4th Year Project
Contents
Chapter 1: Morse functions on Manifolds .................................................................................................................... 2
1.1 - The Morse Lemma .............................................................................................................................................. 2
1.2 – Gradient-Like Vector Fields .............................................................................................................................. 6
Chapter 2: Handlebodies .............................................................................................................................................. 11
2.1 – Handle Decompositions .................................................................................................................................. 11
2.2 – Sliding Handles ................................................................................................................................................ 17
2.3 – Cancelling Handles .......................................................................................................................................... 19
Chapter 3: Handlebodies, Homology and Stability .................................................................................................. 22
3.1 – Homology and Handlebodies ......................................................................................................................... 22
3.2 – Morse-Smale Dynamics ................................................................................................................................... 26
Chapter 4: Morse-Witten Homology ........................................................................................................................... 29
4.1 – Relative Homology Approach ........................................................................................................................ 29
4.2 – Geometrical Interpretation .............................................................................................................................. 32
4.3 – Examples ............................................................................................................................................................ 36
Chapter 5: Poincaré Duality ......................................................................................................................................... 42
5.1 – Cohomology Groups ........................................................................................................................................ 42
5.2 – How to prove Poincaré Duality ...................................................................................................................... 43
5.2.1 The Attaching maps of a Handlebody and Intersection numbers ........................................................ 44
5.2.2 Using Exactness to determine the Sign of an Intersection: ..................................................................... 46
5.3 – The Proof of Poincaré Duality......................................................................................................................... 47
References ....................................................................................................................................................................... 50
0811265
Ian Vincent
Page 1 of 50
Chapter 1: Morse functions on Manifolds
We recall the notion of a manifold. In this document, all manifolds considered will be smooth, and
may or may not have boundary.
We first introduce the concept of Morse functions as follows:
Definition 1.1 (Degeneracy of Critical Points)
Let
be a smooth real-valued function
on a manifold
. Recall the notion of a critical point; that is a point
where
( )
( )
( )
) about . Let ( ) be the Hessian matrix
with respect to a local coordinate system (
). We define the critical point
of at with respect to a coordinate system (
to be a
( )
degenerate critical point if
. Otherwise, the critical point is said to be non-degenerate.
In fact, it is an elementary property that the notion of critical points, and degeneracy is welldefined, that is it does not depend on the choice of local coordinate system;
) and (
) at a critical point , and
Lemma 1.2
Given two coordinate systems (
(
)
let ( ) and
be the Hessians of with respect to these coordinate systems respectively.
( )
Then
( )
.
( ) are related by the formula
( )
( ) ( )( )
) to (
) evaluated at
where ( ) is the Jacobian Matrix of partial derivatives from (
. Then we have
( )
( )
( )
( )
Since the Jacobian ( ) of the coordinate transformation at
has a non-zero determinant and
( )
( ), the above equation implies that
( )
( )
and hence
Proof Observe that
degeneracy of
( ) and
is independent of choice of local coordinate system.
Definition 1.3 (Morse Function)
of is non-degenerate.
A function
is a Morse function if every critical point
Remark
Usually, the critical points of Morse functions are required by definition to have
distinct critical values. I do not assume that here in the definition, and instead in Theorem 1.14, we
know that for any Morse function (as defined here) there exists another Morse function ‘arbitrarily
close by’ with the same critical points but distinct critical values. The precise notion of ‘arbitrarily
)-close and will be discussed later.
close’ is (
1.1 - The Morse Lemma
In short, the Morse Lemma states that for a Morse function , there is a local coordinate system
about any critical point so that locally, can be written in a standard form. This is extremely
useful, and immediately has some important corollaries.
0811265
Ian Vincent
Page 2 of 50
Theorem 1.4 (Morse Lemma for dimension )
Let
be a non-degenerate critical point of
) about
. Then we can choose a local coordinate system (
such that the
coordinate representation of with respect to these coordinates has the following standard form
) and is a
where corresponds to the origin (
( ).
constant with
Remark:
First, we recall Sylvester’s Law from Linear Algebra:
Sylvester’s Law
Any symmetric real matrix A has a diagonalisation of the form
where is a diagonal matrix containing the eigenvalues of , and is an orthonormal
square matrix containing the eigenvectors. The matrix can be written
where
is diagonal with entries
or
matrix
transforms to .
, and
√|
is diagonal with
|. Then the
The number in the statement of Theorem 1.4 corresponds to the number of negative diagonal
entries in the Hessian ( ) after diagonalisation. As ( ) is a symmetric matrix, we apply
Sylvester’s Law to see that does not depend on the way the Hessian is diagonalised; that is is
determined purely by the function and the critical point .
Definition 1.5 (Index of a Critical Point) The number in the statement of Theorem 1.4 is
called the index of a non-degenerate critical point . It should be noted that the index is an integer
between 0 and
.
) at the critical point
Proof of 1.4 Choose a local coordinate system (
, where
(
)
(
)
corresponds to the origin
. We may further assume that
, replacing by
( ) if necessary.
Lemma 1.6
There exist real valued functions
the origin so that,
(
)
defined in a neighbourhood of
(
∑
)
in some neighbourhood of the origin and such that
(
)
(
)
Proof This is a fundamental fact of calculus of several variables; for a proof see any
standard textbook on PDEs or multi-variable calculus.
Since the point
(
) is a critical point,
(
)
Therefore applying Lemma 1.6 again to each of the we can find
a neighbourhood of the origin)
(
)
(
)
(
such that
(
)
∑
(
(
)
for each
smooth functions (defined in
)
)
in a neighbourhood of the origin. Therefore
0811265
Ian Vincent
.
Page 3 of 50
(
Setting
)
we get
(
)
)
(
∑
)
(Schwarz 1993)
(
and
(
∑
)
(
(
∑
)
(
)
).
We shall refer to the representation of (
a “representation by a quadratic form” of .
) in (
) as a “quadratic form representation”, or
The idea of the proof is to change the representation of into the quadratic representation in the
standard form mentioned in the statement of the theorem. We proceed by induction on the
number of terms involved in the quadratic form representing .
Let us now compute second order partial derivatives of (
(
If we assume that the critical point
assume that
)
) at the origin. We obtain
(
)
(
is non-degenerate and so
(
)
)
after a suitable linear transformation of the local coordinate system (
(
)
. Since the function
is continuous, we see that
neighbourhood of the origin.
We introduce a new coordinate system (
), where we define
|(
√|
then we may
∑
). Then we see
is non-zero in a
by
)
We can easily compute and show that the determinant of the Jacobian of the transformation from
(
) to (
) at the origin is not zero, so that (
) is certainly a local
coordinate system. The square of
is computed as follows:
(∑
∑
|
|(
∑
)
(∑
∑
{
Comparing the above with the quadratic form representation (
∑
∑
{
0811265
Ian Vincent
(∑
(∑
)
)
)
if
)
if
) of , we see that
if
(
)
if
Page 4 of 50
In the above, we see that the second terms and thereafter are a sum over
so that this part
is simplified to a quadratic form representation of fewer variables than ( ). Therefore repeating
this process, we proceed by induction on the number of variables to prove that we can represent
in a standard form. Then by permuting the as necessary to get all the minuses at the beginning,
we obtain the required result.
From the Morse Lemma, Theorem 1.4, we easily obtain the following:
Corollary 1.7 A non-degenerate critical point is isolated; that is for a non-degenerate critical point
of , there exists a neighbourhood of , where no other critical points of lie inside .
)) so that
Proof Take a critical point . By the Morse Lemma, we can find a chart ( (
{
} so that in
the in the local coordinate system, there exists
has standard form
In its standard form, we see immediately that the only critical point of
the origin 0, corresponding to the point
.
in the neighbourhood
is
Corollary 1.8 A Morse function defined on a compact manifold admits only finitely many critical
points.
Proof We proceed by deriving a contradiction as follows: assume a Morse function
on a
compact manifold has infinitely many critical points
. By compactness, there is a
convergent subsequence ( ) ( ) of this sequence. Let be its limit point. Consider a local
coordinate system ( ) defined on a neighbourhood
converges to the point
subsequence {
Since
(
is
()
well, since
(
. Since the above subsequence {
( )}
, by choosing a further subsequence if necessary, we may assume that the
is contained the neighbourhood
, its partial derivatives
) and
Therefore
( )}
of
( ))
of
.
( ) depend smoothly on the point
. The derivatives
take the value zero and hence these derivatives take the value zero at
is the limit point of
as
( ).
is a critical point of the function . All the critical points of a Morse Function are non-
degenerate, and so by Corollary 1.8, they are isolated. However, the sequence {
critical points converging to a critical point which is a contradiction with
there can only be finitely many critical points.
( )}
consists of
isolated. Therefore
Of course, the next theorem is extremely important for the rest of this document:
Theorem 1.9 (Existence of Morse Functions)
Let be a closed -manifold and let
be a smooth function defined on . Then there exists a Morse function
arbitrarily close to
. [The precise nature of this ‘close-ness’ is explained below.]
The proof of this theorem is long, beyond the scope of this document and hence omitted. I refer the
reader to (Matsumoto 2002), whose proof is relatively easy to follow.
Remark
We make precise the statement
Theorem 1.9:
0811265
Ian Vincent
and
are “arbitrarily close” mentioned in
Page 5 of 50
{ } of each equipped with local coordinates,
For every
, there exists an open cover
such that for each
the following inequalities hold for all
:
If
(

| ( )

|

|
( )|
( )
( )|
( )
( )|
and are two functions satisfiying the above inequalities for a particular
) are (
)-close.
, we say that
1.2 – Gradient-Like Vector Fields
We recall the notion of gradient vector field:
Definition 1.10 (Gradient Vector Field)
Let
be a function defined in a coordinate
) be its coordinate system. The gradient vector field for in is
neighbourhood and let (
the vector field defined by:
Note that for a Morse function in standard form, its gradient vector field is written as:
As the gradient vector field is only defined in specific coordinate neighbourhoods, the idea of a
Gradient-Like vector field is to globalise the notion of gradient vector field to the entire manifold.
Let be a Morse function defined on a closed -manifold. In the following discussion, we always
( ).
assume that is a smooth vector field on , possibly denoted by
For a Morse function
Due to Corollary 1.8, if
If
∑
, for
, we may use the notation
is compact then
is a finite set.
( ), by
we mean
for the set of critical points of
∑
Definition 1.11 (Gradient-like Vector Field)
We say that
Morse function
if the following two conditions hold:
Ian Vincent
.
( )
is a gradient-like vector field for a
1.
away from the critical points of
2. If
is a critical point of with index , then
has a neighbourhood
) such that has the standard form
coordinate system (
( )
and can be written as its gradient vector field:
0811265
on
with a suitable
Page 6 of 50
Geometrically, the first condition says that away from the critical points of ,
direction into which is increasing.
points in the
Naturally, we need:
Theorem 1.12 Suppose that
exists a gradient-like vector field
is a Morse Function on a compact manifold
for .
. Then there
The proof is beyond the scope of the document and so omitted. The strategy is; given an open
cover of , define a vector field on each
by the gradient vector field and then find
∑
associated
-bump functions . Then after dealing with technicalities, the vector field
is a gradient-like vector field for . See (Matsumoto 2002) once more for details.
Remark
To avoid the business of dealing with
-bump functions, it is possible to define the
Gradient-Like Vector field in a coordinate-free way as follows, at the expense of knowing the
existence of a Riemannian Metric on the manifold:
Alternative Definition (Coordinate-free Gradient-like Vector field)
Given a manifold
, equipped with a Riemannian Metric ⟨ ⟩, for
this determines a map
⟨ ⟩
mapping
[This map is a linear injective map (by definition of the metric ⟨ ⟩. Moreover, since
and
are two vector spaces of the same finite dimension, we conclude that is an
isomorphism].
We then define the gradient-like vector field
( ) to be the unique
such that
⟨ ⟩
, the function mapping any vector
to the directional
derivative of at in the direction .
We now use Gradient-like Vector fields as a useful tool to prove statements to do with Classical
] be a real interval. Use the notation
Morse Theory. Let
be a Morse function and let [
{
}.
( )
[ ]
Theorem 1.13 If has no critical value in the interval [
( ) [ ]
product
], then
[
]
is diffeomorphic to the
Proof Let be a gradient-like vector field for . As
away from critical points for , we
{critical points of } , which is an open subset of
define a new vector field
on
by
Since by hypothesis, [ ] contains no critical points of , it is in
the domain of the vector field . Consider the integral curve
( ) of
( ) . Using the
which starts at a point
of
definition of velocity vector, we obtain:
( ( ))
Thus the curve
0811265
Ian Vincent
( )
( )
( )
( )
( ( )) maintains constant speed 1 and
Page 7 of 50
( ) travels “upward” as defined by the value of . Since it starts at the level
( ) [
will reach the level
at the time
. Define a map
( )
( )
We can show that
is a diffeomorphism by using the facts that
at time
]
[
]
, it
by
( ) depends smoothly on both
and and the two distinct integral curves do not meet (by uniqueness of integral curves).
] as follows: given a point
( ) [
Define a map
[ ]
[ ] we know from the
existence and uniqueness of integral curves that there is a unique integral curve of on [ ]
[
]. Let
( ). Then
( ) and define ( )
passing through at some time
( ). This map is smooth since the integral curve depends smoothly on the point and hence
and both depend smoothly on .
Note that
(
)
( ( ))
(
) and
( )
(
)
( )
so
and hence
is a diffeomorphism.
], and with the observation that
( ) [
Therefore we have proved that [ ]
[
]
( ) [ ], we complete the proof of the theorem.
( )
Notice in the Statement of 1.9, that in some sense, Morse functions are ‘dense’ that is ‘in every
neighbourhood of any function, one encounters Morse functions’. With this notion, we may in fact
assume (with no loss in generality) that given a Morse function , every critical point has a distinct
critical value:
Theorem 1.14 Let
be a Morse function on , and let
be its critical points.
̃
Then there exists a Morse function whose critical points
have the property ̃( )
̃( ) for
)-close.
. Moreover, for any
, we can find such an ̃, so that and ̃ are (
Proof We use a similar argument here as hinted in the proof of the existence of a Morse function
(Theorem 1.12).
( )
We assume that the critical value of at the critical points and are the same, ( )
and try to modify slightly. By the Morse Lemma we can choose a local coordinate system
(
)
about
and
write
in
the
standard
form
Let
be a gradient vector field for
(
)
with respect to this coordinate system. Then
(
)
(
)
(
)
For a sufficiently small
, consider the (closed) -discs of radii and
and their respective
images
and
in the local coordinate system above, centred at . It follows from the above
(
) n the region
equality that
int ( )
Denote by the compact set
and by the interior int
, so that is an open set containing a
). We extend to a
compact set . Consider a
-bump function
with respect to (
smooth function on the entire manifold by setting
outside , and denote the extended
̃
function again (for convenience) by . Define a function by ̃
where
is a small
0811265
Ian Vincent
Page 8 of 50
̃ outside , the critical points of and ̃ coincide there. Similarly, since
on the interior int( ), we see that the point is the only critical point of both and ̃ in
this region.
real number. Since
Therefore the only place where ̃ possibly has a different set of critical points from
between
As
is the region
. We compute differences of the first partial derivatives of
and ̃
̃
|
| |
|
and
is bounded for each by the Extreme Value theorem (applied to the compact region
follows that by making
), it
arbitrary small, the difference
|∑ (
)
∑(
̃
) |
is made arbitrarily small.
Since ∑
(
)
between
∑(
and so ̃ has no critical points between
and
̃
and
, by taking
sufficiently small, then
)
, and so
and ̃ have the same critical points.
Observe that the non-degenerate critical points for and ̃ also coincide, we see that ̃ is a Morse
̃( )
( )
( ) therefore, even though ( )
function. Furthermore, we have ̃( )
̃( ).
( ), ̃( )
By repeating the above argument as necessary for pairs of critical points whose critical values
coincide, (since there are only finitely many critical points for the Morse function ) we obtain that
̃ is a Morse function with distinct critical values.
To prove that ̃ is (
Again, since
)-close to , we see that
̃
( )
( )| |
|
is bounded on the compact set
the difference in the second derivatives of
( )|
, by making
arbitrarily small we can make
and ̃ arbitrarily small.
We finish the chapter with the following useful Lemma:
Lemma 1.15 (Product of two Morse functions)
For compact manifolds and with
and
, let
and
be Morse functions. Define a function
by the formula
(
)(
)
where and are positive numbers. Then for and sufficiently large, is a Morse function
)
whose critical points are of the form (
where is a critical point for and is a
critical point for ; its index is equal to the sum of the indices for and .
Proof Let (
about and
0811265
) be a point of
and let (
respectively. Then we can take (
Ian Vincent
) and (
) be local coordinate systems
) as a local coordinate system
Page 9 of 50
about (
) in
. Observe that
)(
(
) and
(
)(
)
Since and are compact, by the Extreme Value Theorem, and are bounded and so there
| | and
| | . Then the
exists and
large enough in
and
respectively so that
following two conditions become equivalent:
1.
(
2.
( )
)
(
)
for all
for all
and
( )
and
for all
.
is of the form (
From the above we immediately have that a critical point of
points and for and respectively, and that
)(
(
Therefore the Hessian of
) and
(
)(
) for critical
)
is of the form
(
)
(
(
)
( )
(
(
) (
)(
)
( )
( )
and so
Therefore since and are Morse functions, by definition
)(
)
have chosen and sufficiently large so that (
so is a Morse Function.
)
)
( )
( )
. Therefore
(
( ) and we
)
, and
) are the local
Applying the Morse Lemma to , we may assume that (
(
)
) and
coordinates about
to express
in the standard form. It follows that (
(
) are the local coordinates about and to express and in the standard form, and so
) is equal to the sum of the indices of and
by equating both sides, we see that the index of (
.
0811265
Ian Vincent
Page 10 of 50
Chapter 2: Handlebodies
In this chapter, we use Morse functions and gradient-like vector fields to decompose manifolds
into objects called handlebodies. This technique is the main focus of Classical Morse Theory.
2.1 – Handle Decompositions
During this chapter, we will meet the idea of ‘gluing’ two manifolds with boundary along their
boundaries. This leads to some technical issues where the gluing must be done in such a way to
preserve the smooth nature of the resulting manifold. The following theorem makes precise the
process:
Theorem 2.1 (Gluing Manifolds with boundary) Let
and
be manifolds with boundary,
and let
be a diffeomorphism between the boundaries. Then we can construct a new
manifold
by identifying each point
with the point ( )
. The resulting
manifold is unique up to diffeomorphism. (It is permissible to only identify a few of the connected
components of the boundary, instead of the entire boundary.) See the diagram below:
The next theorem is more delicate that its predecessor, which says that this gluing is preserved
under diffeomorphisms of each component:
Theorem 2.2 (Gluing Diffeomorphisms)
Let
and
be the manifolds
obtained by gluing manifolds with boundary (where
and
are
diffeomorphisms). Suppose that we have diffeomorphisms
and
such that
( )
( ) for every point in
. Then there exists a diffeomorphism
obtained by identifying and along the boundary.
( ) so
Note that the identifications form an equivalence class:
( )
( )
equivalence classes in its image; that is
( ( ))
0811265
Ian Vincent
must respect
Page 11 of 50
I do not wish to go into detail with the proofs of these theorems. Intuitively, I think it is clear that
the process can be done, but in order to construct an explicit formula for the diffeomorphism in
theorem 2.2 is a long-winded process. Regardless, for a proof, see (Milnor 1965) page 25.
Now with some technicalities dealt with, we can proceed with the first theorem, which is the
fundamental concept of handle attachments:
Let
be a closed manifold and
a Morse function on . We use the notation
{
}
( )
for a value in the image of . We will investigate how
changes at the parameter is varied.
Theorem 2.3 If
and write
has no critical values in the interval [
.
] then
and
are diffeomorphic,
Proof Since is compact and is smooth, the minimum and maximum values for , and are
obviously critical values and so the interval [ ] does not contain either or . We may therefore
prove Theorem 2.3 under the assumption
.
{
( )
As before, denote the set [ ]
assumption [ ] contains no critical points of .
}. Clearly we have
[
]
. By
By Corollary 1.8,
has only a finite number of critical points. We may therefore assume
that has no critical points in [
] for a small enough positive number . By Theorem 1.13,
[ ] in this case. We also note that [
[
] is diffeomorphic to the product
]
has no critical points in [
[
] so that
] and hence by Theorem 1.13,
[
] is
[
]
diffeomorphic to the product
. Therefore we have a diffeomorphism
[
where we may assume that the restriction of
]
[
]
to the level set
is the identity map.
). We now define another diffeomorphism
-bump function with respect to (
to “glue together” the
[
]
[
] using Theorem 2.2 and
diffeomorphism on [
to get the required diffeomorphism
] and the identity map on
. (I hope the idea is clear; Theorem 2.2 guarantees this will work, but to form an explicit
formula is complicated and provides little in the way of clarity).
Let
be a
From the above theorem, the problem we need to investigate is the change in shape of
has the
parameter passes through a critical value. From Theorem 1.14, we may assume that takes
distinct critical values at distinct critical points. We also notice that has a finite number, say
, of critical points. By permuting the critical values of so that they lie in ascending order, we
can label their corresponding critical points as
; so ( )
.
( ) whenever
Let
( ), the critical value of
when
.
We begin by analysing the changes of
. It is immediate that
around
and
when
, and similarly,
:
By Theorem 1.14, we may assume that is the only point which gives the minimum value. Using
the Morse Lemma, we can write in the standard form
. Observe that the
index of is necessarily zero, because is the minimum of on .
0811265
Ian Vincent
Page 12 of 50
Let
be a sufficiently small number (for example, so
{(
the Morse Lemma we can express
)
). Clearly,
but from
} ; that is
is
diffeomorphic to the -disc
. The standard form shows that takes the minimum value
the centre of the disc, and attains its maximum on the boundary of the disc.
Observe that this argument can be repeated whenever
disc so
is diffeomorphic to the disjoint union
critical point of index
is called an
at
is a critical value of index 0; we add an . The -disc appearing at each
-dimensional -handle, or -handle for short.
Similarly, at the maximum value , by Theorem 1.14 we may assume that
is the only point
(
)
where
. It follows immediately from the Morse Lemma and maximality of that in
standard form,
and hence the index of
is necessarily . Using these
)
local coordinates we see
{(
, which corresponds to the
complement of the
diffeomorphic to
-disc
of radius √
in
. In this case, the boundary
is
.
As increases from
and passes , we see that the boundary of
is capped off with an
-disc, and forms the completion of a compact manifold without boundary. More generally,
whenever passes any critical value of index , the -handle caps off a connected component
of the boundary
; this component is diffeomorphic to
.
We now have a complete idea of what happens around critical points of indices and . Next we
consider the changes of
near a critical point of a general index as passes through the
corresponding critical value .
First, we use the Morse Lemma to take a local coordinate system about the critical point
have in the standard form:
.
The situation will around
idea:
Around
0811265
will be clarified in Theorem 2.5. The image below shows the general
, the darkly shaded area in the above diagram corresponds to
setting
; that is
the inequalities
:
Ian Vincent
and
which we get by
. The lightly shaded area corresponds to
Page 13 of 50
{
where is another positive number much smaller than . This lightly shaded area is called an dimensional handle of index , or more briefly, an -dimensional -handle. This handle is
diffeomorphic to the direct product
.
Definition 2.4 (The core and cocore of a -handle)
) ∑
{(
The -disc
} is called the core of the -handle, and the (
) ∑
{(
)-disc
} intersecting the core is called the cocore.
The name cocore indicates the thickness of the -handle. The core and cocore intersect transversely
at the origin, which is exactly the critical point . See Definition 2.10 later for the concept of
‘transverse intersection’.
We attach a -handle
to
as shown by the thick lines in the diagram above. Then
(
is diffeomorphic to the union
Theorem 2.5 The set
. That is
Strictly speaking, the space
).
is diffeomorphic to the manifold obtained by attaching a -handle to
(
).
with a -handle attached is not “smooth” at the “corners” of the
boundary where the handle meets
. We must “smooth” out these corners to take a
manifold
as shown in the diagram below. A more accurate statement for Theorem 2.5 therefore,
should be that
is diffeomorphic to the smoothed-out manifold .
Notice that in the diagram,
corresponds to the set {(
) ∑
∑
}
Idea of Proof of Theorem 2.5.
The idea is similar to the proof of Theorem 2.3. We again use
a gradient-like vector field for . One can see in the diagram above that the vector field , after
0811265
Ian Vincent
Page 14 of 50
leaving the boundary
reaches the boundary
let
flow along
and
We
of
of
continues to flow upward (that is, with respect to ) until it
. We may multiply the vector field by a suitable function, and
so that it will match
after a certain period of time. This will show that
are diffeomorphic.
will
not
worry too much about the non-smooth edges of the boundary
, and we will pretend that it is in fact a smoothed-out manifold
in our
discussion.
By looking at the standard form of at
and the cocore is diffeomorphic to a (
, we see that the core
)-disc.
is diffeomorphic to a” -disc,
The core of a -handle is -dimensional and the cocore is -dimensional, so that there is no
downward direction; every direction points upward. On the other hand, the core of an -handle is
-dimensional so that any -handle “faces down”.
Definition 2.6 (The attaching map of a -handle.) One attaches a -handle
pasting
along the boundary
to
by
(indicated by the thickened lines in the above
diagram). In order to describe the handle-attaching accurately, one must specify a map
indicating where each point of
corresponds to in the boundary
. The map
is a
smooth embedding which we call the attaching map of the -handle. The boundary
of the core
)-dimensional sphere
disc is a (
, and is called the attaching sphere. [Note that in the case
, we find that
and define instead the “attaching map” to be the disjoint union
.]
An attaching map is an embedding
into the boundary
of a thickened (
)-sphere
.
Example 2.7 (A 3-dimensional 1-handle and a 2-handle). In the diagrams below, a 3-dimensional
1-handle and a 2-handle are depicted, respectively. The picture of the 1-handle justifies the name
“handle”. The 2-handle is depicted as a thickened upside-down “bowl”.
0811265
Ian Vincent
Page 15 of 50
Definition 2.7 (Handlebody) A manifold (with boundary in general) obtained from
by attaching
handles of various indices one after another
(
)
(
) is called an dimensional handlebody.
More precisely, a handlebody is defined in three steps as follows:
1. A disc
is an
-dimensional handlebody.
2. The manifold
attaching map of class
).
by (
(
3. If
(
) obtained from
,
)
by attaching a
, is an
is
an
-dimensional
(
-handle with an
-dimensional handlebody, denoted
handlebody,
then
the
manifold
)
obtained from
by attaching a
-handle
with an attaching map
)
of class
, is an -dimensional handlebody, denoted by (
[Strictly speaking, the manifold is “smoothed out” each time a handle is attached, so that the resulting
handlebodies are always considered smooth manifolds.]
Remark
It is worth noting that the attachment of a -handle is nothing but a disjoint union so in
the case of index
, there is no need to specify the attaching map . Therefore in this case, the
) does not have meaning by itself, but in the
attaching map
in the notation (
notation is traditionally kept as a formality.
Theorem 2.8 (Handle decomposition of a manifold)
When a Morse function
is given
on a closed manifold , a structure of a handlebody on is determined by . The handles of the
handlebody correspond to the critical points of , and the indices of the handles coincide with the
indices of the corresponding critical points.
In other words, can be expressed as a handlebody. When a manifold is expressed as a handlebody,
it is called a handle decomposition.
Proof We may assume that all the critical points of the given Morse function
have distinct
critical values. Permute the critical points in such a way that their critical values are in ascending
order and name them
.
Let be the index of the critical point . Fix a gradient-like vector field on for . The proof now
proceeds by induction on the subscripts of the critical points . Let be the value of at , and we
will show that
is a handlebody.
First, for
, the index of the critical point is , as gives the minimum value of . Therefore,
is diffeomorphic to the -dimensional disc
. From part 1 of Definition 2.7,
is indeed an
-dimensional handlebody, and the statement for the induction is proved for
. In this case,
is a -handle itself.
Next, we make the inductive assumption that
prove that
is a handlebody (
diffeomorphic to a manifold obtained by attaching a
0811265
Ian Vincent
) and we will
is a handlebody (
) . Recall from Theorem 2.5 that
is
-handle. The attaching map
Page 16 of 50
of this handle is determined naturally without ambiguity from the discussion following
Theorem 2.5.
] contains no critical values, so from Theorem 2.3,
The interval [
to
. By consulting the proof, this diffeomorphism is given by letting
gradient-like vector field
until it matches
From the induction hypothesis that
. Let
is a handlebody
be such a diffeomorphism.
(
diffeomorphic to the same handlebody. Therefore the manifold
attaching a
is diffeomorphic
flow along the
), we see that
, which is obtained from
is
by
-handle is also diffeomorphic to a handlebody from part 3 of Definition 2.7.
This completes the proof of Theorem 2.8. However, since it is important, let’s investigate the attaching
map of the new -handle in more detail:
) so that the above
From the induction hypothesis,
is the handlebody (
) to
diffeomorphism
can be seen as a diffeomorphism from (
. Precisely
), but rather is
-handle is not attached directly to the handlebody (
and its attaching map
is naturally determined. Therefore,
) by the diffeomorphism , the is identified with the handlebody (
(
) by the composition map
is attached to the handlebody
{ (
)} If we denote this composition by , then
is
speaking, the
attached to
when
handle
indeed the handlebody
(
)
(
)
Remark
The above proof implies the following: when a handle decomposition of a manifold
from a Morse function
,
is obtained
1. The order of the handles and their indices are determined by the critical points of , and
)of the handles are determined by a gradient-like vector field
2. The attaching maps (
for (since is determined by ).
2.2 – Sliding Handles
In fact, there is some flexibility to the attaching maps of handlebodies; intuitively, ‘sliding the
handles around’ does not change the diffeomorphism type of the handlebody. By ‘sliding’ we
actually mean composing the attaching map of a -handle with an isotopy of the boundary of
the subhandlebody
formed by all the previous handle attachments. First we define ‘Isotopy’,
‘Transversal Intersection’ and make precise the notion of ‘General Position’:
Definition 2.9 (Isotopy)
Let be a -dimensional manifold. The set { } is called a family
of diffeomorphisms if, for each real number in the open interval , a diffeomorphism
is
assigned. The family { } is called an isotopy of if the following two conditions are satisfied:
1. The open interval contains the closed interval [ ], and is constantly the identity map
on for
. Also, for all
, we have
where is some diffeomorphism of
.
2. The map
defined by ( ) ( ( ) ) is a diffeomorphism. That is the
map depends on the parameter smoothly in this sense.
0811265
Ian Vincent
Page 17 of 50
Definition 2.10 (Transversal Intersection) Suppose that there are an -dimensional manifold
and a -dimensional manifold in a -dimensional manifold such that
. We say that
and intersect transversely at a point of , and write
, if for every point
the
tangent spaces
and
have the property
.
In particular, for two transverse submanifolds
( )
Lemma 2.11 (General Position)
Let and
respectively, in a -dimensional manifold .
I.
II.
If
If
and , we have
( )
(
)
be compact submanifolds of dimensions
, then there is an isotopy { } of such that
then there is an isotopy { } of such that
transversely in finitely many points.
and
and ( )
.
and ( ) intersects
Proof
Omitted. See (Milnor 1965), pages 46, 47 for a Proof.
Theorem 2.12 (Sliding Handles)
attaching map
further that an isotopy { } of
Suppose that a -handle
is attached by an
on an -dimensional manifold with boundary, and suppose
is given. Here, we assume that
and
. Then the
new handlebody
of the handle by the isotopy, is diffeomorphic to the original
handlebody
(before the -handle was ‘slid’). Notice
that the only difference between the two handlebodies is that the
attaching map has been replaced by another attaching map
,
where
is some diffeomorphism. In other words, the handle
has been deformed so that it lies somewhere else on the
boundary
. The idea is given in the diagram to the right:
Proof The proof is long and not especially enlightening. Therefore it is
omitted.
From the theorem above, we have the following Corollary, which is the main result of this section:
Corollary 2.13 (Rearrangement of Handles)
Any handlebody can be modified in such a
way that the new one is described as follows:
It is constructed first from a disjoint union of -handles, and then a disjoint union of 1-handles are
attached to them, and then a disjoint union of 2-handles are attached, and so forth, so that handles
are attached in ascending order of indices. Rearranging the handlebody in this manner is
equivalent to modifying the original Morse function
(from which the original
handlebody is constructed) via compositions of isotopies to another Morse function
(from which the modified handlebody with these required properties is constructed.)
The diagram below illustrates the situation of the critical points of the Morse function . As we see,
is obtained from the handlebody
which consists of handles of indices less than or equal to
all attached, by attaching a disjoint union of copies of -handles
0811265
Ian Vincent
Page 18 of 50
at the same time. Here, is the number of handles of index .
Proof (Sketch)
The proof is a sequence of technical Lemmas which all involve performing
isotopies to the gradient-like vector field for . The argument is too long to contain here, and I
refer the reader to (Matsumoto 2002), pages 105-120. As a sketch, the idea is to is to use Theorem
2.12 repeatedly to slide the handles into a particular order:
Let
be the sub-handlebody consisting of handles of index at most . The goal is by sliding, we
ensure that for every
, every -handle is attached to
and all the attachments of handles to
happen disjointly (that is, we slide the -handles off each other). Note that here;
whenever I said ‘sliding’, what I actually meant is ‘composing the attaching maps with an isotopy’,
but in my opinion greater understanding is achieved by thinking of it as ‘sliding’.
2.3 – Cancelling Handles
In this section, we describe a situation where two consecutive handle attachments can be ignored;
that is after attaching two consecutive handles of specific indices under a specific family of
attaching maps, the result is diffeomorphic to the original handlebody.
Definition 2.14 (Belt Sphere)
Consider the situation where a -handle is attached to an
dimensional manifold with boundary. Set
.
In this case, the boundary
of the cocore
of this -handle. By definition, the belt sphere is an (
as in the diagram below:
-
of the -handle is called the belt sphere
)-dimensional sphere embedded in
Theorem 2.15 (Cancelling Handles) Suppose that a manifold
is obtained from an
dimensional manifold with boundary by attaching a -handle, and suppose further that a
)-handle:
manifold
is obtained from
by attaching a (
0811265
Ian Vincent
Page 19 of 50
If the belt sphere
of the -handle and the attaching sphere
handle intersect transversely at a single point in the boundary
of , then
to .
)of the (
is diffeomorphic
Proof Omitted, see (Matsumoto 2002) pages 120-125.
The statement of Theorem 2.15 may seem more reasonable after looking at some examples:
1. The first diagram below illustrates the case where a -handle and a 1-handle are attached to
. In this case, the belt sphere of the -handle is a 2-dimensional sphere
which is the
surface of a 3-dimensional disc
(shown in blue) and the attaching sphere of the 1-handle
is two points (shown in red), one of which lies on the surface of the 0-handle. As we see
from the diagram, the union of the 0-handle and the 1-handle can be squashed into .
2. Next, the diagram below illustrates the situation where a 1-handle and a 2-handle are
attached. The attaching sphere (red) of the 2-handle and the belt sphere (blue) of the 1handle intersect transversely at a single point. In this case, the union of the 1-handle and
the 2-handle can be swallowed into as well.
From the theorems above to do with sliding and cancelling handles, we have the following
theorem, allowing us to assume that there is only one critical point of index and only one of
index :
Theorem 2.16 Let be a closed -dimensional manifold. If is connected, then there is a Morse
function
on with only one critical point of index and one critical point of index .
(The Morse function may have as many other critical points of other indices with
).
Proof By Corollary 2.13, for some Morse function
, we can assume all the critical points
of index 0 take the same critical value , all the critical points of index 1 take the same critical
) and so forth.
value (
0811265
Ian Vincent
Page 20 of 50
The handle decomposition with respect to such a Morse function is of the form:
and
If
(a disjoint union of handles)
by attaching some -handles.
is obtained from
is not connected, then it consists of more than one connected component
then, since we attach handles of indices 2 or higher to obtain
from
, but
, and the attaching
sphere of a handle of index 2 or higher is connected (it is diffeomorphic to
), it must be
attached to a single connected component. Hence the number of connected components does not
increase or decrease after attaching handles of index 2 or higher, and hence if
is
disconnected, then is disconnected. To avoid a contradiction, we must have
connected.
Suppose the handle decomposition of
has more than one connected component. From the
above consideration, they must all be “bridged” together and become connected as a whole after
attaching the 1-handles. One
among the expression above is bridged to another
by a 1(
handle
. Since the attaching sphere
points) of this one handle intersects the
belt sphere
of the 0-handle
at exactly one point, by Theorem 2.15 (Cancelling Handles),
the critical point corresponding to the 0-handle
and the critical point corresponding to the 1handle
are cancelled out together. Repeating the argument, we obtain a Morse function
with only one critical point of index .
Next, we consider the function with the sign reversed,
function obtained from
by setting (– )( )
has index
( ) where
and – , the index of a critical point
sets of critical points are the same for
critical point of – ,
. The function –
is the Morse
( ). Although the
of
is , then as a
.
In terms of handle decompositions, the roles of “core” and “cocore” are interchanged, and a handle of
becomes an (
)-handle of – .
Using the same argument as above and Theorem 2.15, we perturb – and reduce the number of handles to 1. Then it can be considered that the number of -handles of was reduced to . In this
case, the number of -handles of – (that is, the number of -handles of ) stays unchanged and is
1, so that the perturbed has only one 0-handle and one -handle.
0811265
Ian Vincent
Page 21 of 50
Chapter 3: Handlebodies, Homology and Stability
3.1 – Homology and Handlebodies
In the previous chapter, we saw how to decompose a closed manifold into a series of handle
attachments. First, we must clarify the relationship between this handle decomposition (which
gives rise to a cell decomposition of the manifold) and the homology of the manifold. The idea is
that the handle decomposition is akin to dividing up a manifold into a cell complex, which we
then use to compute the homology of the manifold.
In order to prove the theorem, we must first define the notion of Mapping Cylinder:
Definition 3.1 (Mapping Cylinder) For a continuous map
between topological spaces,
[
],
[
],
the space obtained from by attaching
by identifying the point “at the
[ ] and the point ( ) of for each point
bottom” ( ) of the direct product
is called
the mapping cylinder of , and is denoted by
.
Lemma 3.2
The mapping cylinder
of
the inclusion
is a homotopy equivalence.
Proof We construct a continuous map
is homotopy equivalent to . More precisely,
as follows.
[ ], we set ( )
Looking at the structure
for a point of , and set ( )
[ ]. Then it is clear that a continuous map
( ) for a point ( ) of
is defined. We
now show that
and
are homotopy inverses of each other. It is clear that
.
To show
, we construct a homotopy
[
]
from
to
as follows.
[ ].
) for a point ( ) of
Namely, set ( )
for a point of and set (( ) ) (
[ ], and hence, a
) is a point of
In this expression is a parameter of the homotopy and (
point of
.
When
, we easily see that
have shown
.
gives the identity map on
and gives
when
, so we
[ ] in
In short, is a continuous deformation which collapses the product part
down to
{ } gradually. Hence gives a homotopy equivalence
, and also we see that and are
homotopy inverses of each other.
It is best to see Lemma 3.2 in action with a simple example:
0811265
Ian Vincent
Page 22 of 50
Example 3.3 The -disc
is homeomorphic to the mapping cylinder
of the map
{ }, which collapses the (
)-sphere
to a single point . By Lemma 3.2,
is homotopy
equivalent to { } so
is homotopy equivalent to a point { }.
[Of course, this example is so easy the application of Lemma 3.2 seems unnecessary; one could
easily prove
is homotopy equivalent to a point without it]
The main result of this section is the following theorem:
Theorem 3.4 (Handlebodies and Cell Complexes) Let be an -dimensional handlebody. If the
largest index of the handles contained in is , then is homotopy equivalent to an -dimensional
cell complex . More precisely:
1. There exists a continuous map
from the boundary
of to such that is
homeomorphic to the mapping cylinder
of . Therefore by Lemma 3.2,
.
2. There is a one-to-one correspondence between the -handles of and the -cells of .
Proof
The idea is as follows: we reduce the radii of the cocores of handles smaller and smaller, and
finally shrink the handles to their cores. The cell complex is one obtained by attaching the discs
of cores of handles one after another.
By Corollary 2.13 (Rearrangement of Handles), we can assume that
following form:
(
where
)
represents a -handle
Specifically,
(
)
. The notation
(
is a handlebody of the
)
denotes the th -handle.
is constructed from a disjoint union of -handles
by attaching a disjoint
union of -handles and so on.
Theorem 3.4 is proved by induction on the maximal index of handles contained in . If
,
then
,
copies of -discs. If we regard the set of
points {
} as a dimensional cell complex , then by Example 3.3, is homeomorphic to the mapping cylinder
{
of the map
} which collapses each sphere to a
single point.
Assume Theorem 3.4 is proved for handlebodies consisting of handles of indices less than or equal
to
, and prove it for a handlebody whose maximal index of handles contained in is .
Let
be the subhandlebody of
(
form
consisting of all handles of indices less than . Thus
) . By the induction hypothesis, there is a cell complex
continuous map
such that is homeomorphic to the mapping cylinder
simplicity, we now assume that there is only one -handle attached to :
has the
and a
of . For
The handle
can be regarded as a mapping cylinder. Namely, if
{ }
{ }, then
)
denotes the map which sends any (
to the core ( )
is homeomorphic to the mapping cylinder
of . By Example 3.3, regard
as a mapping
0811265
Ian Vincent
Page 23 of 50
cylinder of the second component
by
of this mapping cylinder.
; then
can also be regarded as the product
Our strategy for the proof of Theorem 3.4 is to construct an -dimensional cell complex
{ } and construct a continuous map
and
from and .
from
The attaching map
regarded as a submanifold of
can be
is a smooth embedding, so that
via . We make this identification in what follows.
{ } is also a submanifold of
Then,
{ }
with the attaching map
.
. We denote the restriction of
on
{ }
To simplify the notation, we set
(
of
can be
); then the boundary
decomposed as
Our
desired
continuous map
{ }, where
the map we used when the handle
is defined on the portion
by
{ } is regarded as an -cell in (attached to ). Also, is
is regarded as a mapping cylinder .
)-manifold with boundary, and consider the collar
To define on , we regard as an (
neighbourhood (here it is a closed collar neighbourhood) of the boundary
in
.
Thus we have
[
] and
{ }
.
{ } of the map
may be identified with
with the collar
Let be the mapping cylinder of the restriction |
which we used to identify the handle with a mapping cylinder. Then
, and is homeomorphic to
(which is
attached from outside). Let
(
)
] to
[
be this homeomorphism. Note that by mapping
{ }
[ ], we have a natural map
cylinder
{ }(
[ ]
) we have
) we have
id and on
Now, a desired continuous map
( )
0811265
Ian Vincent
is defined on
{
[
. On
|
] inside the mapping
{ }(
[ ]
.
as follows:
( )
( )
(
[
])
Page 24 of 50
Since the two maps
and coincide on the intersection
RHS, a continuous map | on is well defined.
{ } of the two regions in the
The map
coincides on the intersection
.
defined on each portion of the decomposition of
, so we have obtained the desired continuous map
From the construction of , it is clear that is homeomorphic to the mapping cylinder
, and that
there is a one-to-one correspondence between -cells of the -dimensional cell complex and handles of .
Remark
The cell complex in the proof of Theorem 3.4 is considered to be embedded in the
handlebody from its construction. In particular, if the handlebody is a closed manifold, then
and can be identified. For, if
then the mapping cylinder of the map
is nothing but
itself. For example, as we can see the -sphere
regarded as an -dimensional cell complex
consisting of a single 0-cell (a point) and a single -cell whose boundary is identified to that point.
Now we gone to the effort to prove Theorem 3.4, we can use handlebody theory as a link between
the homology and Morse functions of the manifold. A classical result of Morse Theory is the
following:
Theorem 3.5 (Morse Inequality)
Let be a closed -manifold, and
a Morse function
on . For the number of critical points of index and the -dimensional Betti number ( )
( ( )) of , the following inequality holds:
( )
Proof Let
be a Morse Function on a closed
of critical points of of index .
-manifold
. We denote by
the number
Consider the handle decomposition defined by . Then by Theorem 3.4 (Handlebodies and Cell
Complexes) and the above remark, can be identified with a cell complex , and there is a one-toone correspondence between cells contained in and the handles of . In particular, the number
of -cells of equals the number
of -handles of .
Consider the Cellular Chain Complex of
(
)]:
[here we use the notation as in (Hatcher 2001);
( )→
( )→
→
For each , the rank of ( ) is equal to the number
and ( )
( )
( )
of -cells of . Denote by
( )
Since the th dimensional homology group ( ) is obtained from a subgroup
( )
( ) by taking the quotient by a smaller subgroup ( )
( ), we have
( )
( )
( )
by the identification of and , we have
( )
( )
( )
( ) for all . This proves the Morse inequality.
and therefore we obtain
0811265
Ian Vincent
Page 25 of 50
( )
Observe that the Betti numbers are determined by uniquely; from this inequality we see that
the number of critical points of any Morse function on is restricted by the ‘shape’ of . In
particular, if ( )
, then a Morse function on must have at least one critical point of index .
3.2 – Morse-Smale Dynamics
We now focus our attention away from handlebodies temporarily and back to Morse functions, or
more specifically, to their gradient flows.
As usual, let
denote by
be a Morse function, and let
( ) the flow on
integral curve of –
( ) be a gradient-like vector field of . We
( ) is the
determined by – , that is for fixed , the curve
starting at the point
, so
( )
,
(
( ))
( )) . From the
(
theorem of existence and uniqueness of solutions to ODEs, we see that (
diffeomorphism on
.
)
( ) is a
Example 3.6 To familiarise ourselves with the notion, consider the
sphere
embedded in
in the usual way, and
be the
)
Morse function defined by (
the ‘height’ function. Then
is pictured as shown in blue, and ( ) gives, for a given , the
point on the integral curve (red) of
based at at time .
Definition 3.7 (Stable/Unstable Manifolds of a critical point)
Suppose
( ). Fix a gradient-like vector field on
function and
and
by
the flow of
is a Morse
and as before denote
generated by – . We set
( )
(
)
{
( )
}
(
) is called the stable manifold of
(
) is called the unstable (relative to the
and
}.
)
( ) {
gradient-like vector field ), we set (
The following diagram illustrates the concepts defined above:
0811265
Ian Vincent
Page 26 of 50
Lemma 3.8
Let
and
. For sufficiently small
embedded in the level set {
be the index of a critical point
for a Morse function
) is a sphere of dimension
, the set (
smoothly
}.
Proof Use the Morse Lemma to pick local coordinates
{| |
sufficiently small so that in the neighbourhood
form
∑
In the more concise notation
(
| |
) adapted to . Fix
}, the vector field has the
∑
.
A trajectory, (or flow line) ( ) of – which converges to as
must stay inside for all
sufficiently large (and negative). Inside , the only such trajectories have the form
, and they
}. Moreover, since is strictly decreasing
(
) {
| |
are all included in the disc
(
)
(
) . Being the
on nonconstant trajectories, we deduce that if
then
) is homeomorphic to
boundary of an embedded disc, we conclude that (
.
( ) is a smooth manifold diffeomorphic to
Proposition 3.9
smooth manifold diffeomorphic to
.
Proof We need only prove the statement for the unstable manifold
(
)
(
).
like vector field for
and
Using Lemma 3.8 fix a diffeomorphism
coordinates of , we can define
(
(
). If (
, while
( ) is a
( ) since – is a gradient-
) with
denote the polar
( ) by
)
( ( ))
From the definition of ( ), it is a diffeomorphism. Being the composition of diffeomorphisms,
we therefore see that is a diffeomorphism as required.
The following Theorem allows us to restrict the theory to just dealing with the case where the
unstable manifolds and stable manifolds intersect transversely.
Theorem 3.10 Suppose
gradient-like vector field for
intersects the stable manifold
is a Morse function on a compact manifold . Then there exists a
(
)
such that for any ,
, the unstable manifold
(
) transversally.
Proof Since it is a lengthy proof, we refer the reader to (Nicolaescu 2007), pgs 57-60.
For convenience, we package these above properties into one definition:
Defininiton 3.11 (Morse-Smale Pair, Morse-Smale Vector field) If
is a Morse function
and is a gradient-like vector field for , such that
( )
( )
Then we say that ( ) is a Morse-Smale pair on and that is a Morse-Smale vector field adapted
to .
0811265
Ian Vincent
Page 27 of 50
From Corollary 2.13 (Rearrangement of Handles), in the construction of a handlebody from a
Morse function, we are allowed to attach the handles in ascending order of their index, and under
the conditions of the Lemma, this does not change the diffeomorphism type of the handlebody. In
terms of Morse functions, this Lemma suggest that without loss of generality, we may as well
( ) for all
assume that the Morse function
has the property ( )
whenever
( )
( ).
( ) of extends ‘downwards’ with respect to and in
Intuively then, the unstable manifold
( ) of extends ‘upwards’
particular, ‘away’ from the point . Similarly, the stable manifold
with respect to , and away from the point . By imagining the situation, it seems that the unstable
and stable manifolds of and respectively should be disjoint. This is precisely what we prove
now.
If ( ) is a Morse-Smale pair on and
( ), then
( )
( )
points such that ( )
.
Proposition 3.12
(
Proof Suppose not. By Definition 3.11,
(
(
)
(
))
(
( )
and therefore
(
(
)
) intersects
(
))
(
(
))
(
. Assuming
( )
(
are two distinct critical
(
) transversally we have
(
(
( ))
(
))
( )
)
(
)
( )
, then
))
( )
( ) contains non trivial flow
By definition, if non-empty, then the intersection
lines; the idea is that these lines “diverge” from and “converge” to , and
. In particular,
each flow line has dimension 1. This gives a contradiction; the 0-dimensional subspace
( )
( ) cannot contain a 1-dimensional flow line.
Definition 3.13 (Self-indexing Morse function)
( ) for every
indexing if ( )
.
A Morse function
is called self-
The following theorem is proved using handlebody decompositions in (Matsumoto 2002). The
proof is long, nasty and involves lots of technicalities. In (Nicolaescu 2007), there is a shorter proof
based on (Milnor 1965), pages 37-44 using the theory developed in this section.
Theorem 3.14 (Smale)
Suppose
there exist Morse-Smale pairs ( ) on
is a compact smooth manifold of dimension
such that is self-indexing.
. Then
In fact, the same proof of Theorem 3.14 as in (Nicolaescu 2007) can be used to prove the stronger
(and more useful statement below):
Corollary 3.15
Suppose ( ) is a Morse-Smale pair on the compact manifold
can modify to a smooth Morse function
with the following properties:
and (
1.
2.
(
)
( ) for all
is a gradient-like vector field for .
In particular, (
0811265
)
. Then we
) is a self-indexing Morse-Smale Pair.
Ian Vincent
Page 28 of 50
Chapter 4: Morse-Witten Homology
Armed with the Theory of Handlebodies (Chapter 2) and section 2.2 on Morse-Smale Dynamics,
we can define the Morse-Witten Homology groups for compact manifolds. We look at two
methods. The first is a little more precise and utilises the relative homology of the manifold, while
the second is more geometric.
4.1 – Relative Homology Approach
In this section, we construct the Morse-Witten Homology using handlebodies. This method of
construction is very similar to the construction of Cellular Homology.
For this section, suppose that (
) is a Morse-Smale pair on a compact
such that
is self-indexing. In particular, note that the real numbers
when
. We set
{
}
{
-dimensional manifold
are regular values of
}
Then is a smooth manifold with boundary which can be decomposed into the ‘upper’ and
‘lower’ boundaries
and
respectively:
{
where
}
Recall the following theorems about computing homology groups (as stated in (Hatcher 2001)):
(Excision)
Given subspaces
)
then the inclusion (
all .
Equivalently, for subspaces
)
(
isomorphisms (
(Homology of a Wedge Sum)
an isomorphism
(
such that the closure of is contained in the interior of
) induces isomorphisms
(
)
(
) for
, the inclusion (
which cover
) for all .
For a wedge sum ⋁
⨁ ̃(
⨁
̃ (⋁
)
provided the wedge sum is formed at base points
some neighbourhood
of
in .
(Homology of a Quotient)
quotient map
(
)
(
If
, the inclusions
(
) induces
⋁
induce
)
, such that
for a manifold
)
, with
is a deformation retract of
closed and non-empty the
) induces isomorphisms
(
)
(
)
̃( )
for all
(Where ̃ denotes the reduced homology. In fact, for any topological space ,
. See (Hatcher 2001) for details.
( )
̃ ( ) for
Remark
Hatcher deals with a more general version of this statement; he defines a pair
(
) to be good if is closed, non-empty and is a deformation retract of some neighbourhood
0811265
Ian Vincent
Page 29 of 50
) to be a good pair if
in . The fact that is a manifold in the above statement guarantees (
with closed non-empty; take an open neighbourhood of consisting of a union
arbitrarily small open balls centred about the each point of , then define
. Then is
clearly a deformation retract of , since a ball is a deformation retract of a point, and hence is a
deformation retract of .
of
( )
We define
(
), the th relative homology group of the pair (
).
These groups will form the Morse-Witten Chain groups in the chain complex we are constructing.
Denote by
( )
(
} and for
{
. Observe that for every
( )
{
)
}
{
( ) the unstable disc
, denote by
}, the set
is finite.
We wish to first prove the following:
Proposition 4.1
Proof In the case
union of
( )
The group
, then
(
) is free abelian and finitely generated.
(
, so
)
(
). Since then
( )
-discs, all of which are homotopy equivalent to a point, then we see
and so
is a disjoint
(
)
( ) is free abelian and finitely generated.
For the remainder of the proof assume that
manifold) we know;
. By Theorem 2.8 (Handle decomposition of a
⋃
that is,
is obtained from
sum of |
| -spheres;
by attaching finitely many -handles. Therefore
⋁
. Then recall that
(
)
(
)
for all
is a wedge
and we get the
series of isomorphisms below:
( )
and so
(
̃(
)
)
)
(⋁
⨁ ̃(
)
|
⨁
|
( )is free abelian and finitely generated for all .
In the construction of the long exact sequence of the pair (
homomorphism ̂
(
)
(
) . If
natural homomorphism induced from the quotient map
), we have a boundary
(
)
(
(
)
) is the
(
) then
their composition
̂
(
( )
yields a homomorphism
chain complex we are constructing.
Corollary 4.2 If
0811265
is an
Ian Vincent
)
(
)
(
)
( ). This is the desired boundary homomorphism in the
-dimensional handlebody, then
Page 30 of 50


The inclusion
(
)
induces an isomorphism
(
( ) for all
)
for all
Proof
See (Hatcher 2001) Lemma 2.34 parts (b) and (c) on Cellular Homology. Then apply Theorem 3.4
(Handlebodies and Cell Complexes) so there is a 1-1 correspondence between the -handles of
and the -cells of the cell complex .
We now finish the construction of the Morse-Witten Homology Group:
Theorem 4.3 (Morse-Smale-Witten) Suppose that (
( )
dimensional manifold , and let ( ) and
) is a Morse-Smale pair on a compact
( ) be defined as above. Then:
-
1. the sequence
( )→
( )→
( )→
→
is a chain complex (that is
for all ), and
2. for all , there is a natural isomorphism
( )
( )
where
( ) denotes the th singular homology group of
.
Proof We follow a similar approach as (Hatcher 2001) does for Cellular Homology. For each of
the pairs (
) (
) and (
), we get a long exact sequence. Portions of
these long exact sequences fit together into a diagram as shown below:
1. Notice that by the construction above, I have factored
and
as the compositions
̂ and
̂ respectively. By exactness of the sequence on the diagonal from ‘top-left’ to
‘bottom-right’:
(
0811265
Ian Vincent
)
(
)
̂
(
)
Page 31 of 50
we see that the composition ̂
is zero. Therefore,
̂) (
̂)
(̂
(
)
̂
̂
Since the argument works for any , this proves that the sequence
( )
( )
( )
( )
is a chain complex.
( ) for all .
2. We now prove that
( ) with
By Corollary 4.2, and using exactness of the diagram above, we can identify
(
)
.
(̂ )
(
Since
onto
(
)
̂)
also injective,
, this shows that
is injective and so it maps
(
) isomorphically onto
) and
(̂ )
(
( ̂ ) isomorphically onto
(
). Therefore
(
( )
) and
(
(
( ̂ ) isomorphically
( )
( ̂ ). Since
) and so we have shown
) isomorphically onto
is
maps
. Therefore,
it induces an isomorphism
(
( )
)
( ̂)
as required.
4.2 – Geometrical Interpretation
The following approach follows from section 3.2 on Morse-Smale Dynamics, and we follow the
method outlined in (Salamon, Morse Theory, the Conley Index and Floer Homology 1990):
As before, we denote by
the gradient-like vector field for – and by
( )
the unstable and stable manifolds for
(
)
( )
{
}
.
( )
( )) for every critical point
We first choose an orientation of the tangent space
(
and denote by ⟨ ⟩ the pair consisting of a critical point and this orientation. For every
we then denote by
the free group
⨁ ⟨ ⟩
( )
Claim 4.4
The function being of Morse-Smale type implies that the intersection
( )
( ) has dimension 1 if the indexes satisfy ( )
.
( )
Proof Transversality of the intersection
( )
( ))
(
( )
( )
( )
since
and
0811265
Ian Vincent
( )
( ) means that the codimensions satisfy:
( ))
( ))
(
(
( )
( )
we
have
Page 32 of 50
( )
(
( ))
( )
( ))
(
( )
( )
( )
( )
( ))
As ( )
we have
. As the flow lines equal to the intersection
(
( )
( ), we see that the set of all flow lines between two consecutive critical points has
dimension .
Proposition 4.5
If the difference of the indices ( )
gradient flow lines of
from to .
( )
1, then there are finitely many
Proof Pick local coordinates ( ) for the critical point . In these coordinates, let
( )
centred at with radius
. Then we claim that the intersection
(
be a ball
( )) is
transverse:
(
We apply Lemma 2.11 (General Position). Since
then there is an isotopy { } of
( )
( ))
by Claim 4.4 and
( )
( ))
such that
and (
intersects
transversely in finitely many points. Since isotopies will not change the number of
flow lines, we see that the set is finite.
Each point in corresponds to a gradient flow line of
finitely many gradient flow lines of
from to .
connecting
and ; therefore there are
( )
( )
) by
Since the intersection
( ) is finite if ( )
, define an integer (
assigning a number
or
to every flow line, and taking the sum. We choose whether to assign
+1 or -1 as follows:
( )
Let ( ) be such a flow line (that is ( )
and
( ( )) with
Then the orientation ⟨ ⟩ induces an orientation on the orthogonal complement
|
( )
( )|
in
( ). In the case ( )
( )
, the tangent flow
( )
( )
( )
( ) of
.
induces an
isomorphism from ( ) onto ( ) and we define
to be
or
according to whether this
) ∑
map is orientation preserving or orientation reversing. Then define (
where the sum
runs over all flow lines connecting to . A boundary operator
of the chain complex
defined on the previous page is now given by the formula:
⟨ ⟩
(
∑
)⟨ ⟩
( )
Of course, we must show that this chain complex gives us the same homology groups as the one
as defined in ( ) coincides with
defined in section 4.1. We first show that
Proposition 4.1 we see that
(
)
|
can form ⟨ ⟩, the pair consisting of a critical point
that
(
(
|
(
. Now for every critical point
and an orientation of
) (the unstable manifold of
(
). By
of index , we
( )). Hence we see
actually forms a generator of the group
) as we can see in a diagram in local coordinates ( ) as given by the Morse
Lemma:
0811265
Ian Vincent
Page 33 of 50
(
The bold horizontal line here is a class of
boundary is contained within
whose
. Notice that this is exactly the unstable manifold
respect to the usual Morse-Smale pair (
Lemma 4.6
⟨ ⟩ ∑
), since it is a -cell in
).
⟨ ⟩ with map
Consider the Groups defined by
⨁
(
)⟨ ⟩ as above. Then (
) is a chain complex; that is
is isomorphic to the singular homology
( ) with
( ) of the ambient manifold
defined by
and the quotient
.
Proof (Sketch)
We follow the argument given in the proof of Lemma 3.2 of (Salamon,
Morse Theory, the Conley Index and Floer Homology 1990):
For simplicity, we assume orientability of . [This is unnecessary for the proof but makes things
simpler. (Salamon) gives more information about reducing ourselves to the orientable case.
( )
If
is oriented, then for every regular value
, the level set
is an oriented
( ) will be called positively oriented if
submanifold of . More precisely, a basis
of
( )
( ) since
( )
( ).
defines positively oriented basis of ( )
(
)
(
)
(
)
( )
It follows that the descending sphere
inherits an orientation from
( )
( )
( ) inherits an orientation from
( ). The integer
and the ascending sphere
(
) in ( ) agrees with the intersection number of
( ) and
( ) in
( ).
For every critical point
orientation of
where
of , observe that in the previous observation that the choice of an
( ) determines a generator of the relative homology group
is the index of . Hence
Next, we define
( )
( )
can be identified with
)
).
(
)
( )
( ); the union of orbits connecting to . In the case
, this is a submanifold of dimension
. Define the homomorphism
(
0811265
(
(
Ian Vincent
)
(
)
Page 34 of 50
to be the boundary homomorphism as in Theorem 4.3. We need to show that this coincides with
the map . Recall that ( ) is the flow of the vector field
as before.
Denote by
, and
{
(
{
(
( ))
( )
( ))
}
( )
( )
{
}
{
(
}
( ))
}
These sets are illustrated the diagram below, consisting of a local picture about the critical points
and
. In the diagram, the flow lines
are shown in red, and
respectively
are shown in grey, while
and
are shown respectively in blue:
Note that
is contractible onto
( )
{
} by taking the limit
. Likewise,
} whose width converges to zero as
( ) {
defines a tubular neighbourhood of
.
{
}
(
)
(
)
(
)
Since
and
intersect transversally, it follows that
consists of
(
)
finitely many components
each containing a unique point
. That is, there
exists a diffeomorphism
(its corresponding handle in the handle decomposition of )
( )
(
( )
) { }
with
and
( )
{ } where
. In particular
|
is a
-manifold with boundary
, and the map
(
, diffeomorphic to
induces an isomorphism on homology
)
(
)
(
)
( ) determines a generator of the homology
The given orientation of
(
)
which under the above isomorphism is mapped to a generator
homology class
is determined by the orientation of
via the flow defined isomorphism
one inherited from
where
0811265
(
(
). The
inherited from the orientation of ( )
( ) via the injection
( )
( )
( ) as the first basis vector). Indeed, both orientations agree if and only if
{
} is the sign associated to the connecting orbit
Ian Vincent
)
( ). This orientation may or may not agree with the
( )
(by taking
via
( )
( ).
Page 35 of 50
Now choose a triangulation of the -manifolds and extend it to a triangulation of the
} with boundary
}. Together with the given orientation of
( ) {
( ) {
manifold
( ) this determines a generator
( )
( )
(
)
(
)
(
)
(
The homology class
original triangulation of
agrees with
( )
( )
)
(
) is represented by the
( ) and therefore
together with the orientation inherited from
(
. Using the above isomorphism
∑
(
)
)
and therefore proves that the two boundary maps
results follow.
(
(
and
) we obtain
)
coincide, and by Theorem 4.3, the
4.3 – Examples
Let us look at some examples. The tricky part of the construction to see whether the isomorphism
( )
( ) is orientation preserving or orientation reversing. To avoid mistakes I have
sketched the local picture of the flow lines at each critical point, and given an orientation locally.
Then by performing the flow along each of the flow lines, out of one chart and into another, this
enables us to keep track of the orientations:
Example 4.7 The Torus
Consider a torus embedded in
(tilted by an isotopy), with its
)
usual Morse function defined by (
. Then there are four critical points
with
indices
respectively. Here is a picture, showing the flow lines from to and in red, and
the flow lines of from
to in blue.
Looking locally at the critical points
we see the following local pictures. In the below
diagrams we see the red flow lines from to and the green flow lines from to . The blue flow
( ) and
( ). We orient the tangent spaces ( )
( ))
( )
lines correspond to
(
(
( )) and
0811265
( )
Ian Vincent
(
( )) as shown on the diagrams by ⟨
⟩ ⟨
⟩⟨
⟩ respectively.
Page 36 of 50
Now, the orientation ⟨
( ) of
|
( )
for
( )|
( ) induces an orientation on the orthogonal complements
⟩ on
{
}. Therefore the orientations are
( )
Let
( )
( )
be the tangent flow along the flow line
⟨ ⟩
⟨ ⟩
⟨
⟩
for
⟨
⟩
(
)
⟨ ⟩
( )
⟨
⟩
{
} then we see
⟨ ⟩
⟨
⟩
⟨ ⟩
⟨
⟩
Equivalently, the change of basis matrices are:
( )
Therefore the maps
Therefore by definition,
( )
(
are orientation preserving and
and
, so ⟨ ⟩
)
are orientation reversing.
⟨ ⟩
⟨ ⟩
Similarly, the below diagrams show the local pictures of the flow lines around
The orientation ⟨ ⟩ on
|
( )
( )|
{
for
( ) induces an orientation of the orthogonal complements
The orientation ⟨ ⟩ on
( )
( )|
{
for
( )
( ) induces an orientation of the orthogonal complements
( )
be the tangent flow along the flow line
⟨
⟩
( ) of
} (which consist of single points). Therefore the orientations are
( )
Let
( ) of
} (which consist of single points). Therefore the orientations are
( )
|
:
⟨
⟩
⟨
⟩
for
⟨
⟩
(
)
{
} then we see
⟨
⟩
⟨
⟩
⟨
⟩
⟨
⟩
Equivalently, the change of basis matrices are:
(
Therefore the maps
by definition,
0811265
Ian Vincent
)
(
are orientation preserving and
and
, so ⟨ ⟩
)
(
)
are orientation reversing., and so
⟨ ⟩
⟨ ⟩
⟨ ⟩
.
Page 37 of 50
It follows that all the boundary maps
homology groups
are:
(
in the chain complex (
)
(
) are 0 and so the Morse-Witten
)
(
)
Which coincide with the singular homology groups of the torus.
Example 4.8 The Projective Plane Consider the identification diagram below of
with a Morse-Smale Flow induced by the red arrows:
and with indices ( )
Observe that there are three critical points
( )
, together
and ( )
.
Therefore we form a chain complex:
⟨ ⟩
And we now compute the boundary maps
⟨ ⟩
and
⟨ ⟩
:
From the above diagram, we get the following local pictures:
The red lines denote the flow lines from to , the green lines denote the flow lines from to
( )
and the blue lines denote the flow lines of to . Since ( )
, we do not consider the
( )
( ))
( )
( )) and
green flow lines. We orient the tangent spaces
(
(
( )
( )) as shown on the diagrams by ⟨
(
Now, the orientation ⟨
( ) of
|
( )
for
( )|
⟩ on
{
( ) induces an orientation on the orthogonal complements
}. These orientations are
( )
Also, the orientation ⟨ ⟩ on
|
( )
for
( )|
0811265
{
⟩ ⟨ ⟩ respectively.
⟨ ⟩
( )
⟨
⟩
( ) induces an orientation on the orthogonal complements
}. These the orientations are
Ian Vincent
Page 38 of 50
( ) of
( )
Let
⟨
⟩
be the tangent flow along the flow line
⟨ ⟩ ⟨ ⟩
⟨
⟩ ⟨ ⟩
for
⟨
( )
⟨
{
} then we see
⟨ ⟩
⟨ ⟩
⟩
⟩
⟨
⟩
Equivalently, the change of basis matrices are:
(
Therefore the maps
Therefore by definition,
)
and
Morse-Witten homology groups
(
)
{ }
(
)
(
)
are orientation preserving and
and
, so ⟨ ⟩
(
)
is orientation reversing.
⟨ ⟩ and ⟨ ⟩
⟨ ⟩ so the
are:
(
)
⟨ ⟩
⟨ ⟩
(
)
Which coincide with the singular homology groups of the projective plane.
0811265
Ian Vincent
Page 39 of 50
( )
( ) is, by definition, the set of
Example 4.9 The Special Unitary Group
Recall that
complex matrices satisfying
and
. By standard techniques in Lie Group
( )
Theory, we see that as a manifold,
. To compute its homology, rather than
attempting to decompose it into an -dimensional CW complex in the usual method of computing
the Homology, we compute the Morse-Witten Homology which given a Morse function, we can
form a corresponding Chain complex.
( )
Define
by (
(
)
(Matsumoto 2002), page 99, proves that
critical points:
). The following Lemma from
as defined above is a Morse function, and finds its
( ) is a Morse function, and its critical points
Lemma 4.10 The restriction of to
consist of the diagonal matrices, whose diagonal consists of
s and
s, and whose
determinant is .
Proof The proof is not hard to follow in (Matsumoto 2002), and is out of the scope of the
document, so I omit it from here.
We compute the indices of the critical points as follows:
Writing a critical point of
in the form (
Once more, from (Matsumoto 2002), if
order), the index of the Morse function
If
{
), where
}.
are the subscripts with
(in ascending
at the critical point is given as follows:
( )
, the index is
(
and if
)
(
)
(
)
, then the index is
(
)
(
)
Therefore, simply by considering the possibilities of the diagonal matrices of this form, we see that
the critical points of and their corresponding indices are given below:
Critical Point
(
As we see,
0811265
)
index
(
)
index
(
)
index
(
)
index
has four critical points of indices
Ian Vincent
Index
and . Therefore the chain complex is given by
Page 40 of 50
Fortunately, in the above Chain Complex we conclude that all the boundary maps
if
. Hence the homology groups are: ( ( )) {
otherwise
Critical Point
(
Index
)
index
(
)
(
)
(
)
(
)
(
)
(
)
(
index
index
index
index
index
index
)
index
( ) is given by:
Hence the Chain complex for
→
→
→
→
→
→
→
As before, we can immediately conclude that all the boundary maps
{ }, we immediately see
Therefore for
if
( ( )) {
otherwise
0811265
for all
( ) We can apply much the same technique as above with ( ). First note that
( )
. As before, let
be the function
). By Lemma 4.10 the critical points of are:
Example 4.11
( ( ))
( )
(
To compute
are
(
( )) and
Ian Vincent
(
are , except for
( )), we would need to work out the boundary map
Page 41 of 50
.
.
Chapter 5: Poincaré Duality
While a little off topic, I could not resist showing this proof of Poincaré Duality using Morse
Theory, as presented in (Matsumoto 2002).
5.1 – Cohomology Groups
We quickly introduce the Cohomology Groups
( ).
Consider the chain complex
( )→
( )→
( )→
( )
( )
→
of a cell complex . We may assume using Simplicial or Cellular Homology that the rank of ( )
is equal to the number
of -cells contained in a CW-structure for , and that all the groups
( ) are finitely generated.
Definition 5.1 (Cochains, Coboundary Homomorphism and Cochain Complex)
Let
( ) {
( )
}
( ( ) ) be the set of all homomorphisms from ( ) to ; then
( ) becomes an abelian group with respect to the addition of homomorphisms. Here, the
( ) is defined by (
)( )
( )
( ) for all
( ) . The
addition
for
subtraction is defined similarly.
( ) is
1. The group ( ) is called the -dimensional Cochain Group, and each element
called a -Cochain.
( )
2. For any -cochain
, the composition
( )
( )
( ) is a (
)-cochain. The map
with the boundary homomorphism
( )
( )
where ( )
, is called the Coboundary Homomorphism..
3. The sequence of the cochain groups and coboundary homomorphisms
←
( )←
( )←
( )←
( )←
←
( )←{ }
is called the Cochain Complex of .
Lemma 5.2
For any
.
( ). Then
( )
(
) (
But by definition of the chain complex, we have
is a homomorphism.
Proof Take any
)
(
so
)
( )
since
( ) of
The kernel
is denoted by ( ) and is called the -dimensional cocycle group. This
group ( ) is a subgroup of ( ), and the elements of ( ) are called cocycles. Also, the image
(
) is denoted by ( ) and is called the -dimensional coboundary group, which is a
subgroup of ( ) as well. The elements of ( ) are called coboundaries. By Lemma 5.2, we have
( )
( )
( ).
Definition 5.3 (Cohomology Group)
Cohomology group and denote it by
0811265
Ian Vincent
We call the quotient group
( )
( )
the
-dimensional
( ).
Page 42 of 50
( ) is called a -dimensional cohomology class. The cohomology class to which a
An element of
-dimensional cocycle belongs to is denoted by [ ].
( )
Since a -cochain is a homomorphism
, for any -cochain and any -chain , an
integer ( ) is determined. By the definition of the coboundary homomorphism, we have
(
)( )
(
)
From this we can easily show that the value ( ) for a -dimensional cocycle and a dimensional cycle is determined only by the cohomology class [ ] and the homology class [ ]. (In
the next section on intersection forms, we will go over a similar argument in some detail, when we
prove that an intersection number is determined between homology classes.) Therefore, a ( )
dimensional cohomology class [ ] determines a homomorphism [ ]
.
We set
(
( ) )
( )
{homomorphisms
} and from the above we obtain a map
( )
( ( ) )
( ) to the homomorphism on ( ). This is a homomorphism. The next
by assigning [ ]
theorem is a special case of a weaker version of the theorem called the Universal Coefficient
Theorem.
( )
Theorem 5.4 The homomorphism
( ).
the torsion part of
(
( ) ) is surjective, and its kernel
( ) is
Proof See the Universal Coefficient Theorem in (Hatcher 2001), pages 194-196.
Now we can state the theorem that we will prove in the next section:
Theorem 5.5 (Poincaré Duality)
For
( )
an
orientable closed
{
}
( )
-manifold
,
we
have
5.2 – How to prove Poincaré Duality
First, we make some observations on relations between boundary and coboundary
homomorphisms.
If a cell complex
contains
taken as a natural basis of
, then the oriented cells ⟨
-cells,
( ). A similar basis ⟨
( )
so that the boundary homomorphism
these bases. In fact, if
(⟨
Then
⟩)
⟨
⟩⟨
⟩
⟨
can be represented by a
⟩
For the -dimensional cochain group
0811265
is a homomorphism
Ian Vincent
⟨
⟩
⟨
⟩ can be
⟩ can be taken for
( ),
( ) can be represented by a matrix using
⟨
⟩
integer-valued matrix
(
That is,
⟩
⟩⟨
)
( ), take the dual basis
( )
, of ⟨
⟩⟨
⟩
defined by
Page 43 of 50
⟨
⟩.
⟩)
(⟨
{
when
otherwise
( )
We represent the coboundary homomorphism
basis
( ) and the dual basis
of
(
( ) as follows:
of
(
So that
( ) by a matrix using the dual
)
)
.
( )
( )
Lemma 5.6
The matrix which represents
the transpose of the matrix which represents
( ) with respect to the dual basis is
( ).
⟩. By the
Proof To compute the coefficients
, we evaluate both sides of the above at ⟨
defining equations of the dual basis, the RHS can take a non-zero value only at the th term, and
the value is equal to
.
On the other hand, from the definition of the coboundary homomorphism, the LHS is equal to
(⟨
and we obtain
⟩)
(
⟨
⟩)
(
⟨
⟩
⟨
⟩
⟨
⟩)
.
We now delay the Proof of Theorem 5.5 further, and discuss how the boundary homomorphism
in Cellular homology can be interpreted in terms of handlebodies.
5.2.1 The Attaching maps of a Handlebody and Intersection numbers
Let
be
an
oriented
(
closed
-manifold.
)
We
(
)
use
a
handle
(
decomposition
)
arranged in increasing order of indices. As we have shown in Theorem 3.4 (Handlebodies and Cell
Complexes), can be regarded naturally as an -dimensional cell complex , by regarding the
cores of -handles of this handle decomposition as -cells. Choose and fix an arbitrary orientation
⟨ ⟩ for the core
of the -handle . The we obtain a chain complex
( )→
( )→
→
associated with the handle decomposition of
boundary homomorphism
the -cell
( )→
( )
. Using Cellular Homology for example, the
is represented by the degree
under the attaching map
( )
of how many times
covers
of the cell complex, with the orientations
taken into account. (Here
denotes the -skeleton of
more details on this, see (Hatcher 2001) page 140.
when regarded as a cell complex.) For
When a handlebody is regarded as a cell complex, attaching maps of the cell complex are
essentially the attaching maps of the handlebody. Using this fact, the degree
can be interpreted
in terms of a handle decomposition as follows:
In the handle decomposition
(
let
)
(
)
(
)
be the subhandlebody with all the handles attached up to (and including)
0811265
Ian Vincent
-handles.
Page 44 of 50
Suppose now that ta (
)-handle
is attached to
. The image
sphere embedded in
(
by an attaching map
) of the attaching sphere under
. To simplify the notation, we denote this by
(
is a
-
in what follows:
)
On the other hand, the belt sphere
of a -handle
contained in
is an
(
)-sphere, and also is a submanifold of
. To simplify the notation again, denote this
belt sphere by
The dimension
:
of the attaching sphere
add up to the dimension
of
and the dimension
of the belt sphere
.
Now, we apply Lemma 2.11 (General Position) to the handlebody
{ }
of the boundary
such that
and
(
, so we can find an isotopy
) intersects the belt sphere
transversely at finitely many points. Using Theorem 2.12 (Sliding Handles), the attaching map
can be replaced by
by this isotopy, so that we can assume, with no loss of generality, that
the attaching sphere
and the belt sphere
intersect transversely. (By applying this
argument to the disjoint union of belt spheres, we can assume that any attaching sphere and any
belt sphere intersect transversely in
.)
First we consider a simple situation, where an attaching sphere
intersect at a single point in
. In this case, as we see by making the -handle
thinner to get a corresponding -cell
, the attaching sphere
or negatively. In other words, we obtain
this diagram.]
0811265
Ian Vincent
and a belt sphere
covers
thinner and
exactly once, positively
. [See the diagram below, noting that
Page 45 of 50
in
To study general situations, we use the intersection number. The intersection number is defined
for two oriented submanifolds ⟨ ⟩ and ⟨ ⟩ intersecting transversely at finitely many points in an
oriented manifold ⟨ ⟩. (Here is it assumed that
.)
The sign of the intersection,
or
, is assigned to each intersection point of and , and the
intersection number between ⟨ ⟩ and ⟨ ⟩, ⟨ ⟩ ⟨ ⟩ is the integer obtained by adding up all the
signs of all intersection points. We saw this before in the Geometrical interpretation of MorseWitten Homology, where we defined the boundary homomorphism
⟨ ⟩
∑
(
)⟨ ⟩
Determining the sign of an intersection to be
or
is a question of orientations. There is a nice
way of determining the sign of an intersection using exactness, which we explain now:
5.2.2 Using Exactness to determine the Sign of an Intersection:
Suppose and are two oriented submanifolds of
diagram below:
, and
. Then for
, consider the
where the functions
and are all the inclusion maps. In particular, note that
this diagram we construct an exact sequence in local coordinates:
(
where for
that
( )
(
), ( )
( ( ) ( ))
)
( )
( )
( ( ) ( )) and for (
( )
( )
(
)
. From
)
,
(
)
( )
( ) . Note
, implying exactness.
⟩ for (
) and another ordered basis ⟨
⟩ for (
)
Take an ordered basis ⟨
( ) and ⟨
⟩ is an ordered basis for
. Then for each , take
[here we identify with its image ( ) since is injective]. Call this basis an induced basis for
)
. Declare the sign of this exact sequence to be
if, given positive bases for (
), this induces a positive basis for
and (
. Otherwise, the sign of this intersection
is
.
( ); we know that is surjective (by
Note that for
, there is no unique choice for
exactness) and so at least one preimage exists. Therefore we need to check that the sign is
( ):
independent of the choice of preimage
( )
Suppose that for each
there exist and
with ( )
⟨
possibilities
for
an
induced
basis
are
⟨
⟩. We need to show that
is a positive basis iff
basis; we look at the change of basis matrix between them.
0811265
Ian Vincent
. Then two
⟩
and
is a positive
Page 46 of 50
First notice that
identifying
(
( )
( )
(
) with its image
exist
so that
change of basis matrix from
∑
) so by exactness,
( (
to
; that is
) (here
). This means that there
∑
and so
is of the form
(
and hence its determinant is
))
(
for each . Therefore the
)
is positive iff
is positive.
Via routine arguments, similar to that above, we can check that the sign is independent of all other
) or another positive basis ⟨ ⟩
choices of equivalent bases; another positive basis ⟨ ⟩ for (
) will induce a positive-determinant change of basis matrix on the induced bases.
for (
Since, in our case, we are given orientations of all the attaching spheres and belt spheres of the
handles, applying the method above we can determine (up to sign) the intersection number by
giving it an arbitrary orientation. ⟨
We note that
covers
⟩ ⟨
⟩.
the same number of times as
intersects
consideration, in this case. By choosing all the orientations of ⟨
⟩⟨
with signs taken into
⟩ and ⟨
⟩ “naturally”,
one can compute that
(
Where the
) ⟨
⟩ ⟨
⟩
make up the matrix representing the boundary homomorphism
.
The sign ( ) in front is not essential here, so we omit the details, but we explain how to define
these “natural” orientations. Choose an arbitrary orientation ⟨ ⟩ for the core
of the -handle .
If we denote by
⟨ ⟩. Namely, since
the cocore of
, then an orientation of
is naturally determined from
⟩ is
and
intersect transversely at a single point, the orientation ⟨
⟩
determined in such a way that ⟨ ⟩ ⟨
under the assumption that an orientation of the
whose manifold is specified.
In general, an orientation of the boundary
of a manifold
orientation of . The orientations of attaching spheres
the ones naturally induced from . The equality
is naturally induced from an
and belt spheres
(
) ⟨
⟩ ⟨
are chosen to be
⟩ holds with all these
conventions.
5.3 – The Proof of Poincaré Duality
Finally, after all the preliminary discussions above, we are ready to prove Theorem 5.5.
Proof of Theorem 5.5 (Poincaré Duality) The key point of this proof, which uses Morse Theory
is to turn the manifold “upside down”. For this purpose, we just replace the original Morse
function
by
.
As we have already seen in the proof of Theorem 2.16, the set of critical points of the Morse
function
is identical to the set of critical points of . However, a critical point of index
becomes a critical point of index
with respect to
. This can be seen easily by looking at the
0811265
Ian Vincent
Page 47 of 50
standard form of about a critical point. The roles of the core and cocore of a handle are switched,
)-handle
and a -handle
transforms itself into an (
. The core of
is
, and
the cocore is
.
The handle decomposition associated with
is
(
)
(
) ( )
of -handles, -handles,…, -handles, are, respectively the
)-handles,…, -handles before it was turned upside down.
(
where the numbers
numbers of -handles, (
)
Let
be the cell decomposition of associated with the handle decomposition ( ). From this cell
complex, we obtain a chain complex
(
The basis ⟨
⟩⟨
⟩
)→
⟨
(
⟩ of
(
)
(
)⏞
(
)
) looks like the dual basis of
( ) when we use
intersection numbers. In fact, if we compute the intersection number (in ) between a basis
⟩ of
( ) and from the formulae { ⟨ ⟩ ⟨
⟩
element ⟨
and by the formula
⟨ ⟩ ⟨ ⟩
(
)
⟨ ⟩ ⟨ ⟩}, for any submanifolds
For a basis element ⟨
⟩ of
( ), we obtain
⟨
(
(
Furthermore, since
, we have}
⟩ ⟨
))
(
⟩
{
(
( ))
(
)(
)
if
if
, an isomorphism
)
( )
)
can be defined by assigning ( )(
, the dual basis element
with sign adjusted to a basis
⟩ of
( ). Such an isomorphism
element ⟨
is defined for each dimension .
Lemma 5.7
For any , the following diagram commutes up to sign:
Namely,
Proof
We find a matrix presentation of the boundary homomorphism
basis ⟨
⟩ of
(
) (for all ) and the basis ⟨
⟩ of
with respect to the
(
) (for all ).
Set
(⟨
Let
(
(⟨
⟩)
⟨
⟩
⟨
⟩
be the subhandlebody consisting of all handles from -handle through to
) in the handle decomposition ( ). We see that
boundary
0811265
⟩)
Ian Vincent
of
is equal to the boundary
, and the
of
.Considering the fact
Page 48 of 50
is now the attaching sphere of an (
that the form belt sphere
is now the belt sphere of an (
, and the former attaching sphere
handle
)-handle
in
)-
, we obtain
(
)
⟨
(
by an argument similar to that of
⟩ ⟨
) ⟨
⟩
⟩ ⟨
⟩. They look identical, but we
need to observe a few technicalities.
First of all, the intersection number is counted in ⟨
to apply the same argument as when we computed
⟩(
⟨
⟩) in this case. Also,
, orientations of
and
must
be defined in such a way that
⟨
⟩ ⟨
⟩
This is reason behind the symbol in the notations of orientations above. If we use the
orientations ⟨
sign is (
)(
⟩ and ⟨
)(
)
⟩ based on the “natural” orientations, then the difference in
, from the formula ⟨ ⟩ ⟨ ⟩
(
)
⟨ ⟩ ⟨ ⟩}.
The above considerations imply that if we interchange the order of ⟨
compare with the formula for
⟩ and ⟨
⟩ and
, then we obtain
( )
In this formula, sign ( )
is not important for the proof of Poincaré Duality. In any case,
we have found that the matrix representing
is equal to the transpose of the matrix
which representing
or its negative
.
( )
By Lemma 5.6, the matrix representing
is the transpose of . Hence it is now clear that
( ) and ( ) by the isomorphism
identifying
( ) and
( ) using the isomorphism
.
(
If we use Lemma 5.7, and consider the isomorphism
we find that the chain complex
(
( ) with respect to the dual basis
and
coincide up to sign by
. The same argument applies to
)→
(
)→
( )→
( )→
)
(
( ) in each dimension ,
)
is isomorphic to the cochain complex
Therefore, the homology group
(
)(
( )
( )) of dimension
of the top chain
( )(
( )) of dimension of the lower
complex is isomorphic to the cohomology group
cochain complex. [The difference between
and
in the coboundary homomorphism does not
alter the kernel or image, and so does not affect the structures of the cohomology groups.]
This proves the Poincaré Duality
( )
( ).
Corollary 5.8 If is an orientable, connected, closed -manifold, then the highest dimensional
homology and cohomology groups are both isomorphic to :
( )
( )
0811265
Ian Vincent
Page 49 of 50
( )
Proof By Poincaré Duality we have
is easily proved from definitions that
( )
(
( ) )
( )
Torsion part of
( ) and
( )
( ). On the other hand, it
( )
and by Theorem 5.4 it follows that
as well.
{ }
Giving an orientation of corresponds to specifying a generator of
generator is denoted by [ ], and is called a fundamental class of .
Remark
For any non-orientable connected
-manifold
( )
. Such a specified
( )
, it is known that
{ }.
Corollary 5.9 (Duality of Betti Numbers) If
is an orientable, connected, closed -manifold,
)-dimensional Betti Number coincide:
then the -th dimensional Betti number and the (
( )
( )
Proof By the Universal Coefficient Theorem, we have
(
( ))
( )). Combining
(
with Poincaré Duality, we obtain
( )
(
( ))
(
( ))
(
( ))
( )
as desired.
References
Hatcher, Allen. Algebraic Topology. New York: Cambridge University Press, 2001.
Kozlowski, Andrzej. Morse Smale Flows on a Tilted Torus - Mathematica Demonstrations. n.d.
http://demonstrations.wolfram.com/MorseSmaleFlowsOnATiltedTorus/.
Matsumoto, Yukio. An introduction to Morse Theory. American Mathematical Society, 2002.
Milnor, J. W. Lectures on the h-cobordism Theorem. Princeton University Press, 1965.
Nicolaescu, Liviu. An Invitation to Morse Theory. Springer, 2007.
Salamon, Dietmar. “Lectures on Floer Homology.” University of Warwick, 1 December 1997.
Salamon, Dietmar. “Morse Theory, the Conley Index and Floer Homology.” Bulletin of the London
Mathematical Society, 1990: 113-140.
Schwarz, Matthias. Morse Homology. Basel: Birkhäuser Verlag, 1993.
Tu, Loring W. An Introduction to Manifolds. New York: Springer, 2011.
0811265
Ian Vincent
Page 50 of 50