4.2 Homotopy Groups

We will use what we have learned from group theory and continue the discussion ofproperties of loops (and their higher dimensional counterparts) embedded in topolog-ical spaces to define homotopy groups. These turn out to be very useful topologicalinvariants, and also have numerous important physics applications, for example inthe context of phase transitions in statistical mechanics and quantum field theory.We begin by revisiting loops, and their open counterparts, paths.

Definition: Let X be a topological space, I = [0, 1] ⊂ R.A continuous map α : I → X is a path in X. The path α starts at α0 = α(0) andends at α1 = α(1).If α0 = α1 ≡ x0, then α is a loop with base point x0. We will focus on loops. Noticethat a loop may cross itself arbitrarily many times, that does not violate continuityof the map α.

Figure 7: A path Figure 8: A loop

Definition: A product of two loops α, β with the same base point x0, denoted byα ∗ β, is the loop

(α ∗ β)(t) =

{α(2t) 0 ≤ t ≤ 1

2

β(2t− 1) 12≤ t ≤ 1

In physicists’ language, t could be thought as the time parameter keeping count onhow a loop is traveled. In a product loop the total time 1 is divided in half, with thefirst half spent on traveling through the loop α at double speed 2, and the last halfis spent on traveling through the second loop β, similarly at double speed.

4.2.1 Homotopy

Next we define a notion of continuously deforming a loop α to another loop β. Theintuitive notion is deforming a rubber band without cutting it (recall that loops cancross themselves, not quite like rubber bands). For that, we introduce s ∈ I, a newparameter to keep track of the deformation starting at s = 0 and ending at s = 1.

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Figure 9: A product of two loops.

Let α, β be two loops in X with base point x0. α and β are homotopic, α ∼ β,if there exists a continuous map F : I × I → X, (t, s) 7→ F (t, s) such that

F (t, 0) = α(t) ∀t ∈ IF (t, 1) = β(t) ∀t ∈ IF (0, s) = F (1, s) = x0 ∀s ∈ I.

F is called a homotopy between α and β.

(Note: Actually, this definition of homotopy follows the convention in the book byM. Nakahara, "Geometry, Topology and Physics" (IoP publishing). In mathematics,one often defines a homotopy without the third requirement F (0, s) = F (1, s) =

x0 ∀t ∈ I. Adding the third requirement then defines a homotopy relative to theendpoints {0, 1} of the interval I. We will not make the distinction here and keepfollowing Nakahara’s convention.)Homotopy is an equivalence relation:

1. α ∼ α: choose F (t, s) = α(t) ∀t ∈ I

2. α ∼ β, homotopy F (t, s)⇒ β ∼ α, homotopy F (t, 1− s)

3. α ∼ β, homotopy F (t, s); β ∼ γ, homotopy G(t, s). Then choose

H(t, s) =

{F (t, 2s) 0 ≤ t ≤ 1

2

G(t, 2s− 1) 12≤ t ≤ 1

⇒ H(t, s) is a homotopy between α and γ, so α ∼ γ. First half of thedeformation parameter range is used to deform α to β, the remaining half todeform β to γ, with “deformation speeds” doubled.

The equivalence class [α] is called the homotopy class of α. ([α] = {all loops homotopic with α}).Next, we will try introduce a product of homotopy classes, by setting [α] ∗ [β] =

[α∗β]. The definition may not be well defined, as we could pick other representativesα′ and β′ from the homotopy classes, and it is not immediately clear if [α ∗ β] is thesame as [α′ ∗ β′].

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Lemma: If α ∼ α′ and β ∼ β′, then α ∗ β ∼ α′ ∗ β′.Proof: Let F (t, s) be a homotopy between α and α′ and let G(t, s) be a homotopybetween β and β′. Then

H(t, s) =

{F (2t, s) 0 ≤ t ≤ 1

2

G(2t− 1, s) 12≤ t ≤ 1

is a homotopy between α ∗ β and α′ ∗ β′. This concludes the proof.

By the lemma, a product of homotopy classes [α] ∗ [β] ≡ [α ∗ β] is well defined.

Theorem: The set of homotopy classes of loops at x0 ∈ X, with the product definedas above, is a group called the fundamental group (or first homotopy group) ofX at x0. It is denoted by Π1(X, x0)

Proof:

(0) Closure under multiplication: For all [α], [β] ∈ Π1(X, x0) we have [α] ∗ [β] =

[α ∗ β] ∈ Π1(X, x0), since α ∗ β is also a loop at x0.

(1) Associativity: We need to show (α ∗ β) ∗ γ ∼ α ∗ (β ∗ γ).

Homotopy F (t, s) =

α(

4t1+s

)0 ≤ t ≤ 1+s

4

β(4t− s− 1) 1+s4≤ t ≤ 2+s

4

γ(

4t−s−22−s

)2+s

4≤ t ≤ 1

⇒ [(α ∗ β) ∗ γ] = [α ∗ (β ∗ γ)] ≡ [α ∗ β ∗ γ].

(2) Unit element: Let us show that the unit element is e = [Cx0 ], where Cx0 is theconstant path Cx0(t) = x0 ∀t ∈ I. This follows since we have the homotopies:

α ∗ Cx0 ∼ α : F (t, s) =

{α(

2t1+s

)0 ≤ t ≤ 1+s

2

x01+s

2≤ t ≤ 1

Cx0 ∗ α ∼ α : F (t, s) =

{x0 0 ≤ t ≤ 1−s

2

α(

2t−1+s1+s

)1−s

2≤ t ≤ 1

.

⇒ [α ∗ Cx0 ] = [Cx0 ∗ α] = [α].

(3) Inverse: Define α−1(t) = α(1−t). We need to show that α−1 is really the inverseof α: [α ∗ α−1] = [Cx0 ]. Define:

F (t, s) =

{α(2t(1− s)) 0 ≤ t ≤ 1

2

α(2(1− t)(1− s)) 12≤ t ≤ 1

Now we have F (t, 0) = α ∗ α−1 and F (t, 1) = Cx0 so α ∗ α−1 ∼ Cx0 . Similarlyα−1 ∗ α ∼ Cx0 so we have proven the claim: [α−1 ∗ α] = [α ∗ α−1] = [Cx0 ], quoderat demonstrandum.

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Comment: as is obvious from the above proofs, the bad news is that explicit formulasfor the homotopies F (t, s) are somewhat tedious to construct. For example, in (1)above, the idea is the following. The loops (α ∗ β) ∗ γ and α ∗ (β ∗ γ) look the same,but the time spent on the three loops is distributed differently. In the first product,one half of the time is spent on the product loop α ∗ β, thus one quarter of the timeis spent on α at quadruple velocity. In contrast, in the second product, one half ofthe time is spent on α at double velocity and the remaining half on β ∗ γ. The topformula of the homotopy, F (t, s) = α

(4t

1+s

)slows the speed in traversing through α

from quadruple to double speed, while extending the range of time 0 ≤ t ≤ 1+s4

froma quarter to a half, as the deformation parameter s goes from 0 to 1. Other formulasinvolve similar speed and range adjustments. The good news is that physicists veryrarely need to construct homotopies in detail – usually the question which loops arehomotopic is clear just by intuition and that will be sufficient!

4.2.2 Properties of the Fundamental Group

So far we have defined the fundamental group with reference to the base point x0 ofthe loops. Often one can remove this specification. We first need some definitions:

Definition: If in a topological space X all pairs of points x0, x1 ∈ X can be con-nected with a path, we say that X is path-connected (or pathwise connected).

Proposition: A path-connected space is always connected.

Proof: In this proof we need to know that [0, 1] is connected. Let X be path-connected. Suppose X is not connected, so there exist two nonempty open subsetsX1, X2 with X1

⋂X2 = ∅ such that X = X1

⋃X2. Since X is path-connected, we can

pick two points x1 ∈ X1, x2 ∈ X2 and connect them with a path α : [0, 1]→ X suchthat α(0) = x1 and α(1) = x2. Since α is a continuous function, U1 = α−1(X1) andU2 = α−1(X2) are disjoint open subsets of [0, 1]. One can also verify that U1

⋃U2 =

[0, 1]. Moreover, 0 ∈ U1 and 1 ∈ U2. It then follows that [0, 1] is not connected, whichis a contradiction.

Note that the converse statement is not necessarily true. However, it can be shownthat a connected metric space is also path-connected.

It may not be obvious what a connected but not path-connected space could looklike. One example is given by “the topologist’s sine curve" S, a subset of R2 defined asfollows: S = {(x, y) ∈ R2|y = sin(1/x), x > 0}

⋃(0, 0), i.e. the points on a curve that

oscillates more and more densely as x approaches zero from the positive side, with theorigin added. The subset {(x, y) ∈ R2|y = sin(1/x), x > 0} is connected, and (0, 0)

is its limit point. Thus S is also connected. To prove that S not path-connected, agood strategy would be to establish that there are no paths that would connect the

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origin (0, 0) to the rest of the sine curve points y = sin(x) with x > 0. We leave therigorous proof of this as an exercise.

There is another related notion of connectedness which refers to arcs.

Definition: An arc is a path α which is also a homeomorphism between I and itsimage α(I) ⊂ X.

A path can cross itself, but an arc cannot: otherwise the inverse map to I wouldnot exist (it would not be single-valued, as crossing means that two values of t mapto the same point α(t)).

Definition: In in X all pairs of points x0, x1 can be connected with an arc, we saythat X is arc-connected (or arcwise connected).

Since an arc is also a path, arc-connectedness implies path-connectedness. Theconverse is not necessarily true. It can be shown that if X is a Hausdorff space, thenpath-connected ⇒ arc-connected. Thus, for Hausdorff spaces the two concepts areequivalent. We will use path-connectedness in what follows.

Now we list some facts. Proofs can be found in the abovementioned book byNakahara. (Note that Nakahara mixes arcwise connectedness with pathwise connect-edness.)

1. If x0 and x1 can be connected by a path, then Π1(X, x0) ∼= Π1(X, x1). If X ispath-connected, then the fundamental group is independent of the choice of x0

up to an isomorphism: Π1(X, x0) ∼= Π1(X). (Here’s the basic idea of the proof:let η be a path from x1 to x0. If α is a loop at x0, then joining it with the pathη and its inverse η−1 gives a loop η−1 ∗ α ∗ η defines a loop at x1. Constructthen a map i : Π1(X, x0) → Π1(X, x1), [α] 7→ i([α]) = [η−1 ∗ α ∗ η], and showthat it is well defined and an isomorphism.)

2. Π1(X) is a topological invariant: X ≈ Y ⇒ Π1(X) ∼= Π1(Y ).

3. Examples:

• Π1(R2) = {e} (= the trivial group)

• Π1(R2 \ {a point}) = Z

• Π1(S1) = Z

• Π1(T 2) = Π1(S1 × S1) = Z× Z.

One can show that Π1(X × Y ) = Π1(X)×Π1(Y ) for path-connected spaces Xand Y . In physics literature the trivial group is sometimes denoted by 0, e.g.Π1(R2) = 0. A much more confusing physics literature convention is to use the

53

symbol ⊕ for the direct product group: the notation Π1(X) ⊕ Π1(Y ) appearsfrequently instead of Π1(X)× Π1(Y ).

Definition: A topological space X is simply-connected if and only if X is path-connected and its fundamental group is trivial, Π1(X) = {e}.

For example, R2 \ {a point} is connected and path-connected, but not simply-connected.

The real projective space is defined as RP n = { lines through the origin in Rn+1}. Ifx = (x0, x1, . . . , xn) 6= 0, then x defines a line. All y = λx for some nonzero λ ∈ Rare on the same line and thus we have an equivalence relation: y ∼ x⇔ y = λx, λ ∈R− {0} ⇔ (x and y are on the same line.)So RP n = {[x]| x ∈ Rn+1 − 0} with the above equivalence relation. One can alsodefine the lines by surrounding the origin with an unit n-dimensional sphere Sn andpicking a point on a sphere. But in this case points that are related by a reflectionthrough the origin define the same line. Let Z2 act on Sn by the unit map and thereflection through the origin. Then the lines in Rn+1 are in one-one correspondencewith the points in the quotient space Sn/Z2 (where the opposite points are identified).Thus we can also identify RP n = Sn/Z2.

Example: RP 2 = (S2 with opposite points identified) = S2/Z2.

Π1(RP 2) = Z2. (In fact, Π1(RP n) = Z2 for all n ≥ 2.)

Sometimes for identifying the homotopy group of a topological space it helps toreduce its shape to an essential form. For that purpose, we define some concepts. Inthe following definitions, let Y 6= ∅ be a subspace of a topological space X.

Definition: A continuous map f : X → Y , such that the restriction of f to Y isand identity map, f |Y = idY , is called a retraction. If a retraction exists, we saythat Y is a retract of X.

Example. A unit circle S1 is a retract of R2 \ {(0, 0)} and also a retract of R2 \{(0, 0), (0, 2)}. A possible retraction is f(x) = x

||x|| .

Definition: A subspace Y is a deformation retract of X, if there exists a con-tinuous map

F : X × I → X ; (x, s) 7→ F (x, s)

which satisfies the following three properties:

1. F (x, 0) = x ∀x ∈ X (in the beginning, (at s = 0), F is the identity map on X.)

54

2. F (y, 1) = y ∀y ∈ Y (in the end, (at s = 1), the restriction of F on Y is an identitymap.)

3. F (y, 1) ∈ Y ∀y ∈ X (in the end, (at s = 1), F maps all points to Y )

The properties [2.] and [3.] mean that the identity map inX is continuously deformedto a retraction ofX to Y . (Or, in other words, a retraction is homotopic to the identitymap.)

Note: In Nakahara, the property [2.] is replaced with a stronger requirement:∀t ∈ I : F (y, s) = y ∀y ∈ Y (for all values of s, the restriction of F to Y is theidentity map on Y .)This actually defines what is called a strong deformation retract in mathematics.

Example: Sn is a (strong) deformation retract of Rn+1 \ {0}. A desired map is e.g.

F (x, s) =

((1− s) +

s

||x||

)x ,

where ||x|| is the norm (length) of x.Example: A unit circle S1 is not a deformation retract of R2 \ {(0, 0), (0, 2)}. Thusdeformation retract is a stronger property than a retract. (The above mapping doesnot work now, it is not continuous for all values of s, since the image points have ahole at the image of the point (0, 2)).

Definition: If a deformation retract of X can be a point (Y = {x0}), then X iscontractible.

Example. The Euclidean spaces Rn are contractible.

The following theorem holds:

Theorem: Let Y be a deformation retract of X. Then Π1(X, x0) = Π1(Y, x0) =

Π1(Y ) for all x0 ∈ Y .This theorem often helps to identify fundamental groups. As a simple application,

Π1(R2 \ {a point}) = Π1(S1) = Z, relating the punctured plane to its deformationretract S1.

To find the defomation retracts of e.g. a plane minus n points, and their fundamentalgroups, we introduce a new concept:

Definition: The topological space obtained by gluing together n circles S1 at asingle point is called a rose with n petals. (An alternative name is n-bouquet.)

Note that for n > 1 petals, the roses are not manifolds, because of the crossingpoint. A rose with one petal, S1, is a deformation retract for a plane with 1 point

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Figure 10: A rose with 4 petals.

removed. Similarly, one can deduce see that a rose with n petals is a deformationretract for a plane with n points removed. It is a bit more complicated to see that arose with 2 petals (“figure eight”) is also a deformation retract for a torus with onepoint removed, T 2 \ {1point}.

The fundamental groups of roses with n petals are free groups with n generators.To see that, it is first obvious that a loop can go around a single petal and windaround it n ∈ Z times, with orientation giving the sign, just like in the case of acircle. Further, a loop cannot be continuously moved from one petal to another one.Therefore for n petals there are n basic loops α1, . . . , αn. It may be less obvious thatthe product is non-commutative: for example, α1 ∗ α2 is a loop where first petal 1 istravelled around once, then petal 2. The product loop α2 ∗ α1 describes a differentjourney, traveling through the petals in the opposite order. Since the loops cannotbe continously moved to different petals, the two product loops are not homotopic.This observation generalizes to arbitrary products of loops. Thus, the fundamentalgroup of a rose with n petals consists of the set of generators Xn = {[α1], . . . , [αn]},the homotopy classes of the loops through each petal (with the same orientation),and their arbitrary products (words of arbitrary length), i.e. is the free group with ngenerators F (Xn).

4.2.3 Higher Homotopy Groups

Define: In = {(s1, . . . , sn)| 0 ≤ si ≤ 1, 1 ≤ i ≤ n}∂In = boundary of In = {(s1, . . . , sn)| some si = 0 or 1}

A continuous map α : In → X which maps every point on ∂In to the same pointx0 ∈ X is called an n-loop at x0 ∈ X. Let α and β be n-loops at x0. We say thatα is homeotopic to β, α ∼ β, if there exists a continuous map F : In × I → X suchthat

F (s1, . . . , sn, 0) = α(s1, . . . , sn)

F (s1, . . . , sn, 1) = β(s1, . . . , sn)

F (s1, . . . , sn, t) = x0 ∀t ∈ I when (s1, . . . , sn) ∈ ∂In.

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Homotopy α ∼ β is again an equivalence relation with respect to homotopy classes[α].

Define: α ∗ β : α ∗ β(s1, . . . , sn) =

{α(2s1, s2, . . . , sn) 0 ≤ s1 ≤ 1

2

β(2s1 − 1, s2, . . . , sn) 12≤ s1 ≤ 1.

α−1 : α−1(s1, . . . , sn) = α(1− s1, . . . , sn)

[α] ∗ [β] = [α ∗ β]

⇒ Πn(X, x0), the nth homotopy group of X at x0. (This classifies continuous mapsSn → X.)

Many properties of the fundamental group generalize for the higher homotopygroups:

1. If X is path-connected, then Πn(X, x0) is independent of the base point x0 andone can shorten the notation to Πn(X).

2. If A ⊂ X is a defomation retract of X, then Πn(A, x0) = Πn(X, x0).

3. If X and Y are path-connected topological spaces, then Πn(X × Y, (x0, y0)) =

Πn(X, x0) × Πn(Y, y0) (again, the frequent Physics notation uses ⊕ instead of× for the product group) .

However, the higher homotopy groups are much simpler than the fundamentalgroup: it can be shown that they are always Abelian.

Theorem. The n-dimensional homotopy groups Πn(X, x0) are Abelian for n > 1.

Examples: Π2(S2) = Z. In general, for n-spheres Sn, one can similarly convinceoneself that Πn(Sn) = Z. Likewise, one can argue that every k-loop with k < n canbe contracted to a point. Thus, πk(Sn) = 0 for k < n.

Much less obvious is that even if the dimension of X is strictly less than n, thereis nothing wrong with the definition of n-loops α on X: all that was required in thedefinition was that α is a continous map of In to X. Thus, it make sense to studywhat are the nth homotopy groups of Sk even if k < n. (Insert a discussion ofHopf fibration.) The table below summarizes the results for 1 ≤ k, n ≤ 7 (writtenin the annoying Physics notation convention):

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S1 S2 S3 S4 S5 S6 S7

π1 Z 0 0 0 0 0 0π2 0 Z 0 0 0 0 0π3 0 Z Z 0 0 0 0π4 0 Z2 Z2 Z 0 0 0π5 0 Z2 Z2 Z2 Z 0 0π6 0 Z12 Z12 Z2 Z2 Z 0π7 0 Z2 Z2 Z⊕ Z12 Z2 Z2 Z

4.2.4 Homotopy groups and exact sequences

Given a topological space X, its different homotopy groups πq(X), for q ∈ N, classifythe homotopically inequivalent ways to map a Sq sphere (which can be obtained fromthe q-fold product of intervals [0, 1] ⊂ R:

Iq = [0, 1]× [0, 1]× . . . [0, 1]︸ ︷︷ ︸q factors

⊂ Rq

by identifying all the points on its boundary) to X, where “homotopically inequiva-lent” means “that cannot be mapped into each other by continuous deformations”. Inparticular:

• π0(X) is the set of connected components of X,

• π1(X) is the set of homotopically inequivalent loops in X,

• π2(X) is the set of homotopically inequivalent closed surfaces in X,

et c. (Note, in particular, that, since a generic ZN consists only of N points, it doesnot contain any one-, two- or three-loop or any higher loops either, and the onlypossibly non-trivial homotopy group is π0(ZN) ∼= ZN .)

The homotopy groups of direct products of (group) manifolds are given by

πq(X1 ×X2) = πq(X1)× πq(X2)

Of particular interest are the homotopy groups of N -dimensional spheres. Recallthat: SN = O(N + 1)/O(N) = SO(N + 1)/SO(N) (and, furthermore, for spheres ofodd dimension larger than or equal to three, one also has: S2k+1 = U(k + 1)/U(k));it turns out that:

πq(SN) =

{Z1 for q < N

Z for q = N

(where Z1 = {e} denotes the trivial group, containing only the identity element). Forq > N , in general πq(SN) can be non-trivial; in particular, the simplest non-trivialcase is π3(S2) = Z (which is related to the “Hopf fibration”).

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In order to compute the homotopy groups of various manifolds, it is often usefulto resort to exact sequences of group homomorphisms.

A generic sequence of groups Gi and group homomorphisms fi : Gi → Gi+1:

. . . −→ Gifi−→ Gi+1

fi+1−→ Gi+2 −→ . . .

is said to be “exact” if, for every a (except, possibly, at the end of the sequence), onehas: Im fa ∼= Ker fa+1, where the symbol ∼= denotes group isomorphism.

Furthermore, recall that, due to a fundamental theorem of group homomorphisms,given a group homomorphism f defined on a group G, one has:

Im f = G/Ker f.

Given a Lie group G, and a compact Lie subgroup H ⊂ G, it is possible to provethat the following sequence:

· · · → πq(H)→ πq(G)→ πq(G/H)→ πq−1(H)→ πq−1(G)→ πq−1(G/H)→ . . .

(constructed by repeating the basic block: · · · → πm(H) → πm(G) → πm(G/H) →. . . for values of m which decrease by 1 every time) is exact.

Typically, the determination of non-trivial homotopy groups using exact sequencescan be done by:

• considering a portion of the sequence starting and ending from the trivial groupZ1,

• using the definition of exact sequence,

• using the group homomorphisms’ theorem.

As an example, consider the computation of π3 (SO(3)); since SO(3) ∼= SU(2)/Z2, wecan write:

π3(Z2)f−→ π3 (SU(2))

g−→ π3 (SO(3))h−→ π2(Z2)

First of all, we have: π3(Z2) ∼= π2(Z2) ∼= Z1. Second, note that π3 (SU(2)) ∼= Z,because SU(2) is isomorphic to S3. Next, one can observe that h necessarily mapsevery element of its domain π3 (SO(3)) to the unique element of π2(Z2) ∼= Z1, whichis the identity element of Z1: hence, Ker h ∼= π3 (SO(3)), and, since the sequenceis exact, one obtains: Im g ∼= Ker h ∼= π3 (SO(3)). Due to the homomorphismtheorem, one also has: Im g ∼= π3 (SU(2)) /Ker g. Given that the sequence is exact,one has: Ker g ∼= Im f , but, since the domain of f is isomorphic to Z1, and f is ahomomorphism, f necessarily maps the unique element of its domain to the identityelement in π3 (SU(2)), so Im f ∼= Z1. Thus: Ker g = Z1. Therefore we find that:

π3 (SO(3)) ∼= Ker h ∼= Im g ∼= π3 (SU(2)) /Ker g ∼= π3 (SU(2)) /Im f

∼= π3

(S3)/Im f ∼= π3

(S3)/Z1∼= Z/Z1

∼= Z.

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4.3 Differentiable Manifolds

In this section we add more features to manifolds, so that we can perform calculuson them. We start by introducing coordinates. A familiar setting to think about asa motivation is the surface of the earth, where geographical locations are specified bylatitudes and longitudes. Old folks like us have studied geography at school from abook called World Atlas or something equivalent, which was a collection of maps (orcharts) covering different geographical regions. On each chart there was a coordinategrid showing the latitudes and longitudes. From moving from one region to otherregion, it was helpful that the charts had some overlap, so that moving from onepage to the other one could identify the same location in an overlapping region andcontinue further. So if we simplistically think of the surface of the earth as a two-sphere S2, an atlas was a collection of overlapping regions covering the whole sphere,and for each region there were charts associating coordinates to all points on thatregion of the sphere. What follows next is a sequence of definitions to transform suchconcepts to mathematics.

Definition: Let M be a set (note that we are not assuming it to be a topologicalmanifold). Let U be a subset of M (a "region"). If there exists a bijection x from U

to an open subset of Rn, the pair (U,x) is called a chart. We say that the chart isRn-valued.

Remark: Since x takes values in Rn, it essentially “glues” the coordinates of Rn

onto the region U ofM , associating coordinates with each point on U (and, dependingon the details of the mapping, stretches and “bends” the coordinate grids, as would benecessary if M would be e.g. the two-sphere). It is common to call U a coordinateneighborhood and x a coordinate function.

Definition: Let (U, x) be a chart on M , and p a point in U . If x(p) = 0 ∈ Rn, thechart is centered at p.

Remark: This means that we choose the origin of the coordinate grid to be atthe point p. As a geographical exmaple, on the surface of the Earth, the origin ofcoordinates (0,0) or 0◦ N 0◦ E has been set to a fictional island called the Null Is-land in the Gulf of Guinea. For example, the point (10,2) can then be found in Benin.

We are now ready to cover the whole set M with a collection (Uα,xα) of over-lapping charts, indexed by an index set A 3 α , to define a "World Atlas" of M .We can control how smoothly moving from one chart to another happens: note thatthe composite map xα ◦x−1

α can be defined in a non-empty intersection of two chartslabelled by α, β, and this map is a mapping from a subset of Rn (the image of Uα) to

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another subset of Rn. Such maps are called transition functions, overlap maps,or change of coordinate maps. Recall that a map f : Rn ⊃ U → Rn is class Cr

(1 ≤ r ≤ ∞) if it is continuous, all partial derivatives up to order r

∂rf l

∂(x1)r1 · · · ∂(xn)rn, f = (f 1, . . . , fn),

l = 1, . . . ,m

r1 + r2 + . . .+ rn = k

exist and are continuous. A class r = ∞ map has derivatives up to arbitrary orderand is also called a smooth (or infinitely differentiable) map. We can use this toclassify the smoothness of the composite maps xα ◦ x−1

α .

Definition: A collection A = {(Uα, xα)}α∈A on M is an Rn-valued atlas of classCr (or a Cr atlas), if the following conditions are satisfied:

(i) The collection covers all of M : ∪α∈A = M .

(ii) The intersections in Rn are open sets: xα(Uα ∩ Uβ) are open in Rn for all α, β

(iii) The overlap maps are class Cr: whenever Uα ∩ Uβ 6= ∅, xα ◦ x−1β is a Cr diffeo-

morphism.

Definition: Let A1,A2 be two Cr-atlases. They are equivalent if A1 ∪ A2 is alsoa Cr-atlas. This is an equivalence relation, the equivalence class is called a differen-tiable structure.

Now we are going to shortcut some details2. Note that so far M was just aset. Given a Cr differentiable structure, we can use it to induce a topology on M

(inherited from Rn through the coordinate functions xα). Now, suppose that M wasalready a topological manifold with a topology τ . Is this topology the same as thetopology induced by the Cr structure? A criterion is that if the topological manifoldis equipped with a Cr-atlas, then if all coordinate functions are homeomorphismswith respect to the topology τ , the topology induced by the Cr structure is the sameas τ . With this in mind, we are ready to define what is a differentiable manifold.

Definition: A topological spaceM is an n-dimensional differentiable manifoldof class Cr if

(i) M is provided with a collection A = {(Uα,xα)}, where {Uα} is an open cov-ering of M :

⋃α Uα = M , and every xα : Uα → U ′α ⊂ Rn, U ′α open, is a

homeomorphism.2A detailed discussion can be found in Jeffrey M. Lee, "Differential and Physical Geometry",

section 1.2.

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(ii) The collection A is a Cr-atlas. (All overlap maps ψαβ ≡ xα ◦ x−1β are Cr

diffeomorphisms.)

The number n is the dimension of the manifold: we denote dim M = n.In what follows, we will focus on class C∞ differentiable manifolds, often calledsmooth manifolds. (Sometimes in physics context, these are also called differ-entiable manifolds.)Note that a given smooth manifold M can have several different differentiable struc-tures: for example S7 has 28 and R4 has infinitely (!) many differentiable structures.

Examples of smooth manifolds: Sn

Let us realize Sn as a subset of Rn+1: Sn = {x ∈ Rn+1|∑n

i=0(xi)2 = 1}.One possible atlas consists of the

• coordinate neighborhoods (the labels α are now called i±)

Ui+ ≡ {x ∈ Sn|xi > 0}Ui− ≡ {x ∈ Sn|xi < 0}

• coordinate functions:

xi+(x0, . . . , xn) = (x0, . . . , xi−1, xi+1, . . . , xn) ∈ Rn

xi−(x0, . . . , xn) = (x0, . . . , xi−1, xi+1, . . . , xn) ∈ Rn

(so these are projections onto the plane xi = 0.)

The transition functions are (i 6= j, α = ±, β = ±),

ψiαjβ =xiα ◦ x−1jβ ,

(x0, . . . ,xi, . . . , xj−1, xj+1, . . . , xn)

7→ (x0, . . . , xi−1, xi+1, . . . , xj−1, β

√1−

∑k 6=j

(xk)2, xj+1, . . . , xn)

these are class C∞.There are other compatible atlases, e.g. the stereographic projection.

Definition: As with topological manifolds, we can define a differential manifoldwith a boundary of class Cr by restricting the coordinates into the half-space: werequire xα : Uα → U ′i ⊂ Hm, where U ′i is open in Hn. Again, points with coordinatexm = 0 belong to the boundary of M (denoted by ∂M). The transition functionsxαβ : xβ(Uα ∩ Uβ) → xα(Uα ∩ Uβ) must now be Cr in an open set of Rn whichcontains xβ(Uα ∩ Uβ).

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4.4 Calculus on Manifolds

4.4.1 Differentiable Maps

Definition: Let M,N be smooth manifolds with dimensions dim M = m anddim N = n. Let f be a map f : M → N, p 7→ f(p). If for every chart (V,y) withf(p) ∈ V there exists a chart (U,x) with p ∈ U such that f(U) ⊂ V and the compos-ite map y ◦ f ◦ x−1 is of class Cr, we say that the map f is of class Cr at p. If themap is of class Cr at every point p ∈M , we call the map class Cr.

Note that the definition is independent of the choice of charts, because by as-sumption the transition functions are smooth. If we choose another pair of charts(U ′,x′), (V ′,y′), we can decompose

y′ ◦ f ◦ x′−1 =

C∞︷ ︸︸ ︷y′ ◦ y−1

Cr︷ ︸︸ ︷y ◦ f ◦ x−1 ◦

C∞︷ ︸︸ ︷x ◦ x′−1

so we see that y′ ◦ f ◦ x′−1 is also of class Cr.

A special case is when N = R. A function f : M → R is of class Cr if it is of classCr at every point p ∈M with a chart (U,x) with p ∈ U . The set of C∞ functions onM is denoted by F(M).

Definition: Let M and N be smooth manifolds. A homeomorphism f : M → N

is called a Cr-diffeomorphism, if f and f−1 are both class Cr maps. A C∞ diffeo-morphism is called simply a diffeomorphism. With the composition of maps, the setof all diffeomorphisms from M to itself is a group denoted Diff(M). (Respectively,Cr-diffeomorphisms form a group Diffr(M).)

Example: Consider the two-sphere S2 in R3 defined by x2 + y2 + z2 = 1. The maprθ : S2 → S2, rθ(x) = (x cos θ − y sin θ, x sin θ + y cos θ, z) is a diffeomorphism.

Definition: Two smooth manifolds M and N are diffeomorphic if and only if thereexists a diffeomorphism f : M → N .

Example: The open disk B(0, 1) = {(x, y) ∈ R2|x2 + y2 < 1} is diffeomorphic tothe plane, for example the map f : B(0, 1)→ R2, f(x, y) = ( x

1−√

1−x2−y2, y

1−√

1−x2−y2)

is a diffeomorphism.

The difference between a homeomorphism and a diffeomorphism is that the for-mer is a continuous deformation whereas the latter is a smooth deformation which

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is a stronger requirement. For example, a plane is homeomorphic to a cone, but notdiffeomorphic to it, as the the mapping is not smooth at the apex of the cone. (Asimplified version: consider the following map of a line to a V-shape curve: f(x) = |x|,which is not differentiable at the origin.)

Note: Manifolds being diffeomorphic is an equivalence relation, he equivalenceclasses are called diffeomorphism classes. There is also a local version of this relation:

Definition: A map f : M → N between two smooth manifolds of the same dimen-sion is a local diffeomorphism if and only if every point p ∈ M there exists an opensubset Up, p ∈ Up, such that f |Up : Up → f(Up) is a diffeomorphism onto an opensubset of N .

Example: Consider S2/Z2 where Z2 acts as a reflection through the origin, sothat the antipodal points ±(x.y.z) on the sphere are identified. The map S2 →S2/Z2, (x, y, z) 7→ ±(x, y, z) is a local diffemorphism, but not a diffeomorphism sincethe two antipodal points on S2 map to the same point.

Definition: Let M and M be smooth manifolds. A surjective local diffeomorphismφ : M → M is called a smooth covering if every point p ∈ M hs an open connectedneighbourhood U such that each connected component Ui of the inverse image φ−1(U)

is diffeomorphic to U via the restrictions φUi : Ui → U . The space M is called thecovering space of M . If M is simply connected (the fundamental group is trivial), itis called a universal covering space.

Example: Let us consider S1 as the unit circle on the complex plane. The mapR→ S1, x 7→ eix is a smooth covering, and R is a universal covering space of S1. Bya similar argument, Rn is an universal covering space of the n-torus T n. The previousexample shows that S2 is a universal covering space of S2/Z2 = RP 2.

4.4.2 Tangent Vectors

Suppose that you are traveling in Rn along a path x(t) = (xi(t)) (specifying yourlocation as a function of time) in a region where, say, the temperature varies gentlyand is specified by a function T (x). Then, along your path, the local change oftemperature as a function of time is given by using the chain rule,

dT (x(t))

dt=∑i

dxi(t)

dt

∂T (x)

∂xi. (12)

Recalling that the velocity vector v(t) = (vi(t)) = dx(t)/dt is tangential to the path,the above is a directional derivative of the temperature function

dT (x(t))

dt= vi(t)

∂T (x)

∂xi= ∇vT (x) (13)

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with respect to the velocity vector of along the path. In the above we used theEinstein summation convention:

• When an index twice, it is understood to be summed over. For examplexµy

µ ≡∑m

µ=1 xµyµ = x1y

1 + . . .+ xmym.

So the tangent vector (velocity) v along the curve gives rise to an operator ∇v actingon functions.

More generally, suppose we have a vector valued smooth function V(x) = (V i(x))

in Rn (each component V i is a smooth function Rn → R). Associated with it, we cancompute the directional derivative of a smooth function f : Rn → R by acting on itwith the directional derivative ∇V, to obtain a smooth function ∇Vf . On the otherhand, as is familiar from the theory of differential equations, solving the first orderequations

dx(t)

dt= V(x(t)) (14)

gives rise (at least locally) to a family of integral curves, which are uniquely specifiedonce an initial condition x(t0) = x0 is specified. The vector V is then tangential tothese integral curves at every point x.

Tangent vectors are defined using curves. Let c : (a, b) → M be a curve (we canassume 0 ∈ (a, b) ). Denote c(0) = p and let f : M → R be a function.The rate of change of f along the curve c at point p is

df(c(t))

dt

∣∣∣∣t=0

=∂f

∂xµdxµ(c(t))

dt

∣∣∣∣t=0

,

where xµ(p) = ϕµ(p) are local coordinates and

∂f

∂xµ≡ ∂(f ◦ ϕ−1(x))

∂xµ.

Also we have introduced the Einstein summation convention:

• When an index appears once as a subscript and once as a superscript, it is under-stood to be summed over. For example xµyµ ≡

∑mµ=1 xµy

µ = x1y1 + . . .+ xmy

m.

In other words, df(c(t))dt

is obtained by acting on the function f with the differentialoperator

Xp ≡ Xµp

(∂

∂xµ

)p

, where Xµp =

dxµ(c(t))

dt

∣∣∣∣t=0

.

The operator Xp is called a tangent vector of M at p. It depends on the curve,but several curves can give rise to the same tangent vector Xp. We can see that twocurves c1 and c2 give the same Xp if and only if

(i) c1(0) = c2(0) = p

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