Homotopy and Path Homotopy
The invariants studied so far — dimension, compactness, connectedness, the number of components —
answer coarse questions about a space. 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
\[
H(x, 0) = F_0(x) \quad\text{and}\quad H(x, 1) = F_1(x) \qquad \text{for all } x \in X.
\]
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 \(t \in I\) — that is, the
deformation 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 being 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. We first record the notion of a path in the topological
generality needed here.
Definition: Path
Let \(X\) be a topological space. A path in \(X\) is a continuous map \(f : I \to X\); the
points \(f(0)\) and \(f(1)\) are its endpoints, and \(f\) is said to be a path
from \(f(0)\) to \(f(1)\). When \(X\) is a metric space this reduces to the
metric-space notion of a path,
the present definition merely dropping the requirement that the topology arise from a metric.
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
\(\{0, 1\}\): there is a continuous map \(H : I \times I \to X\) with
\[
\begin{aligned}
H(s, 0) &= f_0(s), & H(s, 1) &= f_1(s), && s \in I; \\
H(0, t) &= p, & H(1, t) &= q, && t \in I.
\end{aligned}
\]
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 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. If \(f \sim f'\) and \(g \sim g'\) — with matching endpoints
throughout, so that the products are defined — then \(f \cdot g \sim f' \cdot g'\): 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: \((f \cdot g) \cdot h\) and \(f \cdot (g \cdot h)\)
traverse the same three paths but allot them different portions of the parameter interval, so 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 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.
If \(\alpha\) is a path from \(q\) to \(q'\), then sending a loop \(f\) based at \(q\) to the loop
\(\bar{\alpha} \cdot f \cdot \alpha\) based at \(q'\) — go from \(q'\) back to \(q\) along \(\alpha\),
traverse \(f\), and return — induces a map on path classes that is a group isomorphism
\(\pi_1(X, q) \to \pi_1(X, q')\), with inverse given by conjugation along \(\bar\alpha\). 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: there is a point \(c \in U\) such that for
every \(x \in U\) the line segment from \(c\) to \(x\) lies 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
\[
H(s, 0) = f(s), \qquad H(s, 1) = c, \qquad H(0, t) = H(1, t) = (1-t)\, c + t\, c = c,
\]
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.
\(\blacksquare\)
In particular every 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 \(F(q)\); and 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
\[
F_* : \pi_1(X, q) \to \pi_1\bigl(Y, F(q)\bigr), \qquad F_*[f] = [F \circ f].
\]
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.
\(\blacksquare\)
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 — 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.
We state this without proof; the argument requires tracking how a homotopy, as opposed to an equality, of maps
affects the induced homomorphism, and is carried out in full in the standard topology references. Its force 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; and 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 — a fact we exploit in
the next section.
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 rotations that return to the identity 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 \(n > 1\), the sphere \(\mathbb{S}^n\) is simply connected.
We state these without proof. 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 \(n > 1\) 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. Both facts are proved in the standard
references by the covering-space methods sketched at the end of this page.
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
\[
\pi_1\bigl(X_1 \times \cdots \times X_k,\, (q_1, \ldots, q_k)\bigr) \;\cong\;
\pi_1(X_1, q_1) \times \cdots \times \pi_1(X_k, q_k).
\]
The isomorphism sends a path class \([f]\) to the tuple of path classes of its projections,
\(([p_1 \circ f], \ldots, [p_k \circ f])\), the point being that a loop into a
product
is continuous if and only if each of its components is, so a loop in the product is exactly a tuple of loops in
the factors, and path homotopies likewise decompose coordinatewise. 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 dictated by their identification with familiar
spaces.
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
\[
\pi_1(\mathrm{SU}(2)) \;\cong\; 1, \qquad
\pi_1(\mathrm{SO}(3)) \;\cong\; \mathbb{Z}/2\mathbb{Z}.
\]
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 identification with the
real projective space
\(\mathbb{RP}^3 = \mathbb{S}^3 / \{x \sim -x\}\) realizes the antipodal quotient
\(\mathbb{S}^3 \to \mathbb{RP}^3\) as a covering map with two sheets, whose total space \(\mathbb{S}^3\) is
simply connected. For such a covering the fundamental group of the base is in bijection with a single fiber,
which here has two points; carried out with the lifting properties of covering maps, this yields
\(\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2\mathbb{Z}\). The lifting machinery that makes this rigorous is the
content of the theory of covering spaces, beyond the present page; we record the conclusion as the
computation toward which the constructions of this page point.
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. Traversing it 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 \(\mathbb{Z}/2\mathbb{Z}\). 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
fundamental groups of spheres and rotation groups 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: 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: 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 \(\mathbb{S}^3\) is the same mechanism producing the group \(\mathbb{Z}/2\mathbb{Z}\). The theory of
covering maps, the systematic relationship between covering spaces of a base and subgroups of its fundamental
group, and the role 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.