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π1 (T ) ∼ Z — Z, π1 (C) = Z — {0} . (41)
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 de¬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 de¬ning higher dimensional analogs of
the (one dimensional) loops. For technical reasons, one does not choose directly
spheres and starts with n’cubes which are de¬ned as

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

whose boundary is given by

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

Loops are curves with the initial and ¬nal points identi¬ed. Correspondingly,
one considers continuous maps from the n’cube to the topological space X

± : In ’ X

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

± : In ’ X for s ∈ ‚I n .
, ±(s) = x0

±(I n ) is called an n’loop in X. Due to the identi¬cation of the points on the
boundary these n’loops are topologically equivalent to n’spheres. One now
proceeds as above and introduces a homotopy, i.e. continuous deformations of
n’loops
F : In — I ’ X
and requires

F (s1 , s2 , . . . , 0) = ±(s1 , . . . , sn )
F (s1 , s2 , . . . , 1) = β(s1 , . . . , sn )
(s1 , . . . , sn ) ∈ ‚I n ’ ±∼β.
F (s1 , s2 , . . . , t) = x0 , for

The homotopy establishes an equivalence relation between the n’loops. The
space of n’loops is thereby partitioned into disjoint classes. The set of equiv-
alence classes is, for arcwise connected spaces (independence of x0 ), denoted
by
πn (X) = {±|± : I n ’ X, ±(s ∈ ‚I n ) = x0 } .
As π1 , also πn can be equipped with an algebraic structure. To this end one
de¬nes a product of maps ±, β 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
±
 1
 ±(2s1 , s2 , . . . , sn ) 0 ¤ s1 ¤
,
± —¦ β(s1 , s2 , . . . , sn ) = 2
 β(2s ’ 1, s , . . . , s ) , 1
 ¤ s1 ¤ 1
1 2 n
2
±’1 (s1 , s2 , . . . , sn ) = ±(1 ’ s1 , s2 , . . . , sn ) .
After de¬nition of the unit element and the inverse respectively
±’1 (s1 , s2 ...sn ) = ±(1 ’ s1 , s2 ...sn )
e(s1 , s2 ...sn ) = x0 ,
πn is seen to be a group. The algebraic structure of the higher homotopy groups
is simple
πn (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
πn (S n ) ∼ Z. (43)
In this case the integer n characterizing the mapping generalizes the winding
number of mappings between circles. By introducing polar coordinates θ, • and
θ , • on two spheres, under the mapping
θ = θ, • = •,
the sphere S 2 is covered once if θ and • wrap the sphere S 2 once. This 2-loop
belongs to the class k = 1 ∈ π2 (S 2 ). Under the mapping
θ = θ, • = 2•
S 2 is covered twice and the 2-loop belongs to the class k = 2 ∈ π2 (S 2 ). Another
important result is
πm (S n ) = 0 m < n, (44)
a special case of which (π1 (S 2 )) has been discussed above. There are no simple
intuitive arguments concerning the homotopy groups πn (S m ) for n > m, which
in general are non-trivial. A famous example (cf. [2]) is
π3 (S 2 ) ∼ Z , (45)
a result which is useful in the study of Yang“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 ∈ S 2 , which are regular points in the
(di¬erentiable) mapping
f : S3 ’ S2
i.e. the di¬erential df is 2-dimensional in y1 and y2 . The preimages of these points
M1,2 = f ’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 di¬erent ¬elds of physics and are frequently of
complex structure. Most commonly, such non-trivial topological spaces are ob-
tained by identi¬cation of certain points which are elements of simple topological
spaces. The mathematical concept behind such identi¬cations is that of a quo-
tient space. The identi¬cation of points is formulated as an equivalence relation
between them.
De¬nition: Let X be a topological space and ∼ an equivalence relation on X.
Denote by
[x] = {y ∈ X|y ∼ x}
the equivalence class of x and with X/∼ the set of equivalence classes; the
projection taking each x ∈ X to its equivalence class be denoted by

π (x) = [x] .

X/∼ is then called quotient space of X relative to the relation ∼. The quotient
space is a topological space with subsets V ‚ X/∼ de¬ned to be open if π ’1 (V )
is an open subset of X.

• 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 identi¬cation. Let the equivalence relation be de¬ned
by:
X = R, x, y ∈ R, x ∼ y if x ’ y ∈ Z.
R/∼ can be identi¬ed with

S 1 = {z ∈ C||z| = 1} ,

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

π (x) = e2iπx .

The circle is the topological space in which the phase of the Higgs ¬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 speci¬ed by points
on a circle. In ¬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 .
• 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
de¬ne

x∼y (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 di¬erentiable manifold of dimension n. Alternatively, the
projective spaces can be viewed as spheres with antipodal points identi¬ed

RP n = {x|x ∈ S n , x ∼ ’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 identi¬cation of
head and tail, the n’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.
• The n’spheres are the central objects of homotopy; physical systems in
general are de¬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 {∞} to Rn
Rn ∪ {∞} = S n . (47)
This procedure is called the one-point (or Alexandro¬ ) compacti¬cation of
Rn ([21]). Geometrically this is achieved by the stereographic projection with
the in¬nitely remote points being mapped to the north-pole of the sphere.
For this to make sense, the ¬elds which are de¬ned in Rn have to approach
a constant with |x| ’ ∞. Similarly the process of compacti¬cation of a disc
D2 or equivalently a square to S 2 as shown in Fig. 5 requires the ¬eld (phase
and modulus of a complex ¬eld) to be constant along the boundary.




D2
D2



Fig. 5. Compacti¬cation of a disc D2 to S 2 can be achieved by deforming the disc and
¬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 ∈ Y is a regular value of f , the set f ’1 (y0 ) consists of only a ¬nite number
28 F. Lenz

of points x1 , ...xm . Denoting with xβ , y0 the local coordinates, the Jacobian
±
i
de¬ned by
±
‚y0
Ji = det =0
‚xβi

is non-zero.
De¬nition: The degree of f with respect to y0 ∈ Y is de¬ned as

degf = sgn (Ji ) . (48)
∈f ’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 ’ S n , are homotopic i¬ their degrees coincide.
For illustration, return to our introductory example and consider maps from
the unit circle to the unit circle S 1 ’ S 1 . As we have seen above, we can picture
the unit circle as arising from R1 by identi¬cation of the points x + 2nπ and
y + 2nπ respectively. We consider a map with the property

f (x + 2π) = f (x) + 2kπ ,

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 · x
and with the choice y0 = π we have f ’1 (y0 ) = { k π, k π, ...π}.
1 2

Since ‚y0 /‚x x=l/(kπ) = 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:

1 df
degf = k = dx .
2π dx
0

Many of the homotopy invariants appearing in our discussion can actually be
calculated after identi¬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 ¬eld generated by circular current is given by the linking
number lk (1) of these two curves. The topological invariant lk can be identi¬ed
with the degree of the following map [22]

s1 (t1 ) ’ s2 (t2 )
T 2 ’ S2 : (t1 , t2 ) ’ ˆ12 =
s .
|s1 (t1 ) ’ s2 (t2 )|
Topological Concepts in Gauge Theories 29

The generalization of the above integral representation of the degree is usually
formulated in terms of di¬erential forms as

f — ω = degf ω (49)
X Y

where f — is the induced map (pull back) of di¬erential forms of degree n de-
¬ned on Y . In the Rn this reduces to the formula for changing the variables of
integrations over some function χ
± ±
‚y0 ‚y0

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