Download The Fundamental Group

The Fundamental Group
Brendan Fong
∗
April 18, 2009
Abstract
This paper forms notes for an introductory presentation on algebraic
topology. This is done via discussion of the first homotopy group, or fundamental group. We show the strength of this approach by computing
the fundamental group of the circle, and using this to provide interesting
proofs of Brouwer’s fixed point theorem in dimension 2 and the fundamental theorem of algebra.
1
Introduction and Motivation
Classification of topological spaces is one of the central objectives of topology [1,
p.120]. Perhaps unexpectedly, the use of algebraic means—creating algebraic
images of topological spaces that retain key properties under homeomorphisms
of the topological space—has been found to be especially effective in this aim.
Here we discuss the fundamental group, a basic way of forming an algebraic
image of a topological space.
We motivate this idea with an example taken from [2, Chapter 1]. Consider
the two spaces X, and Y , both formed by taking the complement in R3 of two
circles. In the case of X the two circles are disjoint (we depict circle A lying
‘above’ circle B), while in the case of Y the two circles A0 and B 0 are interlinked.
This is displayed in Figure 1.
Figure 1: Spaces X and Y are the complements in R3 of the circles A, B and A0 , B 0
respectively.
We now introduce a third circle, C and C 0 respectively, which can be considered as a closed path, or loop, in the spaces X and Y . This path runs into
the page through A, out of the page through B, out of the page through A, and
into the page through B. We show this in Figure 2.
∗ Undergraduate Student, Department of Mathematics, National University of Singapore
and Department of Mathematics, Australian National University.
1
Figure 2: Spaces X and Y with the loops C and C 0 included.
Now in the case of the space X, we see that this path C creates an interlinked
set of rings, none of which can be separated from the others. These are in fact
called the Borromean rings, and have the special property that if any one circle
is removed then the two remaining circles are not interlinked. On the other
hand, we see that in the space Y the circle C 0 is actually disjoint from the other
two circles, as we can slide it downwards to separate it from A0 and B 0 while all
the time remaining in the space Y .
Thus we see that the drawing a loop in the ‘same’ manner in each space
produces two different results. This is made especially evident when we allow stretching and sliding of the loop. Extending this, we wish to study the
set of all loops—up to stretching or sliding, or what we shall call ‘continuous
deformation’—in a given space, in the hope that this will allow us to say when
two spaces are indeed ‘different’. Note that this is often a nontrivial problem.
Although sometimes it is intuitively easy to see that two spaces are not homeomorphic, rigourously proving that a homeomorphism does not actually exist
often requires some work.
This paper formalises these ideas in the notion of the fundamental group.
In the next section we define and discuss the idea of homotopy, which allows
us to say when two loops are ‘essentially the same’. From this we build the
fundamental group (Section 3). In Section 4 we then find the fundamental
group of the circle, and we conclude by discussing two interesting applications
in Section 5. We assume a little familiarity with elementary point-set topology
and theory of groups. Material is chiefly taken from Croom [1, Chapter 9] and
Hatcher [2, Section 1.1]. The book [3], by May, also provides a good, if concise,
introduction.
2
Homotopy
This section discusses the notion of homotopy, which is used to define the equivalence of two paths in a topological space. We begin with the already familiar
notion of a path in a topological space. Let I = [0, 1] and X denote a topological space throughout this report. A path in a space X is then a continuous
function α : I → X. A loop is a path α such that α(0) = α(1); it is a path that
begins and ends at the same point. We call this point the basepoint. We define
2
a relation between two paths as follows:
Definition 2.1. A homotopy of paths is a family of functions ft : I → X,
0 ≤ t ≤ 1 such that ft (0) and ft (1) are constant as functions of t and the map
F : I × I → X defined by F (s, t) = ft (s) is continuous.
We call ft (0) and ft (1) the fixed endpoints of the homotopy. Two paths α
and β are said to be homotopic if there exists a homotopy such that f0 (s) = α(s)
and f1 (s) = β(s). We denote this α ∼ β. Note that α and β must have common
endpoints. Intuitively, two paths are homotopic if we can continuously deform
one into the other.
Our first task to develop tools to distinguish between paths that are not
homotopic. While it is clear how to show two paths are homotopic—just give a
homotopy—it is less straightforward to prove that two paths are not homotopic.
As an example, the loops α and β with basepoint equal to x0 on the annulus
in Figure 3 look intuitively homotopic, and can indeed be shown to be if we
consider this annulus as a subspace of R2 . In this case we may write down the
homotopy ft (s) = (t − 1)α(s) + tβ(s) between the two loops. On the other
hand, the loops α and γ are intuitively not homomorphic, as γ circumscribes
the ‘hole’, and any continuous deformation of it should also circumscribe the
‘hole’, while α can be continuously deformed to a point. One of the aims of this
report is to prove this intuition.
Figure 3: Example: Loops on an annulus.
We first show that path homotopy is an equivalence relation. This also
justifies our use of the notation ∼ for the relation of homotopy.
Proposition 2.1. The relation of path homotopy forms an equivalence relation
on paths of fixed endpoints in a topological space.
Proof. Reflexive: Clearly path homotopy is reflexive, as any path α is homotopic
to itself by the constant homotopy ft = α for all 0 ≤ t ≤ 1.
3
Symmetric: If the path α is homotopic to the path β via the homotopy ft ,
where f0 = α, f1 = β, then the homotopy gu = f1−u is a path homotopy from
β to α. Thus β is homotopic to α.
Transitive: Let α be homotopic to β via the homotopy ft , and β be homotopic to γ via the homotopy gt . We show α is homotopic to γ. Construct then
the homotopy:
(
ft (2s)
0 ≤ s ≤ 12
.
ht (s) =
gt (2s − 1) 21 ≤ s ≤ 1
This is continuous by the gluing lemma, and hence a homotopy from h0 = f0 = α
to h1 = g1 = γ. Thus α is homotopic to γ.
We thus conclude that the relation of path homotopy is in fact an equivalence
relation.
Note that loops are themselves paths, albeit with both endpoints equal, so
loop homotopy is also an equivalence relation on the loops at a fixed point. We
denote the homotopy class of the path (or loop) α by [α].
Definition 2.2. The loop c : I → X at basepoint x0 ∈ X defined by c(s) = x0
is called the constant loop at x0 . We call any loop equivalent to a constant loop
nullhomotopic.
Returning to the above figure, we now see we are done if we can show γ is
not nullhomotopic. This will be done in the final section, after we compute the
fundamental group of the circle and discuss the idea of a retract. First, however,
we must define the fundamental group.
3
The Fundamental Group
In this section we show that we can put a group structure on the set of homotopy
classes of loops at a given basepoint. For convenience, we take all loops to be
loops at a common basepoint x0 , unless stated otherwise. We first define the
group operation:
Definition 3.1. The product α · β of two loops α, β in a space X is given by
(
α(2s)
0 ≤ s ≤ 21
.
α · β(s) :=
β(2s − 1) 12 ≤ s ≤ 1
This is essentially composition of paths: we first trace out the loop α, and
then the loop β (albeit ‘twice as fast’). The result is clearly again a loop. We
show that it preserves homotopy classes.
Lemma 3.1. If α ∼ α0 , β ∼ β 0 , then α · β ∼ α0 · β 0 .
Proof. Let ft be a homotopy such that f0 = α, f1 = α0 , and gt be a homotopy
such that g0 = β and g1 = β 0 . We claim that:
(
ft (2s)
0 ≤ s ≤ 21
ht =
gt (2s − 1) 21 ≤ s ≤ 1
Then h0 is precisely α · β, h1 is α0 · β 0 , and H(t, s) is continuous by the gluing
lemma. Thus ht shows that α · β ∼ α0 · β 0 .
4
Thus we can also consider · an operation on homotopy classes, defined by
[α] · [β] = [α · β]. This is our group operation.
Definition 3.2. The fundamental group Π1 (X, x0 ) of X at x0 is the set of all
homotopy classes of loops with basepoint x0 under the product operation · for
homotopy classes of loops.
The fundamental group is also called the first homotopy group. We check
that this is indeed a group. We have shown the operation is well defined. It
remains to check the properties of identity, invertibility and associativity.
Identity: Note that the identity element is the equivalence class of the constant loop c(s) = x0 for all 0 ≤ s ≤ 1. Intuitively, this corresponds to the idea
that tracing a reparametrisation of a path—tracing it out at different ‘speed’—
does not affect homotopy class.
Formally, c · α ∼ α by the homotopy
(
x0
0 ≤ s ≤ 1−t
2 ,
ft (s) =
2
α( 1+t s) 1−t
≤
s
≤
1
2
so c is a left identity. The fact that c is a right identity follows similarly.
Invertibility: The inverse element of the class [α], the class [α∗ ], where
α∗ (s) = α(1 − s).
This formalises the idea that tracing out a path, and then retracing it precisely
but in the reverse direction, is clearly homotopic to just staying fixed at the
basepoint. To show that αα∗ is homotopic to the constant map c, we can use
the homotopy:


0 ≤ s ≤ 1−t
α(2s)
2
1−t
1+t
ft (s) = α( 2 ) = α∗ ( 2 ) 1−t
.
≤ s ≤ 1+t
2
2

 ∗
1+t
α (2s − 1)
2 ≤s≤1
Since (α∗ )∗ = α, this shows that α∗ is both a left and right inverse of α.
Associativity: It thus remains to show that the operation · is associative.
Let α, β, γ be loops with basepoint x0 . We want to show α · (β · γ) is simply
a reparametrisation of (α · β) · γ, and hence that the two loops are homotopic.
This is clear if we write out the two functions:


0 ≤ s ≤ 21
α(2s)
α · (β · γ) = β(4s − 2) 21 ≤ s ≤ 43


γ(4s − 3) 34 ≤ s ≤ 1
and


0 ≤ s ≤ 41
α(4s)
(α · β) · γ = β(4s − 1) 14 ≤ s ≤ 21 .


γ(2s − 1) 12 ≤ s ≤ 1
Reparametrising by the continuous, piecewise linear function:

1

0 ≤ s ≤ 12
2s
1
1
3
,
r(s) = s − 4
2 ≤s≤ 4


3
2s − 1 4 ≤ s ≤ 1
5
we see (α · (β · γ))(r) = ((α · β) · γ)(s), and hence the two are homotopic via
(α · (β · γ))(rt ), where rt = (1 − t)r(s) − ts. Thus Π1 (X, x0 ) is a group.
As an example, note that the fundamental group Π1 (R, 0) is trivial, since
any loop α at 0 is homotopic to the constant loop via the homotopy ft : I → R,
s 7→ (1 − t)α(s) for 0 ≤ t ≤ 1.
We now consider the dependence of Π1 (X, x0 ) on the basepoint x0 .
Proposition 3.2. If two points x0 and x1 are connected by a path p : I → X
then the groups Π1 (X, x0 ) and Π1 (X, x1 ) are isomorphic.
Proof. Let γ : I → X be a path from x0 to x1 . We define the change–of–
basepoint map Φγ : Π1 (X, x0 ) → Π1 (X, x1 ) by Φγ ([α]) = [γ ∗ ·α ·γ]. This clearly
takes a loop with basepoint x0 to a loop with basepoint x1 , as we conjugate the
loop with a path from x0 to x1 .
This is a homomorphism as we have shown that γ · γ ∗ is homotopic to the
constant map, so
Φγ (α · β) = [γ ∗ · α · β · γ] = [(γ ∗ · α · γ) · (γ ∗ · β · γ)] = Φγ (α) · Φγ (β).
Likewise, however, the map Φγ ∗ : Π1 (X, x1 ) → Π1 (X, x0 ) is a homomorphism
from the homotopy group at x1 to that at x0 . Since
Φγ ◦ Φγ ∗ ([α]) = Φγ ([γ · α · γ ∗ ]) = [γ ∗ · γ · α · γ ∗ · γ] = [α],
and similiarly
Φγ ∗ ◦ Φγ = [α],
Φγ is invertible and hence an isomorphism.
Thus the fundamental group of a path connected space is independent of the
choice of basepoint is the space X is path connected. In this case we just write
Π1 (X).
Note that a homeomorphism maps homotopic loops to homotopic loops. In
the case that the fundamental group is independent of basepoint, this shows
that the fundamental group is a topological invariant, and, if two spaces have
different fundamental groups, hence can indeed be used to say that the two
spaces are not homeomorphic.
We call a space simply connected if it is path connected and has trivial
fundamental group. These are characterised by the following result:
Proposition 3.3. A space X is simply connected if and only if there is a unique
homotopy class of paths connecting any two points in X.
Proof. Suppose X is simply connected. Then it is path connected and has trivial
fundamental group. Path connectedness gives the fact that there is at least one
homotopy class of paths connecting any two points in X. Choose two points
x0 and x1 , and suppose there exists two paths α, and β from x0 to x1 . Note
that αβ ∗ and β ∗ β are both loops in X, so since Π1 (X) is trivial they are both
homotopic to constant loops. This gives
α ∼ α(β ∗ β) ∼ (αβ ∗ )β ∼ β.
Since α and β are arbitrary paths connecting arbitrary points in X, we see that
any two paths connecting the same two points are homotopic, and so there is a
unique homotopy class of paths connecting any two points in X.
6
For the converse, note that by hypothesis there exists a path connecting any
two points in X, so X is path connected. Furthermore, there exists a unique
homotopy class of loops connecting the basepoint x0 to itself. Thus all loops at
x0 are homotopic to the constant loop, and so the fundamental group Π1 (X) is
trivial. Thus the space X is simply connected.
Note that this is a strengthening of the notion of path connectedness to
spaces that have an essentially unique path connecting any two points. Such
spaces include the Euclidean spaces, the unit n-disk and n-ball.
4
The Circle
In this section we compute a non-trivial fundamental group, that of the circle
S 1 . While such a computation is not straightforward, we see the pay-off in the
two elegant applications in the next section. Although we will not discuss it
here, Van Kampen’s theorem allows us to decompose a complex space into ones
with known fundamental groups, and construct the fundamental group of the
more complex space from the known fundamental groups of the other spaces.
This further enhances the utility of this result. For more details see Hatcher [2,
Section 1.2].
Covering Paths
The notion of covering spaces is key in computation of fundamental groups. In
essence, this is a space that projects onto our base space with the property that
there exists an open cover of the base space such that for any set in the open
cover, each connected component of the preimage of this set maps homeomorphically onto the set. Interestingly, there exists a correspondence between covering
spaces over a base space and the fundamental group of the base space similar to
the Galois correspondence between field extensions and automorphism groups.
This subsection discusses the real line R as a covering space of the base space
S1.
We consider the circle S 1 as the unit circle in R2 . Note that S 1 is path
connected. Thus the fundamental group is independent of the basepoint and
we may take the convention that any loop in S 1 has basepoint (1, 0).
Define the covering projection p : R → S 1 of R over S 1 by
p(s) = (cos 2πs, sin 2πs).
Definition 4.1. Let α : I → S 1 be a path in S 1 . Then a function α̃ : I → R
such that pα̃ = α is called a covering path of α. Likewise, a covering homotopy
of a homotopy ft : I → S 1 is a homotopy f˜t : I → R such that the associated
function F̃ has the property that pF̃ = F .
We prove two lemmas. The key point here will be that there exists an open
cover O of S 1 such that each U ∈ O is path connected and the projection map
p : R → S 1 maps each path component of p−1 (U ) homeomorphically onto U . In
essence, this says that, locally, S 1 looks like R. Note that one such cover is the
set {Ua , Ub } where Ua is the anticlockwise open arc on S 1 beginning at (1, 0)
and ending at (0, −1), and Ub is the anticlockwise open arc on S 1 beginning at
7
(−1, 0) and ending at (0, 1). We assume then, for the sake of convenience, that
O consists of just two open sets.
Lemma 4.1. For each path α : I → S 1 starting at a point x0 ∈ S 1 and each
x̃0 ∈ p−1 (x0 ) there is a unique covering path α̃ : I → R starting at x̃0 .
Proof. Let be a Lebesgue number for the open cover {α−1 (Ua ), α−1 (Ub )} of I.
(This exists as I is compact.) We can thus choose a finite sequence of numbers
0 = t0 , t1 , . . . , tn−1 , tn = 1 such that consecutive terms differ by less than , and
hence the image of each interval α([ti , ti+1 ]) lies completely in Ua or Ub .
Fix x̃0 ∈ p−1 (x0 ). We want to construct a covering path α̃ of α starting
at x̃0 . Now consider t0 = 0. We have α(t0 ) = x0 , and [t0 , t1 ] ⊆ U0 for some
U0 ∈ {Ua , Ub }. By choice of our open cover {Ua , Ub } of S 1 , each path component
of p−1 (U0 ) maps homeomorphically onto U0 . In particular, there exists a unique
path component P0 of p−1 (U0 ) such that x̃0 ∈ P0 . Restricting the covering
projection p to this path component, we have a homeomorphism between P0
and U0 . We can then define α̃ on the interval [t0 , t1 ] by
α̃(s) := ((p|P0 )−1 ◦ α)(s).
This lifts the path α to a covering path α̃ on [t0 , t1 ]. We the proceed by induction
to define α̃ on the entire interval I.
Suppose α̃ has been defined on the interval [t0 , ti ]. We define it on [t0 , ti+1 ]
in the following manner. By construction, α([ti , ti+1 ]) lies completely in Ui for
Ua or Ub . Let Pi be the unique path component of p−1 (Ui ) containing α̃(ti ).
Then, noting that p|Pi is a homeomorphism between Pi and Ui , we define α̃ on
[ti , ti+1 ] by:
α̃(s) := ((p|Pi )−1 ◦ α)(s).
By the gluing lemma, this defines a continuous covering path α̃ on the interval
[t0 , ti+1 ].
Since there are only finitely many ti , continuing inductively allows us to define our covering path α̃ on the entire interval I. Note that each path component
Pi is determined uniquely by α̃(ti ), so α̃ is unique. This proves the lemma.
Lemma 4.2. For each homotopy ft : I → S 1 of paths starting at x0 and each
x̃0 ∈ p−1 (x0 ) there is a unique covering homotopy f˜t : I → R of paths starting
at x̃0 .
Proof. This is proved in a similar manner to the previous lemma. Considering
the map F : I × I → S 1 defined by F (s, t) = ft (s), we wish to lift this to a
continuous covering map F̃ : I ×I → R. This will allow us to define the covering
homotopy f˜t = F̃ (·, t).
Now the above argument uses a Lebesgue number of an open cover to divide
I into small subintervals each of which are mapped by α into a single set in the
open cover, before defining a covering path piece by piece on these subintervals.
This argument generalises to I × I in the obvious manner: we use a Lebesgue
number to divide I × I into small subrectangles [si , si+1 ] × [ti , ti+1 ] each of
which are mapped by F into a single set in the open cover, before defining our
covering map F̃ piece by piece on these subrectangles. In this case, however,
we first define F̃ on I × [t0 , t1 ]—that is, we define F̃ on the entire interval [0, 1]
for small t—before using the same process to define F̃ for all t, completing the
8
definition of the covering homotopy f˜t . The map F̃ is again continuous by the
gluing lemma and the fact that we make the unique choice at each stage to
ensure the definition of F̃ agrees on the edges of the subrectangles. This choice
also ensures the uniqueness of F̃ , and hence of the family of functions of our
covering homotopy ft .
The Fundamental Group of the Circle
We now finally come to the proof of the main theorem.
Theorem 4.3. The fundamental group Π1 (S 1 ) is isomorphic to the additive
group of integers Z.
Proof. We prove this by explicit construction of an isomorphism. In particular,
we shall show:
Φ : Z → Π1 (S 1 )
n 7→ [ωn ],
where ωn (s) := (cos 2πns, sin 2πns) is a loop in S 1 based at (1,0), is an isomorphism of groups. We call ω̃n : I → R, s 7→ ns the ‘lift’ of ωn ; it is the path that
travels from 0 to n at constant speed. Note that the map Φ can be interpreted as
a taking an integer n to the homotopy class [pf˜] = [ωn ] for any path f˜ with endpoints 0 and n in R. This is because the homotopy f˜t (s) = (1 − t)f˜(s) + tω̃n (s)
shows that [f˜] = [ω̃n ].
We show Φ is (i) a homomorphism, (ii) surjective, and (iii) injective.
(i) Homomorphism property: We wish to show that Φ(m + n) = Φ(m) · Φ(n).
Let τm : R → R be a translation by m, so τm (x) = x + m. Thus ω̃m · τm ω̃n
is a path in R with endpoints 0 and m + n, consisting of tracing out the
path ω̃m and then using the endpoint m of that path as a point to begin
the path ω̃n . Thus
Φ(m+n) = [ωm+n ] = [p(ω̃m ·τm ω̃n )] = [ωm ·ωn ] = [ωm ]·[ωn ] = Φ(m)·Φ(n),
so Φ is a homomorphism.
(ii) Surjectivity: Take any loop f ∈ S 1 with basepoint (1, 0). We wish to
show Φ maps onto [f ]. By Lemma 4.1, there exists a unique covering path
f˜ : I → R such that pf˜ = f and f starts at 0 ∈ p−1 ((1, 0)). Since (1, 0) is
also the endpoint of the loop, and hence path, f , the path f˜ must end at
some n ∈ Z = p−1 ((1, 0)). Thus Φ(f˜) = [pf˜] = [f ], and so Φ is surjective.
(iii) Injectivity: Suppose Φ(m) = Φ(n). Then ωm ∼ ωn . Let ft : I → S 1 be
a homotopy from f0 = ωm to f1 = ωn . By Lemma 4.2, we have a unique
covering homotopy f˜t : I → R such that ft (0) = 0 for all t. But since this
covering homotopy is a homotopy, the lifted paths ω̃m and ω̃n must have
the same endpoint. Thus m = n, and Φ is injective.
This proves Φ is an isomorphism of groups, and Z ∼
= Π1 (S 1 ).
We identify Π1 (S 1 ) with Z, so we state ‘the fundamental group of the circle
is the additive group of integers’. We call the homotopy class of loops [ω1 ] the
generator of Π1 (S 1 ).
9
5
Applications
This section borrows heavily from the discussion in Hatcher [2, pp. 31-2]. We
use the previous result to give proofs of two well-known theorems, Brouwer’s
Fixed Point Theorem in Dimension 2 and the Fundamental Theorem of Algebra,
showing the potency of the concept of the fundamental group.
The Brouwer Fixed Point Theorem in Dimension 2
Let D2 be the two dimensional unit disc. The Brouwer fixed point theorem in
dimension 2 states:
Theorem 5.1. Every continuous map f : D2 → D2 has a fixed point.
We prove this theorem by showing that if there is a map without a fixed
point, then we can find a homotopy between the constant loop and any loop in
S 1 , contradicting the fact that Π1 (S 1 ) is nontrivial. This is done by constructing
what we will call a retraction of D2 onto S 1 . We first define this idea rigourously.
Definition 5.1. A subspace A of a topological space X is a retract of X if
there is a continuous function f : X → A such that f |A is the identity function.
The map f is called a retraction.
As an example, observe that an annulus can be retracted onto, say, its outer
circle by mapping any point in the annulus to the point on the outer circle
whose radius runs through that point. Since a retract is continuous, it is not
to hard to see that any homotopy between loops on the annulus is a homotopy
between the image of the loops on the circle. Referring back to Figure 3 then,
it is clear that α is not homotopic to γ, as their images in S 1 are homotopic to
that of the constant path c and ω1 , respectively. The proof of Brouwer’s fixed
point theorem uses similar ideas.
Proof of Theorem 5.1. We prove by contradiction. Suppose f : D2 → D2 does
not have a fixed point, so f (x) 6= x for all x ∈ D2 . We then can define a function
r : D2 → S 1 such that r(x) is equal to the point of intersection between the
ray from f (x) through x and S 1 , the boundary of the disk D2 (see figure 4).
The continuity of f ensures that r is continuous, as small changes of x can
only result in small changes of f (x) and hence r(x). Note also that r|S 1 is the
identity function. Thus r is a retraction of D2 onto S 1 .
Figure 4: The map r(x) is defined by the intersection of the ray from f (x) through x
and S 1 .
10
Let α be any loop in S 1 . Since D2 is simply connected, there exists a
homotopy ht of α = h0 to the constant loop at (1, 0) in D2 . But the composition
rht is then a family of continuous functions S 1 → S 1 such that rh0 = h0 = α
and rh1 = h1 is the constant loop at (1, 0). This shows that [α] = 0 for all
paths α in S 1 , clearly contradicting the fact that S 1 has nontrivial fundamental
group.
Recall that the fixed point property is a topological invariant. This theorem
then says that many ‘nice’ subspaces of R2 , such as any closed polygon, has the
fixed point property.
The Fundamental Theorem of Algebra
Theorem 5.2. Every nonconstant polynomial with coefficients in C has a root
in C.
Proof. Let
p(z) = z n + a1 z n−1 + · · · + an
be a polynomial with coefficients in C. Note that the degree of p is n.(Since
every polynomial is a constant multiple of a monic polynomial, it suffices to
consider just these.) Further assume that p has no root in C. We show that it
must be constant. This is done by using the polynomial to define loops on the
unit circle, which we know has fundamental group isomorphic to Z by Theorem
4.3.
Since p has no roots in C, p(re2πis ) is nonzero for each real number r ≥ 0
and each s ∈ I. This allows us to define
fr (s) :=
p(re2πis )/p(r)
,
|p(re2πis )/p(r)|
a family of loops with basepoint 1 on the unit circle. Now since polynomial
functions are continuous, we see fr ∼ f0 for all r ≥ 0, and so fr is nullhomotopic
for all r.
Now fix r such that r > |a1 | + · · · + |an | and r > 1. Then for |z| = r we have
|z n | = r · |z n−1 | > (|a1 | + · · · + |an |)|z n−1 | ≥ |a1 z n−1 + · · · + an |.
Thus |z n | > t|a1 z n−1 + · · · + an | for all 0 ≤ t ≤ 1, and so the polynomial
pt (z) = z n + t(a1 z n−1 + · · · + an ) has no roots on the circle |z| = r for all
0 ≤ t ≤ 1. This allows us to define a homotopy between fr and ωn via
fr,t :=
pt (re2πis )/pt (r)
,
|pt (re2πis )/pt (r)|
where r is fixed. (Observe that fr,1 = fr and fr,0 = ωn .)
Thus ωn is nullhomotopic. But [ωn ] is an element of the group Π1 (S 1 ), and
so is nullhomotopic if and only if n = 0. Thus the degree of p is 0, and hence p
must be a constant polynomial. This proves the theorem.
Note that, counting multiplicities, this implies that a polynomial p(x) of
degree n has precisely n roots, as for each root α of p(x), we may apply the
11
theorem to the polynomial p(x)/(x − α) of degree n − 1. Continuing inductively,
we see that the number of roots is equal to the degree.
This is a striking application of the fundamental group, not just because
here we see an algebraic result proved using topological means (as opposed to
the previous result, where algebraic means serve to prove a topological result),
but because of the significance of the theorem proved.
In broader terms, however, it is perhaps unsurprising that the fundamental
group can be used to prove results such as these. One of the key uses of the
fundamental group is, as discussed, to show that there cannot exist a certain
map (that is, a homeomorphism) in certain cases. On the other hand, both
of these theorems just proved are statements of the nonexistence of certain
maps: the first a nonconstant polynomial with no root in C, the second a map
on the disc without the fixed point property. Since most of the work proving
‘nonexistence’ is done in the construction of the relevant fundamental groups
(those of the circle and the plane/disc), we can provide elegant proofs of these
results by simple, but clever, application of the fundamental group.
Acknowledgement
Thanks go to Professor Wayne Lawton for his help researching and writing
this paper, but also more generally for his willingness to discuss interesting
mathematics at every opportunity.
References
[1] F. H. Croom: Principles of Topology, Thomson Learning (1989).
[2] A. Hatcher: Algebraic Topology, Cambridge University Press (2002).
[3] J. P. May: A Concise Course in Algebraic Topology, University of Chicago
Press (1999).
12