Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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102
Chapter
5. Homology
and
Cohomology
where
w
is a p-form
and
C is a
(p
+
I)-chain.
In
this way we have defined a
pairing
of
forms
wand
chains C:
(w,C)
=
fcw
(5.53)
where
wE
f\P(M)
and
C E
Cp(K).
Stokes
theorem
says
that
(dw,
C)
=
(w,
8C).
(5.54)
Thus
we
have
obtained
the
fundamental
result
that
the
boundary
operator
8
and
the
exterior derivative
d
are
dual
to
each
other.
We
can
use
this
pairing
to
associate
the
de
Rham
cohomology classes
and
the
simplical homology classes
(with real coefficients)
to
each other.
Let
[w]
be
an
element
of
HP(M)
(the
square
bracket
is used
to
denote
an
equivalence class).
The
class
[w]
contains some
particular
closed form
w
satisfying
dw
=
0,
along
with
all
other
forms which differ from
w
by
an
exact
form
dj,
where
j
is
an
arbitrary
(p
-
1) form.
Thus
[w]
=
{w,w+dh,w+dh,
...
}
(5.55)
where
ji
are
all
the
elements of
f\p-l
(M).
This
is
what
a
particular
de
Rham
cohomology class looks like.
Next, let
[C]
be
an
element
of
Hp(M),
namely, a homology class.
Thus
[C]
contains a given cycle
C
with
8C
=
0,
and
all
others
which differ by
the
addition
of
boundaries
8¢,
where
¢
is
any
(p
+
I)-chain. Thus,
[C]
=
{C,C+8¢1,C+8¢2,
...
}
(5.56)
where
¢i
are
all
the
elements
of
Cp+1(M).
Now
the
question is
whether
the
pairing
(w,
C)
=
Ie
w
of
forms
and
chains
induces
a pairing
[w],
[C]
of
de
Rham
cohomology classes
and
simplicial homol-
ogy classes.
In
other
words, if we define
([w],
[C])
=
(w,
C),
(5.57)
where
w
is
any
element of
[w],
and
C
any
element of
[C],
is
this
independent
of
the
representatives w
E
[w],
C E
[C]
that
we chose?
This
will
be
true
if
and
only if
(w
+
dj,
C
+
8¢)
=
(w,
C)
where
j
is a
(p
-
I)-form
and
¢
is a
(p
+
I)-chain,
both
arbitrary.
The
desired result is
true
by
linearity
and
Stokes's theorem:
(w
+
dj,
C + 8¢)
=
(w,
C) +
(w,
8¢) +
(dj,
C) +
(dJ,8¢)
=
(w,
C) +
(dw,
¢) +
(1,8C)
+
(d
2
j,
¢)
=
(w,
C)
(5.58)
(5.59)
5.3.
Harmonic
Forms
and
de
Rham
Cohomology
103
since
dw
=
0,
ec
=
0
and
d
2
f
=
O.
Thus
we have exhibited
the
duality
between de
Rham
cohomology
and
(real) simplical homology for
smoothly
triangulated
manifolds.
This
is
the
con-
tent
of
de
Rham's
theorem. (This
actually
requires
many
technical
points
to
be
demonstrated
in
addition
to
the
above simple calculation,
but
we
omit
these
here.) Thus,
(5.60)
where
we
recall
that
the
left
hand
side,
with
p
as a subscript, denotes
the
pth
simplicial homology group, while
the
right
hand
side,
with
p
as a superscript,
denotes
the
pth
de
Rham
cohomology group.
It
may
seem a
bit
surprising
that
the
de
Rham
groups, defined
in
terms
of
intrinsically
local
C=
forms,
capture
only global
information
about
the
topology
of
a manifold.
In
fact
the
following
lemma
tells us
this
must
be
so.
Poincare's
lemma:
Every
closed form is
locally exact.
In
other
words, given
W
such
that
dw
=
0,
on
any
finite
patch
of
the
manifold we
can
find
f
such
that
W=
df.
Therefore
the
construction
of
closed forms modulo
exact
forms
contains
no lo-
cal information.
Whatever
information
it
does
carry
must
therefore
be
global,
concerning
the
topology
of
the
space.
Example:
If
the
whole manifold is homeomorphic
to
an
open
disc in
IR
n
,
then
it
follows
that
all closed forms
are
exact,
and
HP(M)
=
0 for
p
>
O.
In
electrodynamics (which we usually do
on
IR
3
),
dF
=
0 implies
F
=
dA,
so
there
always exists a gauge
potential.
If
there
is a pointlike
magnetic
monopole,
then
fields
are
singular
at
its location so we
must
work
in
the
modified space
IR3
-
{O}
where
the
location
of
the
monopole
has
been
removed. Now one
can
still have
dF
=
0,
but
for
F
corresponding
to
a monopole field
there
is no globally defined
A
for which
F
=
dA.
Thus
one ends
up
either
with
a Dirac
string
singularity
(by which
the
space is changed from
R3
-
{O}
to
R3
-
{infinite half-line}
and
then
again
F
=
dA,
or
else we
must
allow for multi-valued gauge potentials.
Several physical effects, such as
the
Bohm-Aharonov
effect, follow from similar
considerations.
5.3
Harmonic
Forms
and
de
Rham
Cohomology
We have seen
that
in
de
Rham
cohomology,
an
element
of
HP(M)
is
an
equiva-
lence class
of
closed forms which differ
by
exact
forms. We
may
use
the
Hodge
decomposition
theorem
as a convenience
to
uniquely specify a
particular
ele-
ment
of
each class. Recall
that
according
to
this
theorem
(valid
on
compact
manifolds) ,
Wp
=
do
+
J{3
+
I
(5.61)
104
Chapter
5.
Homology
and
Cohomology
where
6.'"'(p
=
0,
and
this
decomposition is unique. Now suppose
w
is closed, so
dw
=
O.
Then
(5.62)
Since
d
2
a
=
0,
and
also
d'"'(
=
0 (every
harmonic
form is necessarily closed), we
have
d(5{3)
=0
(5.63)
This
implies
({3,
d5{3)
=
(5{3,
5{3)
=
O.
which in
turn
implies
5{3
=
O.
So for a
closed form
w,
w=da+'"'(
(5.64)
with
6.'"'(
=
O.
Thus
any
closed form is
equal
to
a
harmonic
form plus
an
exact
form.
This
gives a convenient way
to
label
any
element
of
HP(M).
Among all
the
elements
in
a cohomology class,
only one
will
be
harmonic.
Then
the
set
of
classes which form
the
cohomology
group
HP
is equivalent
to
the
set
of
distinct
harmonic
p-forms
on
M,
denoted
HarmP(M).
So we
can
say
that
HP(M,
lR)
'"
HarmP(M,
lR)
(5.65)
This
result
is
particularly
useful for
the
following reason.
The
harmonic
p-forms
are
just
the
p-forms which solve Laplace's equation,
6.w
=
0,
(5.66)
so
they
span
the
kernel
of
the
differential
operator
6..
Now
6.
is self-adjoint
and
positive semi-definite,
and
it
is a
theorem
that
such
an
operator
on
a
compact
manifold
has
a finite-dimensional kernel. So
there
are
finitely
many
solutions
to
the
above equation.
This
shows
that
HarmP(M)
and
hence
HP(M)
are
finite-
dimensional
on
a
compact
manifold.
Chapter
6
Fibre
Bundles
6.1
The
Concept
of
a
Fibre
Bundle
A fibre
bundle
is a
kind
of
topological space
with
additional
structure
on
it.
Its
main
property
is
that
locally,
but
not
always globally,
it
looks like
the
product
of
two
distinct
spaces, one
of
which is
to
be
thought
of
as a "fibre" above
the
other,
which is
the
"base".
This
concept is
of
great
importance
in
physics, where
the
"base" is usually
the
spacetime manifold, while
the
fibre represents
dynamical
variables. Some
subtle
and
important
physical
phenomena
can
be
conveniently
represented in
the
language
of
fibre bundles, where
they
appear
more
natural
and
can
be
associated
with
topological
properties
of
the
bundle.
Let us first discuss
the
concept
of
product space
and
product manifold.
The
product
of
two sets 5
1
,5
2
is
the
set
(6.1)
.
If
51
and
52
are
topological spaces,
then
let
U
1
=
{ud,
U
2
=
{Uj}
be
their
respective topologies. We
may
define a topology, called
the
pfoduct
topology,
on
5
1
05
2
:
(6.2)
It
is easy
to
check
that
this
defines a topology
on
the
set
5
1
05
2
.
For example,
the
usual topology
generated
by
open
intervals
on
R
leads
to
a
product
topology
on
IR2
=
IR
0
IR
where
the
open
sets
are
open
rectangles:
{ (x,
y)
E
R2
I
a
<
x
<
b,
c
<
y
<
d }
(6.3)
This
is equivalent
to
the
usual
topology
of
open
disks
on
IR2.
If
the
spaces 51
and
52
are
also differentiable manifolds,
then
so is
their
product.
To prove this, we
construct
an
atlas
as follows.
51
has
an
atlas
(Uo:,
cPo:)
,
while 52
has
an
atlas
(V,e,
'lj;(3).
Then
the
product
manifold 5
1
05
2
has
an
atlas
(Uo:
0
Vf3,
cPo:
0
'lj;(3)
where
Uo:
0
Vf3
are
the
usual
products
of
sets,
106
Chapter
6.
Fibre Bundles
y
d
----------,------,
c
----------t-------i
a
b
x
Figure
6.1:
The
product
topology
on
R2.
while
<Pa
0
'ljJf3
are
the
homeomorphisms:
(6.4)
Clearly, if 8
1
and
8
2
have dimensions
ml,
m2,
then
8
1
0 8
2
has dimension
ml
+
m2·
An
example
of
a
product
space is
the
cylinder, 8
1
0
[0,1].
Since
[0,1]
is a
manifold
with
boundary, so is 8
1
0
[
0,
1
].
This
space is
illustrated
in Fig. 6.2.
Figure 6.
2:
The
cylinder, 8
1
0
[
0,
1
].
The
shaded
region is
U
x
[0,1]
where
U
is
an
open
interval in 8
1
.
Consider now
the
Mobius
strip
(Fig. 6.3), a well-known
object
that
one
can
construct
by folding a
strip
around
and
gluing
the
two ends after giving one of
them
a twist.
It
is definitely
not
a
product
man
ifold. Yet, if we
take
some
open
set U in 8
1
,
then
the
shaded
region
of
the
Mobius
strip
does resemble
the
direct
product
space
U
0
[
0,
1]. Thus,
we
may
say
that
the
Mobius
strip
locally
looks
like a direct
product
manifold.
Let us consider
another
simple
examp
le
of
a space which locally looks like a
direct
product.
Take
the
infinite helix, or spiral,
and
compare
it
with
the
direct .
6.1.
The
Concept
of
a
Fibre
Bundle
107
Figure
6.
3:
The
Mobius strip, which locally looks like a
product
space.
product
space 8
1
®
71...
Figure 6.4:
The
infinite helix
and
the
space 8
1
®
Z.
We
can
define
the
helix,
illustrated
in
the
first
picture
in Fig. 6.4, as a subset
of 3-dimensional space.
It
is
parametrised
by (cos
21ft,
sin
21ft,
t)
,
t
E
lR.
On
the
other
hand,
·
the
space 8
1
®
71..,
illustrated
in
the
second
picture
in Fig. 6.4,
can
be
parametrised
as
(cos
21ft,
sin
21ft,
n),
t
E
lR,
n
E
71..
and
is quite different -
it
is
an
infinite collection of circles, labelled
by
integers.
Consider now
the
open
sets
marked
on
both
diagrams. Clearly
they
are
homeomorphic
to
each
other
,
and
each is homeomorphic
to
the
product
space
(open interval) x
71...
But
the
whole helix
is
not
homeomorphic
to
8
1
®
71..!
In
fact,
it
is a connected space while 8
1
®
71..
is not. We will see below
that
the
two
spaces
in
Fig. 6.4 are distinct fibre bundles,
but
with
the
same base space
and
fibre.
Let us
restate
the
situation
above somewhat differently.
First
of all,
the
circle 8
1
can
be
thought
of as
the
quotient space lR/
71..
(the
space
obtained
by
identifying all sets of real numbers differing
by
21fn).
Thus
the
second space
in
the
figure is homeomorphic
to
(lR/
71..)
®
71...
On
the
other
hand
,
the
first one
(the
helix) is simply homeomorphic
to
lR,
and
this
is certainly
not
the
same as
(lR/71..)
®
71..,
despite
the
fact
that
they
have homeomorphic subsets.
This
is why
108
Chapter
6.
Fibre Bundles
we
say
the
two spaces
are
locally,
but
not
globally, equivalent.
Definition:
A
fibre bundle
is a space
E
which is locally like
the
product
of
two
spaces
X
and
F
and
possesses
the
following properties:
(i)
E
is a topological space, called
the
total
space.
(ii)
X
is a topological space, called
the
base space,
and
there
is a surjective,
continuous
map
7f
:
E
---7
X
called
the
projection
map.
(iii)
F
is a topological space, called
the
fibre.
(iv)
There
is a
group
G
of
homeomorphisms
of
F
onto
itself, called
the
structure
group
of
the
fibre bundle.
(v)
There
is a collection
¢
of
homeomorphisms:
(6.5)
such
that
if
(x,
1)
is a
point
of
Un
(>9
F
then
(6.6)
(Un
is
any
open
set
of
the
base
X).
We now
explain
all
these
properties
in
some detail,
and
then
illustrate
them
through
examples.
Let
us first discuss
the
projection
map.
This
simply
associates
to
each
point
on
the
fibre
bundle
a
corresponding
point
on
the
base
space
X.
This
tells us
that
every
point
in
the
bundle
E
"lies above" a
unique
point
in
the
base
X.
7f
is generally a
many-to-one
mapping,
and
the
inverse
image
7f-l(X)
for a given
x
E
X
is
the
set
of
all
points
in
E
above
the
point
x.
Next,
the
homeomorphisms
¢n
give a specific way
of
regarding
the
points
in
E
"above"
the
open
set
Un
C
X
as being
points
in
the
product
space
Un
(>9
F.
This
is achieved
by
homeomorphic
ally
mapping
7f-l(U
n
)
to
Un
(>9
F.
The
role
of
the
group
G
appears
when
we consider
points
in
X
which lie
in
the
intersection
of
two
open
sets
Un
and
U(3.
We have
the
homeomorphisms:
¢n :
7f-l(U
n
n
U(3)
---7
(Un
n
U(3)
(>9
F
¢(3
:
7f-l(U
n
n
U(3)
---7
(Un
n
U(3)
(>9
F
Combining these, we find homeomorphisms:
(6.7)
(6.8)
Now for
each
fixed x
E
(Un
n
U(3),
these
can
be
regarded
as homeomorphisms,
(6.9)
We require
that
these
homeomorphims
form a
group
of
transformations
of
the
fibre
F;
this
is called
the
structure
group
G.
If
the
topological spaces
are
actually
differentiable manifolds,
then
we
get
(with
a few
assumptions
about
differentiability) a
differentiable fibre bundle.
6.1. The Concept of
a
Fibre Bundle
109
Let
us see how
this
works in
the
example
of
the
helix defined earlier. As a
topological space,
the
total
space
E
is
the
real line
JR.
Chose
the
base
space
X
to
be
the
circle
Sl,
and
the
fibre
F
to
be
the
set
of integers
lL
.
The
projection
map
just
takes
each
point
on
the
helix
to
the
corresponding
point
on
the
circle:
(6.10)
This
is a surjective, continuous
map
E
-->
X,
and
(cos
27ft,
sin
27ft)
parametrises
points
in
Sl,
with
t
being
a
parameter
that
ranges from 0
to
1 along
the
circle.
Given
an
open
interval
U
a
=
{t
I
a
<
t
<
b}
of
S1,
7f-l(U
a
)
is
the
set
of
all
points
(cos
27ft,
sin
27ft, t
+
n),
where
t E
U
a,
nEll
(note
that
t
now
has
a
restricted
range
, unlike
the
parameter
t
introduced
earlier while defining
the
helix.
This
is
because
t
now
represents
a
point
in
an
open
set
of
the
circle
Sl).
Define
the
homeomorphism:
(6.11)
by
CPa
((
cos
27ft,
sin
27ft, t
+
n))
=
(cos
27ft,
sin
27ft,
n)
(6.12)
This
explicitly shows
that
the
inverse image,
under
the
projection
map
,
of
an
open
set
in
the
base
space
, is
homeomorphic
to
the
direct
product
of
the
open
set
with
the
fibre.
3/4
Figure
6.5:
Two
overlapping
open
sets
U
1
,
U
2
on
the
circle
Sl.
To
complete
this
example
, consider two overlapping
open
sets
of
Sl
defined
by:
1 3
U
1
: - -
E
<
t
< -
+
E
4 4
3 5
U
2 :
4 -
E
<
t
<
4
+
E
(6.13)
110
Chapter
6.
Fibre
BW1dles
Then
cPi(U
1
n
U
2
),
cP2(U
1
n
U2)
are
given
on
the
upper
and
lower overlapping
regions as follows.
Lower overlap:
~
-
E
<
t
<
~
+
E.
¢l
(cos
21ft,
sin
21ft,
t
+
n)
=
(cos
21ft,
sin
21ft,
n)
¢2
(cos
21ft,
sin
21f, t
+
n)
=
(cos
21ft,
sin
21ft,
n)
Clearly
the
homeomorphism
cP2
.
cPl
1
is
just
the
identity
on
this
overlap.
(6.14)
The
upper
overlap is
more
subtle, since
with
respect
to
the
patch
U
1
the
region is
i -
E
<
t
<
i
+
E,
while
with
respect
to
U
2
it
is
~
-
E
<
t
<
~
+
E.
It
is convenient
to
make
the
two ranges coincide, in which case
n
gets replaced by
n -
1 in
th
e image
of
cP2.
Then
,
Upper overlap: i -
E
<
t
<
i
+
E:
cPl
(cos
21ft,
sin
21ft, t
+
n)
=
(cos
21ft,
sin
21ft,
n)
cP
2
(cos,
21ft,
sin
21ft, t
+
n)
=
(cos
21ft,
sin
21ft,
n -
1)
(6.15)
Thus
(6.16)
is given by
(6.17)
For fixed
t,
this
can
be
thought
of
as
the
map
cP2
.
cPl
1
:
71..
---)
71..
(6.18)
defined by
cP2
.
cPl
1
(n) = n -
1
(6.19)
This
map
,
under
composition
with
its
elf
and
its
inverse,
generates
the
group
G
=
71..
of
all integers
under
addition.
Thus the
structure
group
of
th
e
bundle
in
this
example is
71...
In
this
way we have identified all
the
ingredients
in
our
definition of fibre
bundle,
in
the
simple example
of
the
helix.
The
one special
property
of
this
example is
that
the
structure
group
G
and
the
fibre
F
are
the
same,
namely
,
71...
We will see
later
that
this
example corresponds
to
a
principal bundle.
We close
this
section
with
a
rather
obvious definition:
Definition:
A fibre
bundle
with
total
space
E,
base space
X
and
fibre
F
is
called a
trivial bundle
if E
=
X
@
F .
In
this
case
the
homeomorphisms
cPj
.
cPi
1
are
all equal
to
the
identity
and
the
structure
group
is trivial.
6.2.
Tangent
and
Cotangent
Bundles
111
6.2
Tangent
and
Cotangent
Bundles
In
a previous
chapter
we have defined
the
tangent
space
to
a manifold
M
at
a
point
p,
denoted Tp(M).
This
space has
the
structure
of
an
n-dimensional
vector space, where n is
the
dimension of
the
manifold. Let us now use
it
to
define a
particular
fibre
bundle
over
the
base space
X
=
M.
The
fibre
is
F
=
Tp(M)
rv
JR
n
,
and
the
total
space is
(6.20)
(It
is always
true
that
the
total
space of a bundle is
the
union of
the
fibres above
each
point
.)
This
is
called
the
tangent bundle
of
the
manifold M.
The
projection
1l':
E
----+
X
is
easy
to
visualise in
this
case.
It
is
the
map
1l':
T(M)
----+
M defined by
1l'(V
E
Tp(M))
=
p.
So
it
associates
the
tangent
space
at
a
point
p
of
the
manifold,
with
the
point
p
itself.
Next
we
must
specify a homeomorphism:
(6.21)
where
U
a
is
an
open
set of
M.
This
is provided by
the
local coordinates
on
Ua.
If
p
E
U
a
and
its coordinates are
Xi(p)
E
JR
n
,
then
a
tangent
vector
Vp
E Tp(M)
is
an
element of
1l'-1
(U
a),
and
it
can
be represented as
. a
Vp
=
a'(x(p))-a
.
x'
(6.22)
This
defines a
map
V
----+
(p,ai(x(p)))
which
is
the
desired homeomorphism
from
1l'-1(U
a
)
to
U
a
®
JR
n
.
Now consider two different
coordinate
systems
Xi
and
yi
in patches
U
a
and
U(3
respectively.
In
the
overlap
U
a
U
U(3
of
the
two patches, we have
the
two
homeomorphisms
¢a
:
Vp
----+
(p,
ai(p)),
¢(3
:
Vp
----+
(PY(p)),
and
¢(3
.
¢-;;l
relates
ai(p)
and
bi(p)
by
in coordinates
Xi
in coordinates
yi
¢(3
.
¢-;,l
:
ai(p)
----+
bi(p) =
~~;
Ip
a
j
(p)
(6.23)
(6.24)
Thus
the
structure
group
G
consists of all real nonsingular n x n matrices, in
other
words G
=
GL(n,
JR).
One
can
similarly
construct
the
cotangent bundle
UpT;(M),
and
the
tensor
bundles
Up
[Tp(M)
®
Tp(M)
®
...
®
T;(M)
®
T;(M)
®
...
T;(M)]
(6.25)
All these have
structure
group G
=
GL(n,
JR),
although
the
fibres are different
in each case.