ńņš. 7 |

24 F. Lenz

3.2 Higher Homotopy Groups

The fundamental group displays the properties of loops under continuous defor-

mations and thereby characterizes topological properties of the space in which

the loops are deļ¬ned. With this tool only a certain class of non-trivial topologi-

cal properties can be detected. We have already seen above that a point defect

cannot be detected by loops in dimensions higher than two and therefore the

concept of homotopy groups must be generalized to higher dimensions. Although

in R3 a circle cannot enclose a pointlike defect, a 2-sphere can. The higher ho-

motopy groups are obtained by suitably deļ¬ning higher dimensional analogs of

the (one dimensional) loops. For technical reasons, one does not choose directly

spheres and starts with nā’cubes which are deļ¬ned as

I n = {(s1 , . . . , sn ) | 0 ā¤ si ā¤ 1 all i}

whose boundary is given by

ā‚I n = {(s1 , . . . , sn ) ā I n | si = 0 or si = 1 for at least one i}.

Loops are curves with the initial and ļ¬nal points identiļ¬ed. Correspondingly,

one considers continuous maps from the nā’cube to the topological space X

Ī± : In ā’ X

with the properties that the image of the boundary is one point in X

Ī± : In ā’ X for s ā ā‚I n .

, Ī±(s) = x0

Ī±(I n ) is called an nā’loop in X. Due to the identiļ¬cation of the points on the

boundary these nā’loops are topologically equivalent to nā’spheres. One now

proceeds as above and introduces a homotopy, i.e. continuous deformations of

nā’loops

F : In Ć— I ā’ X

and requires

F (s1 , s2 , . . . , 0) = Ī±(s1 , . . . , sn )

F (s1 , s2 , . . . , 1) = Ī²(s1 , . . . , sn )

(s1 , . . . , sn ) ā ā‚I n ā’ Ī±ā¼Ī².

F (s1 , s2 , . . . , t) = x0 , for

The homotopy establishes an equivalence relation between the nā’loops. The

space of nā’loops is thereby partitioned into disjoint classes. The set of equiv-

alence classes is, for arcwise connected spaces (independence of x0 ), denoted

by

Ļn (X) = {Ī±|Ī± : I n ā’ X, Ī±(s ā ā‚I n ) = x0 } .

As Ļ1 , also Ļn can be equipped with an algebraic structure. To this end one

deļ¬nes a product of maps Ī±, Ī² by connecting them along a common part of the

Topological Concepts in Gauge Theories 25

boundary, e.g. along the part given by s1 =1

ļ£±

ļ£“ 1

ļ£² Ī±(2s1 , s2 , . . . , sn ) 0 ā¤ s1 ā¤

,

Ī± ā—¦ Ī²(s1 , s2 , . . . , sn ) = 2

ļ£“ Ī²(2s ā’ 1, s , . . . , s ) , 1

ļ£³ ā¤ s1 ā¤ 1

1 2 n

2

Ī±ā’1 (s1 , s2 , . . . , sn ) = Ī±(1 ā’ s1 , s2 , . . . , sn ) .

After deļ¬nition of the unit element and the inverse respectively

Ī±ā’1 (s1 , s2 ...sn ) = Ī±(1 ā’ s1 , s2 ...sn )

e(s1 , s2 ...sn ) = x0 ,

Ļn is seen to be a group. The algebraic structure of the higher homotopy groups

is simple

Ļn (X) is abelian for n > 1 . (42)

The fundamental group, on the other hand, may be non-abelian, although most

of the applications in physics deal with abelian fundamental groups. An example

of a non-abelian fundamental group will be discussed below (cf. (75)).

The mapping between spheres is of relevance for many applications of homo-

topy theory. The following result holds

Ļn (S n ) ā¼ Z. (43)

In this case the integer n characterizing the mapping generalizes the winding

number of mappings between circles. By introducing polar coordinates Īø, Ļ• and

Īø , Ļ• on two spheres, under the mapping

Īø = Īø, Ļ• = Ļ•,

the sphere S 2 is covered once if Īø and Ļ• wrap the sphere S 2 once. This 2-loop

belongs to the class k = 1 ā Ļ2 (S 2 ). Under the mapping

Īø = Īø, Ļ• = 2Ļ•

S 2 is covered twice and the 2-loop belongs to the class k = 2 ā Ļ2 (S 2 ). Another

important result is

Ļm (S n ) = 0 m < n, (44)

a special case of which (Ļ1 (S 2 )) has been discussed above. There are no simple

intuitive arguments concerning the homotopy groups Ļn (S m ) for n > m, which

in general are non-trivial. A famous example (cf. [2]) is

Ļ3 (S 2 ) ā¼ Z , (45)

a result which is useful in the study of Yangā“Mills theories in a certain class of

gauges (cf. [20]). The integer k labeling the equivalence classes has a geometric

interpretation. Consider two points y1 , y2 ā S 2 , which are regular points in the

(diļ¬erentiable) mapping

f : S3 ā’ S2

i.e. the diļ¬erential df is 2-dimensional in y1 and y2 . The preimages of these points

M1,2 = f ā’1 (y1,2 ) are curves C1 , C2 on S 3 ; the integer k is the linking number

lk{C1 , C2 } of these curves, cf. (1). It is called the Hopf invariant.

26 F. Lenz

3.3 Quotient Spaces

Topological spaces arise in very diļ¬erent ļ¬elds of physics and are frequently of

complex structure. Most commonly, such non-trivial topological spaces are ob-

tained by identiļ¬cation of certain points which are elements of simple topological

spaces. The mathematical concept behind such identiļ¬cations is that of a quo-

tient space. The identiļ¬cation of points is formulated as an equivalence relation

between them.

Deļ¬nition: Let X be a topological space and ā¼ an equivalence relation on X.

Denote by

[x] = {y ā X|y ā¼ x}

the equivalence class of x and with X/ā¼ the set of equivalence classes; the

projection taking each x ā X to its equivalence class be denoted by

Ļ (x) = [x] .

X/ā¼ is then called quotient space of X relative to the relation ā¼. The quotient

space is a topological space with subsets V ā‚ X/ā¼ deļ¬ned to be open if Ļ ā’1 (V )

is an open subset of X.

ā¢ An elementary example of a quotient space is a circle. It is obtained by an

equivalence relation of points in R and therefore owes its non-trivial topolog-

ical properties to this identiļ¬cation. Let the equivalence relation be deļ¬ned

by:

X = R, x, y ā R, x ā¼ y if x ā’ y ā Z.

R/ā¼ can be identiļ¬ed with

S 1 = {z ā C||z| = 1} ,

the unit circle in the complex plane and the projection is given by

Ļ (x) = e2iĻx .

The circle is the topological space in which the phase of the Higgs ļ¬eld or of

the wave function of a superconductor lives. Also the orientation of the spins

of magnetic substances with restricted to a plane can be speciļ¬ed by points

on a circle. In ļ¬eld theory such models are called O(2) models. If the spins

can have an arbitrary direction in 3-dimensions (O(3) models), the relevant

manifold representing such spins is the surface of a ball, i.e. S 2 .

ā¢ Let us consider

X = Rn+1 \ {0} ,

i.e. the set of all (n+1) tuples x = (x1 , x2 , ..., xn+1 ) except (0, 0, ..., 0), and

deļ¬ne

xā¼y (y 1 , y 2 , ..., y n+1 ) = (tx1 , tx2 , ..., txn+1 ) .

if for real t = 0

Topological Concepts in Gauge Theories 27

The equivalence classes [x] may be visualized as lines through the origin.

The resulting quotient space is called the real projective space and denoted

by RP n ; it is a diļ¬erentiable manifold of dimension n. Alternatively, the

projective spaces can be viewed as spheres with antipodal points identiļ¬ed

RP n = {x|x ā S n , x ā¼ ā’x}. (46)

These topological spaces are important in condensed matter physics. These

are the topological spaces of the degrees of freedom of (nematic) liquid crys-

tals. Nematic liquid crystals consist of long rod-shaped molecules which spon-

taneously orient themselves like spins of a magnetic substances. Unlike spins,

there is no distinction between head and tail. Thus, after identiļ¬cation of

head and tail, the nā’spheres relevant for the degrees of freedom of magnetic

substances, the spins, turn into the projective spaces relevant for the degrees

of freedom of liquid crystals, the directors.

ā¢ The nā’spheres are the central objects of homotopy; physical systems in

general are deļ¬ned in the Rn . In order to apply homotopy arguments, often

the space Rn has to be replaced by S n . Formally this is possible by adjoining

the point {ā} to Rn

Rn āŖ {ā} = S n . (47)

This procedure is called the one-point (or Alexandroļ¬ ) compactiļ¬cation of

Rn ([21]). Geometrically this is achieved by the stereographic projection with

the inļ¬nitely remote points being mapped to the north-pole of the sphere.

For this to make sense, the ļ¬elds which are deļ¬ned in Rn have to approach

a constant with |x| ā’ ā. Similarly the process of compactiļ¬cation of a disc

D2 or equivalently a square to S 2 as shown in Fig. 5 requires the ļ¬eld (phase

and modulus of a complex ļ¬eld) to be constant along the boundary.

D2

D2

Fig. 5. Compactiļ¬cation of a disc D2 to S 2 can be achieved by deforming the disc and

ļ¬nally adding a point, the north-pole

3.4 Degree of Maps

For mappings between closed oriented manifolds X and Y of equal dimension

(n), a homotopy invariant, the degree can be introduced [2,3]. Unlike many other

topological invariants, the degree possesses an integral representation, which is

extremely useful for actually calculating the value of topological invariants. If

y0 ā Y is a regular value of f , the set f ā’1 (y0 ) consists of only a ļ¬nite number

28 F. Lenz

of points x1 , ...xm . Denoting with xĪ² , y0 the local coordinates, the Jacobian

Ī±

i

deļ¬ned by

Ī±

ā‚y0

Ji = det =0

ā‚xĪ²i

is non-zero.

Deļ¬nition: The degree of f with respect to y0 ā Y is deļ¬ned as

degf = sgn (Ji ) . (48)

āf ā’1 (y

xi 0)

The degree has the important property of being independent of the choice of the

regular value y0 and to be invariant under homotopies, i.e. the degree can be

used to classify homotopic classes. In particular, it can be proven that a pair of

smooth maps from a closed oriented n-dimensional manifold X n to the n-sphere

S n , f, g : X n ā’ S n , are homotopic iļ¬ their degrees coincide.

For illustration, return to our introductory example and consider maps from

the unit circle to the unit circle S 1 ā’ S 1 . As we have seen above, we can picture

the unit circle as arising from R1 by identiļ¬cation of the points x + 2nĻ and

y + 2nĻ respectively. We consider a map with the property

f (x + 2Ļ) = f (x) + 2kĻ ,

i.e. if x moves around once the unit circle, its image y = f (x) has turned around

k times. In this case, every y0 has at least k preimages with slopes (i.e. values

of the Jacobian) of the same sign. For the representative of the k-th homotopy

class, for instance,

fk (x) = k Ā· x

and with the choice y0 = Ļ we have f ā’1 (y0 ) = { k Ļ, k Ļ, ...Ļ}.

1 2

Since ā‚y0 /ā‚x x=l/(kĻ) = 1, the degree is k. Any continuous deformation can

only add pairs of pre-images with slopes of opposite signs which do not change

the degree. The degree can be rewritten in the following integral form:

2Ļ

1 df

degf = k = dx .

2Ļ dx

0

Many of the homotopy invariants appearing in our discussion can actually be

calculated after identiļ¬cation with the degree of an appropriate map and its

evaluation by the integral representation of the degree. In the Introduction we

have seen that the work of transporting a magnetic monopole around a closed

curve in the magnetic ļ¬eld generated by circular current is given by the linking

number lk (1) of these two curves. The topological invariant lk can be identiļ¬ed

with the degree of the following map [22]

s1 (t1 ) ā’ s2 (t2 )

T 2 ā’ S2 : (t1 , t2 ) ā’ Ė12 =

s .

|s1 (t1 ) ā’ s2 (t2 )|

Topological Concepts in Gauge Theories 29

The generalization of the above integral representation of the degree is usually

formulated in terms of diļ¬erential forms as

f ā— Ļ = degf Ļ (49)

X Y

where f ā— is the induced map (pull back) of diļ¬erential forms of degree n de-

ļ¬ned on Y . In the Rn this reduces to the formula for changing the variables of

integrations over some function Ļ

Ī± Ī±

ā‚y0 ā‚y0

ńņš. 7 |