde{g}\), and in particular
\(\widetilde{f}(1) = \widetilde{g}(1)\).
Path lifting and monodromy are the engine that turns a covering into a computation of \(\pi_1\). A loop in the
base lifts to a path in the total space whose endpoint, well-defined by monodromy, records the loop's class up
to the classes of loops whose lifts from the same starting point are again loops. We now make this precise in
three steps. The first is that a covering has a well-defined number of sheets.
Proposition (Number of Sheets)
Let \(\pi : E \to X\) be a covering map. All fibers \(\pi^{-1}(p)\), \(p \in X\), have the same
cardinality. This common cardinality is the number of sheets of the covering, and \(\pi\)
is called \(k\)-sheeted when it equals \(k\).
Proof:
Let \(U \subseteq X\) be an evenly covered open set. Each sheet over \(U\) is mapped bijectively onto
\(U\), so it meets each fiber \(\pi^{-1}(p)\) with \(p \in U\) in exactly one point, and the sheets are
disjoint and exhaust \(\pi^{-1}(U)\). Hence for every \(p \in U\) the fiber \(\pi^{-1}(p)\) is in
bijection with the set of sheets over \(U\), and the cardinality of the fiber is constant on \(U\).
Consequently, for each cardinal \(\kappa\), the set of points of \(X\) whose fiber has cardinality
\(\kappa\) is open, and so is its complement, the union of the corresponding sets for the other
cardinals. Since \(X\) is connected by the definition of a covering map and every fiber is nonempty by
surjectivity, exactly one of these sets is all of \(X\).
The second step identifies the number of sheets inside the fundamental group of the base. Fix a point
\(e \in E\) and put \(p = \pi(e)\). The
induced homomorphism
\(\pi_*\) carries \(\pi_1(E, e)\) onto a subgroup of \(\pi_1(X, p)\), consisting of the classes of loops
at \(p\) that are projections of loops at \(e\). The fiber over \(p\) is counted by the
cosets
of this subgroup.
Theorem (Number of Sheets and Index)
Let \(\pi : E \to X\) be a covering map, let \(e \in E\), put \(p = \pi(e)\), and let
\(H = \pi_*\bigl(\pi_1(E, e)\bigr) \subseteq \pi_1(X, p)\). For a loop \(f\) at \(p\), let
\(\widetilde f_e\) be its lift starting at \(e\). Then the map
\[
\Phi : \pi_1(X, p) \to \pi^{-1}(p), \quad \Phi[f] = \widetilde f_e(1),
\]
is well defined and surjective, and \(\Phi[f] = \Phi[g]\) if and only if \(H[f] = H[g]\). Consequently
the number of sheets of \(\pi\) equals the index of \(H\) in \(\pi_1(X, p)\), the number of cosets
of \(H\).
Proof:
Path lifting provides \(\widetilde f_e\) for every loop \(f\) at \(p\), and by monodromy its endpoint
depends only on the path class \([f]\), so \(\Phi\) is well defined. For surjectivity, let
\(e' \in \pi^{-1}(p)\). The space \(E\) is connected and locally path-connected, hence
path-connected,
so there is a path \(\gamma\) in \(E\) from \(e\) to \(e'\). Then \(f = \pi \circ \gamma\) is a loop at
\(p\) and \(\gamma\) is a lift of \(f\) starting at \(e\), so by uniqueness of lifts
\(\Phi[f] = \gamma(1) = e'\).
Now let \(f\) and \(g\) be loops at \(p\) with lifts \(\widetilde f\) and \(\widetilde g\) starting at
\(e\). A product of lifts is a lift of the product, and the reverse of a lift is a lift of the reverse.
Suppose first that \(\widetilde f(1) = \widetilde g(1)\). Then \(h = \widetilde f \cdot \bar{\widetilde g}\)
is a loop at \(e\) with \(\pi \circ h = f \cdot \bar g\), so
\([f][g]^{-1} = [f \cdot \bar g] = \pi_*[h] \in H\). Conversely, suppose \([f][g]^{-1} \in H\), so that
\(f \cdot \bar g\) is path-homotopic to \(\pi \circ h\) for some loop \(h\) at \(e\). The lift of
\(\pi \circ h\) starting at \(e\) is \(h\) itself, which ends at \(e\), so by monodromy the lift of
\(f \cdot \bar g\) starting at \(e\) also ends at \(e\). That lift is \(\widetilde f\) followed by the
lift of \(\bar g\) starting at \(\widetilde f(1)\). Reversing the second piece gives a lift of \(g\)
starting at \(e\) and ending at \(\widetilde f(1)\), which by uniqueness is \(\widetilde g\). Hence
\(\widetilde g(1) = \widetilde f(1)\).
Since \([f][g]^{-1} \in H\) is equivalent to \(H[f] = H[g]\), the map \(\Phi\) induces a bijection from
the set of right cosets of \(H\) onto \(\pi^{-1}(p)\). The assignment \(Ha \mapsto a^{-1}H\) is a
bijection from right cosets onto left cosets, so the number of cosets does not depend on the side, and
it equals the cardinality of the fiber, which is the number of sheets.
The third step decides when an arbitrary map into the base can be lifted. The argument that made
\(\Phi\) well defined answers this as well.
Theorem (Lifting Criterion)
Let \(\pi : E \to X\) be a covering map, let \(Y\) be a connected and
locally path-connected
space, and let \(F : Y \to X\) be continuous. Let \(y \in Y\) and \(e \in E\) satisfy
\(\pi(e) = F(y)\). Then there is a lift \(\widetilde F : Y \to E\) of \(F\) with \(\widetilde F(y) = e\)
if and only if
\[
F_*\bigl(\pi_1(Y, y)\bigr) \subseteq \pi_*\bigl(\pi_1(E, e)\bigr).
\]
Such a lift is unique. In particular, if \(Y\) is simply connected and locally path-connected, every
continuous map \(F : Y \to X\) has a unique lift with \(\widetilde F(y) = e\).
Proof:
Uniqueness is the unique lifting property, since \(Y\) is connected. If a lift \(\widetilde F\) with
\(\widetilde F(y) = e\) exists, then \(F = \pi \circ \widetilde F\), and functoriality gives
\(F_* = \pi_* \circ \widetilde F_*\), whose image lies in \(\pi_*(\pi_1(E, e))\). The condition is
therefore necessary. The final statement is the special case in which \(\pi_1(Y, y)\) is trivial.
Conversely, assume the condition. The space \(Y\) is path-connected, being connected and locally
path-connected. For \(z \in Y\), choose a path \(\gamma\) in \(Y\) from \(y\) to \(z\), and define
\(\widetilde F(z)\) to be the endpoint of the lift of \(F \circ \gamma\) starting at \(e\). This does not
depend on \(\gamma\). If \(\gamma'\) is another such path, then \(\gamma \cdot \bar\gamma'\) is a loop at
\(y\), and by hypothesis \(F \circ (\gamma \cdot \bar\gamma') = (F \circ \gamma) \cdot
\overline{F \circ \gamma'}\) is path-homotopic to \(\pi \circ h\) for some loop \(h\) at \(e\). As in the
preceding proof, the lift of this product starting at \(e\) then ends at \(e\), which says that the lifts
of \(F \circ \gamma\) and \(F \circ \gamma'\) starting at \(e\) have the same endpoint. Thus
\(\widetilde F\) is well defined, \(\pi \circ \widetilde F = F\), and \(\widetilde F(y) = e\), the
constant path at \(y\) lifting to the constant path at \(e\).
It remains to show that \(\widetilde F\) is continuous. Fix \(z \in Y\), let \(U\) be an evenly covered
neighborhood of \(F(z)\), and let \(S\) be the sheet over \(U\) that contains \(\widetilde F(z)\). Since
\(Y\) is locally path-connected and \(F\) is continuous, there is a path-connected open neighborhood
\(V\) of \(z\) with \(F(V) \subseteq U\). For \(z' \in V\), choose a path \(\delta\) in \(V\) from
\(z\) to \(z'\) and use the path \(\gamma \cdot \delta\) to compute \(\widetilde F(z')\). The lift of
\(F \circ (\gamma \cdot \delta)\) starting at \(e\) runs along the lift of \(F \circ \gamma\) to
\(\widetilde F(z)\) and then along the lift of \(F \circ \delta\) starting at \(\widetilde F(z)\). The
latter is \((\pi|_S)^{-1} \circ F \circ \delta\), since this path lies in \(S\), projects to
\(F \circ \delta\), and starts at \(\widetilde F(z)\). Hence
\(\widetilde F(z') = (\pi|_S)^{-1}\bigl(F(z')\bigr)\), so \(\widetilde F = (\pi|_S)^{-1} \circ F\) on \(V\),
which is continuous.
These three results are the first layer of a systematic correspondence between the coverings of a
base and the subgroups of its fundamental group. The complete classification of coverings, and the
construction of the simply connected universal cover, belong to the next stage of algebraic topology
and to the study of Lie groups, where the universal cover of a Lie group carries the same Lie algebra
while resolving the topology of its fundamental group.