Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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92
Chapter
5. Homology
and
Cohomology
orientation
to
each face.
This
can
be
motivated
as follows: if
[VA,
V
1
]
is
an
ordered
I-simplex,
then
the
boundary
(denoted
by
8)
must
satisfy:
(5.11)
At
the
same
time,
the
boundary
will
naturally
contain
just
the
two
end
points
of
this
one-dimensional simplex, namely
[Va]
and
[V1]'
Clearly,
the
only possibility satisfying
these
conditions is:
(5.12)
(or
the
opposite signs
on
the
right,
but
that
is
an
overall convention).
The
sum
of
two oriented O-simplices
on
the
right
hand
side is a formal one,
and
we will
give
it
a precise meaning below.
Generalising
this
idea
to
one higher dimension, we see
that
the
boundary
of
an
oriented 2-simplex is
(5.13)
The
rule is evidently
to
eliminate one
vertex
at
a
time
and
to
choose a sign
corresponding
to
the
location
of
the
deleted vertex.
Thus
for a general p-simplex,
we have:
Definition:
The
boundary
of
an
oriented p-simplex is
p
8[V
O
,V
1
,···
,Vp]
=
2:)-l)j[V
o
,'
"
,"Cj"
..
,Vp]
j=O
where
"Cj
means
that
the
vertex
Vj
is
omitted
in
that
term.
(5.14)
At
this
stage
we need
to
define
what
the
formal sums used above really
mean. Actually, all we will do is
to
give a
name
to
such
sums
of
simplices,
and
enlarge
our
notion
of
simplices
to
include such formal sums.
This
will give a
useful
group
structure
to
the
study
of
simplicial complexes.
Definition:
Let
K
be
an
n-dimensional simplicial complex, containing
lp
p-
simplices. A
p-chain
of
K
is a formal
sum
of
oriented simplices
with
integer
coefficients.
Thus,
if
af
(i
=
1,
...
,lp)
are
oriented p-simplices,
then
lp
C
p
=
2:
n
wf,
(5.15)
i=1
is a p-chain. So in fact
the
boundary
of a p-simplex, as defined above, is a
(p
-
I)-chain.
In
terms
of chains, a simplex
can
be
thought
of
as
the
special
case
of
a chain where
ther
e is only one
term
in
the
sum
.
The
boundary
operator
on
p-chains is easily defined in
terms
of
the
bound-
ary
operator
on
each simplex
in
the
chain:
(5.16)
5.1.
Simplicial
Homology
93
The
collection
of
p-chains forms
an
abelian group.
The
group
axioms
are
estab-
lished as follows:
(i)
C
p
=
2::
nicrf,
Dp
=
2::
micrf
=}
C
p
+
Dp
=
2::(ni
+
mi)crf.
(ii)
The
identity
chain 0 is
obtained
by picking all
ni
=
O.
(iii)
If
C
p
=
2::
nwf
is a chain
then
-C
p
==
2::(
-ni)crf
is
another
chain,
with
C
p
+
(-C
p
)
=
O.
(iv) Associativity obviously holds.
This
is called
the
free Abelian group
generated
by
the
p-simplices of
K,
de-
noted
Cp(K,
Z)
or
simply
Cp(K).
(The
Z
denotes
that
we only consider integer
multiples, which is
part
of
the
meaning
of
free Abelian group).
Summarising
the
discussion above,
we
have defined a
map
8,
called
the
boundary
map,
on
p-chains. (To
be
precise, we will label
the
map
8
p
when
it
acts
on
p-chains,
though
this
may
seem a
bit
pedantic.
The
reader
may
have
realised
that
even for
the
exterior derivative
operator
d,
which we discussed
earlier, one could assign a label
p
when
it
acts
on
p-forms.
It
is necessary
to
stress
this
point
only when confusion might otherwise arise).
This
map:
has
the
following properties:
(i) 8
p
(2::
nwf) =
2::
ni(8
p
crf)
(
ii)
8
cr
P
=
8
[Vr
o
VI
...
V;]
=""P
(-l)j[Vr
o
···
V ...
V;]
p
p",
p -
~J=o
"
J'
,p
(iii)
8
p
[V
o
]
=
0,
since a O-simplex
cannot
have a boundary.
(iv)
8
p
is a
homomorphism of
the
Abelian groups
Cp(K)
---+
Cp_I(K).
For
the
last
point, one
can
check for example
that
8
p
(
-C
p
)
=
-8
p
(
C
p
),
8
p
(
C
p
+
Dp)
=
8
p
C
p
+
8
p
Dp,
and
so on. However,
it
is
important
to
note
that
this
homomorphism
is
not
an
isomorphism, because in general
it
is many-to-one.
:For
example, two different chains
may
have a vanishing boundary,
in
which case
they
are
both
mapped
to
0
E
C
p
-
I
.
More generally, two
or
more chains
can
have
the
same
boundary.
As
an
example,
any
2-chain
ofthe
form
[Vo,
VI]
+
[VI, V
2
]
+
[V2'
Vol
satisfies
=0
(5.17)
Now we will prove
an
important
theorem
about
the
boundary
operator
8:
its
square
is zero,
just
as for
the
exterior derivative.
Theorem:
8
2
=
0 (this really means
8
p
-
1
8
p
=
0
in
our
pedantic
notation).
94
Chapter
5.
Homology
and
Cohomology
Proof:
The
proof
is
straightforward
if slightly tedious
to
write
down.
It
basi-
cally relies
on
a kind
of
antisymmetry. We have:
p
-
"'(-l)j
a
1
[Vro
...
v·
...
V,]
-
~
p-"
J,
,
p
j=O
P
(j-l
=
f;(-l)j
i;(_1)i[Vo,
...
,Vi,···
,Vj,
...
,Vp]
+
~
(_l)i-l[VrO
...
v
...
II;
...
V,])
~
"J"
2,
,p
i=j+l
-
"'(-l)i+j
[Vro
...
11;...
v
...
V,]
-
L-t
"2,,
J'
'P
i<j
L
+"
1
~ ~
+
(
_1)'
J-
[Vio
...
v.
...
Vi
...
V,]
,
'J'
,2,
,p
i>j
=
2:
((-l)i+j
+
(_l)i+j-l)
[Vo,··· ,
Vi,···
,
Vj,
... ,
V
p
]
'<J
=
0 (5.18)
The
last
equality follows since
(-l)i+j
+
(_l)i+j-l
=
o.
Using these properties
of
the
boundary
operator,
we
can
identify two useful
kinds
of
chains.
(i)
If
ac
p
=
0
then
the
chain
C
p
is called a
cycle.
(ii)
If
C
p
=
aCp+l
for some chain
C
p
+
1
,
then
C
p
is called a boundary.
Clearly, if
C
p
=
aC
p
+
1
then
ac
p
=
O.
Thus
every
boundary
is a cycle. How-
ever,
the
reverse is
not
true
in general, as we will see. Indeed, if
the
boundary
of
a chain is zero,
that
does
not
necessarily imply
that
this
chain is a
boundary
of
some
other
one.
This
property
turns
out
to
capture
a lot
of
information
about
the
manifold which will
be
triangulated
using simplicial complexes.
We have
already
noted
that
the
p-chains
of
K
form
an
Abelian
group
Cp(K).
It
is easy
to
see
that
cycles
and
boundaries
form Abelian subgroups
of
this
group.
Define
Zp(K)
==
{
C
p
E
Cp(K) I
ac
p
=
0 }
Bp(K)
==
{
Cp
E
Cp(K)
I
C
p
=
aCp+l }
(5.19)
Thus
Zp(K)
and
Bp(K)
are, respectively,
the
groups formed by
the
cycles
and
the
boundaries
of
K,
and
clearly,
Bp(K)
C
Zp(K)
c
Cp(K)
(not
just
as sets
but
also as groups).
5.1. Simplicial Homology
95
Now
we
are
in a position
to
define
the
concept
of
a homology group.
This
is again
an
Abelian group, in which
the
elements
are
the
cycles
up
to
possible
addition
of boundaries.
Definition:
The
p-th
homology group
Hp(K)
of
the
simplicial complex K
with
integer coefficients, is
the
quotient
of
the
group
of
cycles by
the
group
of
bound-
aries:
Hp(K)
=
Zp(K)j
Bp(K)
Thus,
the
elements
of
Hp(K)
are
the
equivalence classes
of
cycles which differ
by boundaries.
It
is obvious
that
H
is itself
an
Abelian group. Evidently, for
an
n-dimensional simplicial complex
K,
we have
Hp(K)
=
0 for
p
>
n
since
there
are
no chains
of
dimension larger
than
that
of
the
complex, by definition.
Thus
there
is a finite
number
of homology groups for each manifold, one for
each integer
p
between 0
and
n.
Let
us make this definition still
more
explicit. Suppose
h
E
Hp(K).
Then
h
is
an
equivalence class
of
p-cycles:
h
=
[C~,
C;,
C2,
...
J.
Any
pair
of
cycles
C~,
C~
in
this
class have
the
property
that
their
difference is a
boundary:
C;-
C~
=
oC
p
+
l
for some chain
Cp+!o
A different element
h'
E
Hp(K)
will represent
a
distinct
equivalent class:
h'
=
[D~,
D;,
D~,
..
.
J.
If
h'
is
not
the
same
class as
h,
it
means
that
there
is
no C
p
+!
such
that
C;
-
D~
=
oC
p
+
l
'
We
can
equivalently define
Hp(K)
in
terms
of
an
"exact sequence".
Definition:
If
f:
G
--->
H
is a
homomorphism
from
the
group
G
to
the
group
H,
then:
(i)
The
kernel
of
f,
denoted
ker(f),
is
the
subgroup
of
elements
of
G
which get
mapped
to
the
identity
in
H:
ker(f)
==
{
x
E
G
I
f(x)
=
0
E
H }
(5.20)
(ii)
The
image
of
f,
denoted
im
(f),
is
the
subgroup
of elements
of
H
which
come from some element of
G
under
the
map:
im(f)
==
{
y
E
H
I
y
=
f(x),x
E
G }
(5.21)
Now clearly,
the
group
Zp(K)
of
cycles
is
the
same
as
ker(op),
while
the
group
Bp(K)
of
boundaries
is
the
same
as
im(op+!).
In
fact,
the
boundary
operator
0
provides a sequence
of
homomorphisms
(5.22)
This
is called
an
exact sequence,
which means
that
the
kernel
of
the
map
at
each stage is contained in
the
image
of
the
subsequent
map.
This
is obviously
a consequence
of
Op-lOp
=
0 for each
p.
In
this
language,
the
homology groups
Hp(K)
defined earlier
are
just
the
quotients
of
the
kernel by
the
image:
(5.23)
96
Chapter
5.
Homology and Cohomology
So far,
the
homology
group
appears
to
be
a
property
associated
to
a given
simplicial complex (equivalently,
polyhedron)
K,
which specifies a
triangulation
of
the
manifold.
But
the
following
result
tells us
that
it
is
independent
of
the
triangulation.
Theorem:
If
two
polyhedra
K,
L
are
homeomorphic
as topological spaces,
then
Hp(K)
=
Hp(L).
(We
omit
the
proof.)
Since every
compact
manifold
M
can
be
smoothly
triangulated
by
some
polyhedron
K,
we
can
define Hp(M),
the
homology
group
of
the
manifold,
to
be
the
homology
group
Hp(K)
for
the
polyhedron
triangulating
it.
This
is
an
important
way
to
characterize
the
topology
of
a manifold.
Example:
Let
us find
Ho
and
Hl
for
the
circle,
Sl.
The
circle
can
be
triangu-
lated
as
in
Fig. 5.6.
Thus,
(5.24)
o
Figure
5.6:
Triangulation
of
Sl.
Now let us look for
the
I-cycles which
constitute
Zl(K).
The
most
general
I-chain
is:
(5.25)
where
a,
b,
c
are
integers.
This
chain
will
be
a cycle if
Bel
=
O.
This
leads to:
(5.26)
It
follows
that
a
= b =
c.
So
Zl(K)
is
the
group
of
elements:
(5.27)
Next, we look for
I-boundaries.
By
definition,
these
must
bound
2-chains,
but
there
are
no
2-chains
in
our
complex, since
it
is I-dimensional.
It
follows
that
Bl
(K)
=
O.
Thus,
Hl(K)
==
Zl(K)/Bl(K)
=
Zl(K)
= {
a([Vo,
V
l
]
+
[Vl'
V
2
]
+
[V2,
Va])
I
a
Ell.}
(5.28)
which is isomorphic
to
the
group
71..
of
integers.
Thus
we
can
write:
(5.29)
5.1. Simplicial Homology
97
For
Ho(K),
we
start
by
finding
Zo(K).
This
is defined
by
(5.30)
But
this
is always
true
since
8[Vil
=
0 by definition. (A O-chain
cannot
have a
boundary.)
So
Zo(K)
= {
a[vol
+
b[Vll
+
c[V2l
I
a,
b,
c
E
71..
} (5.31)
which is isomorphic
to
71..
EB
71..
ill
71...
Next
we find
the
O-boundaries:
Bo(K)
= {
Co
E
Co(K)
I
Co
=
8C
1
}
(5.32)
Take a general I-chain, as above,
then
its
boundary
is
Thus,
an
element
of
Bo
(K)
is specified
by
three
integers
(a
-
b),
(b
- c),
(c -
a).
But
these
are
not
independent
(they
sum
to
zero!). However
any
two of
them
are
independent,
so
Bo(K)
=
71..
EB
71..
(5.34)
The
quotient
is
obtained
as follows: a
general
element of
Zo(K)
is:
=
(a
+
b
+
c)
[Vo
1
(mod
Bo(K))
(5.35)
since
the
second
term,
((
-b
-
c)
[Vol
+
b[V1l
+
c[V2l),
is
an
element
of
Bo(K)
as
we
have shown above. So
Ho(K)
is labelled
by
a single integer
(a
+
b
+
c),
and
hence
(5.36)
This
completes
the
calculation
of
the
simplicial homology
groups
for
Sl.
The
answer
that
we found for
HO(Sl)
is
actually
a special case
of
a general
result:
Theorem:
If
K
is a
connected
polyhedron
then
Ho(K)
=
71...
(The
proof
is
straightforward.
)
The
homology
groups
defined above were
built
up
using formal
sums
of
sim-
plices
with
integer coefficients, so
they
should
actually
be
denoted
Hp(M,
71..).
Instead
of
integer coefficients, we could have considered formal
sums
of
sim-
plices
with
real
coefficients, which define homology groups which
are
denoted
Hp(M,
lR).
These
groups
are
related,
but
not
in
a one-to-one fashion,
to
Hp(M,
71..).
98
Chapter
5.
Homology
and
Cohomology
For this, first
note
that
for
any
manifold,
Hp(M,
7L.)
must
be
a direct
sum
of
Abelian groups
of
integers, which means
it
has
the
general form
(5.37)
where
7L.
is
the
group
of
integers
under
addition, while
7L.p
is
the
finite
group
of
integers
under
addition
modulo
p,
known as
the
cyclic group
of
order p.
The
finite groups
ZPi
appearing
in
Hp(M,7L.)
are
known as
torsion
sub-
groups
of
the
homology group.
If
we
consider now
the
real,
rather
than
integer,
cohomology,
the
result
turns
out
to
be:
Hp(M,
lR)
=
lR
EEl
...
EEl
lR (5.38)
where
the
number
of
copies
of
lR
in
H
p
(M,
lR) is
the
same
as
the
number
of
copies
of
7L.
in
Hp(M,
7L.).
In
other
words,
in
the
real
cohomology,
the
torsion subgroups
are
ignored,
and
only
the
7L.
factors
in
the
integer homology contribute.
Definition:
The
number
of
7L.
factors in
Hp(M,
7L.),
or
equivalently
the
dimen-
sion
of
Hp(M,
lR), is called
the
p-th
Betti
number
of
the
manifold
M,
which we
denote
pp.
The
Betti
numbers
are
topological invariants
of
a manifold, which means
that
any
two manifolds
that
are
homeomorphic
to
each
other
will necessarily
have
the
same
Betti
numbers. However
the
converse is
not
true: having
the
same
Betti
numbers
does
not
tell us
that
the
two manifolds
are
homeomorphic.
The
sum
of
all
the
Betti
numbers
with
alternating
signs is called
the
Euler
characteristic
X(M)
of
the
manifold:
dimM
X(M)
=
L
(-l)Ppp
(5.39)
p=o
The
Euler
characteristic, like
the
Betti
numbers
in
terms
of which we de-
fined it,
captures
something of
the
topology of a manifold. A very interesting
theorem
tells
us
that
the
same
topological information
can
be
captured
merely
by
counting
the
simplices in a
triangulation
of
the
manifold.
Theorem:
For
an
n-dimensional simplical complex
K,
let
a
p
denote
the
number
of p-simplices
of
K.
Then
n
X(K)
=
L(-l)pa
p
(5.40)
p=o
Thus
one
can
deduce
the
Euler
characteristic
of a manifold merely
by
trian-
gulating
it,
without
having
to
actually
compute
the
homology groups!
It
is
not
the
case in general
that
a
p
=
Pp
for each
p,
so
that
counting simplices does
not
determine
the
individual
Betti
numbers. Indeed,
the
number
of
simplices in a
5.1. Simplicial Homology
99
triangulation
is
not
a topological invariant of
the
manifold, since
there
are
many
different ways
to
triangulate
a manifold
with
various numbers
of
simplices.
The
theorem
above tells us
that
however
we
triangulate
the
manifold,
the
alternating
sum
of
the
number
of 'simplices will come
out
the
same
and
will
be
a topological
invariant equal
to
the
Euler characteristic.
Example
:
Consider
the
2-sphere, 8
2
.
This
is homeomorphic
to
a
tetrahedron,
for which we easily find
(5.41)
from which
it
follows
that
(5.42)
This
example is a special case
of
a famous relation, valid for
the
surface of
any
polyhedron in 3· dimensional space. Any polyhedron
P
is homeomorphic
to
8
2
, so
X(P)
=
2. Now let
V
be
the
number
of vertices,
E
the
number
of
edges
and
F
the
number
of
faces of
the
polyhedron.
In
principle these numbers
cannot
yet
be
related
to
ai,
i
=
0,1,2
since
the
a's
refer
to
the
simplices in a
triangulation,
while a polyhedron
may
have faces which are
not
triangles. Nev-
ertheless,
any
polygonal face of a
tetrahedron
can
be
subdivided into triangles,
and
it
is easy
to
check
that
the
number
V - E
+
F
for
the
polyhedron is equal
to
ao
- a1
+
a2 as
obtained
after subdividing all faces into triangles.
It
follows
that
for any polyhedron in 3 dimensions,
V-E+F=2
(5.43)
which is a classic result in
mathematics
known as
the
Euler relation.
Figure
5.7: "Triangulation" of a 2-
torus
(by squares).
Example:
Another
convenient case
to
study
is
the
two-dimensional
torus
8
1
®
8
1
=
T2.
This
can
be
easily "triangulated"
by
squares, as shown in Fig. 5.7.
From
the
discussion above,
it
is possible
to
treat
the
squares as 2-simplices even
though
the
latter
should really
be
triangles.
Studying
the
figure (including
the
100
Chapter
5.
Homology
and
Cohomology
part
of
it
that
is
not
visible!), we find:
and
hence
O!o
=
32
O!l
=
24
+
24
+
12
+
4
=
64
0!2
=
8
+
8
+
12
+
4
=
32
X(T2)
=
32 - 64
+
32
=
0
So
we
learn
that
the
2-torus
has
vanishing
Euler
characteristic.
(5.44)
(5.45)
Exercise:
Consider a general 2-dimensional manifold, which
has
the
form
of
a surface
with
many
"handles"
(the
sphere
and
torus
have 0
and
1
handle
respectively).
By
breaking
it
up
into
tori
and
triangulating
each
torus,
show
that
a surface
~g
with
9
handles
has
Euler
characteristic
(5.46)
5.2
De
Rham
Cohomology
We now
introduce
a
method
to
characterise
the
topology of a manifold in
terms
of
the
properties
of
differential forms.
Although
individual differential forms
depend
on
the
local
properties
of a manifold,
it
turns
out
that
one
can
define
a
structure
very analogous
to
the
homology discussed above,
on
the
space of
forms,
and
this
reproduces
the
same
topological invariants
(the
Betti
numbers)
that
we
obtained
through
simplicial homology.
Recall
that
the
exterior algebra
U~=o
/IF
(M)
is
the
space
of
all p-forms
on
M,O
::;
p ::;
n
(where
n
is
the
dimension
of
the
manifold). Now each
p-form
is
mapped
onto
some
(p
+
1)-form (possibly zero) by
the
exterior
derivative
d,
which satisfies
d
2
=
O.
Thus
we
have a sequence
of
maps:
d
p
_2
p-l
(M)
d
p
_
1
p
(''')
d
p
p+l
(M)
d
p
+
1
.
..
-+
1\
-+
1\
lV1
-+
1\
-+
...
(5.47)
where
d
p
is
the
d
operator
on p-forms, viewed as a
group
homomorphism
between
the
vector spaces
I\P
and
I\P+
1.
In
perfect analogy
with
the
definitions
of
cycles
and
boundaries, we now
define two special kinds of forms:
(i)
If
dw
p
=
0
then
the
form
wp
is said
to
be
closed.
(ii)
If
wp
=
dw
p
-
1
for some form
Wp-l,
then
wp
is said
to
be
exact.
Moreover,
the
p-forms
on
M form
an
Abelian group, indeed a real vector space
I\P(M).
Closed
and
exact
forms
span
subs paces of
this
vector space,
or
in
other
words
they
are
Abelian
subgroups of
this
Abelian group. We define
ZP(M)
==
{
wp
E
I\P(M)
I
dw
p
=
0 }
BP(M)
==
{
wp
E
I\P(M)
I
wp
=
dWp-l }
(5.48)
5.2.
De
Rham
Cohomology
101
Thus
ZP(M)
and
BP(M)
are, respectively,
the
groups formed by
the
closed
and
the
exact
forms
on
M.
All
exact
forms
are
closed, by
virtue
of
d
2
=
0,
therefore
BP(M)
c
ZP(M)
c
N(M)
(as vector spaces, hence as groups).
A close analogy is
apparent,
in
structure
and
in
notation,
between
these
concepts for
the
differential forms
and
the
corresponding notions for simplicial
complexes.
Some
important
differences should
be
noted. While for simplices
we
started
by
taking
linear combinations
with
integer coefficients, for forms
we
are
forced
at
the
outset
to
allow linear combinations
with
real coefficients,
since forms
are
intrinsically real-valued.
Another
point
is
that
simplices were
themselves
an
auxiliary
construction
allowing
us
to
define topological invariants
for a manifold, while
in
the
case
of
forms we work directly
on
the
manifold itself.
An
important
point
about
notation
is
that
for differential forms,
the
label
p
on
the
groups
of
closed
and
exact
forms is always
written
as a superscript, by
convention, whereas for p-chains
it
is a subscript.
After all this,
the
following definition should come as no surprise.
Definition:
The
p-th
de
Rham cohomology gmup HP(M)
of
the
manifold
M,
is
the
quotient
of
the
group
of closed forms
by
the
group
of
exact
forms:
(5.49)
Thus,
the
elements
of
HP
(M)
are
the
equivalence classes
of
closed forms
which differ by
exact
forms. Again we clearly have
HP(M)
=
0 for
p
>
n
(where
n
is
the
dimension of
the
manifold) since by
antisymmetry
there
are
no forms
of
dimension larger
than
that
of
the
manifold. So
there
is a finite
number
of
de
Rham
cohomology groups for each manifold, one for each integer
p
between 0
and
n.
As we
noted
above, forms
are
intrinsically defined
with
real
(rather
than
integer) coefficients, so
this
is a
real
cohomology,
and
each
group
has
the
form:
HP(M) =
IR
EEl
...
EEl
IR
(5.50)
Suppose now
that
the
manifold
M
has
been
smoothly
triangulated.
Thus
there
is a
homeomorphism
between
the
whole
of
M
and
a polyhedron.
Pick
an
open
p-simplex
(5P
in
the
polyhedron.
This
will
be
homeomorphic
to
an
open
submanifold
of
M,
so
we
can
integrate
a p-form
wp
over
this
set
(using
techniques described
in
the
previous
chapter)
and
call
it
the
integral
of
Wp
over
the simplex
(5P.
Thus
we have defined
Ja
p
wp.
This
is easily generalized
to
the
integration
of
forms over chains. Given a
p-chain
C
p
=
Li
nwf,
we
define
(5.51 )
An
obvious generalization of Stokes'
theorem
tells us
that
{dw={w
J
c
Jac
(5.52)