The principles of Homological Algebra as used in Topology will be developed here.
Let be a ring and
and
be right (or left)
-modules. Then the sequence
is exact iff
.
More generally, is an exact complex if im
ker
.
The homology
is
.
Let be points in
. The set of points satisfying
where
forms an
-simplex.
The demarcation of a topological space as a simplicial complex is called the triangulation of
. A simplicial complex satisfies the following two properties:
1. Any face of a simplex in is also a simplex in
.
2. The intersection of two simplices and
is a face of both the simplices.
For example, dividing the unit square into two triangles (cutting along the diagonal) would form a simplicial complex. However, Cutting the square into small rectangles would not form a simplicial complex, as the small rectangles are not simplices.
For a triangulated space , we can form a free
-module generated by the
-simplices. For example, for the unit square, we can form a free abelian group generated by the two triangles that the unit square has been divided into.
Now we move on to the signs. It is easy to see that . In general, we have
,where
is the number of transpositions.
Now we shall talk about the mapping . The simplex
is mapped to
, where
.
It is easy to see, on working out some examples on your own, that . I recommend working on
, as it gives the simplest triangulation.
The homology
here is equal to
.
We know move on to the following important proposition: . It is important to note that dimension implies the number of generators of the free module, and not any other geometric interpretation that you may have in mind.
Proof: . So for example if we had
, this would be equal to
.
If we assume that maps
to
, we can conclude that
. Also, we know that
. Hence, we get
. This method can now be generalised for any
.
Now we come to the Euler characteristic. Let be the
homology group of
. The Euler characteristic is defined as
.
How would you interpret this statement? Take the unit square, and divide it into triangles (or triangulate it in any other fashion you like). The -modules
for
and
are all
, as these are no
-simplices in the unit square for those values of
. Hence, the Euler characteristic of the unit square would be #(Triangles)-#(Edges)+#(Vertices), where # denotes cardinality. The Euler characteristic of the closed interval
would be
.
Morse Theory
Now we come to Morse Theory.
Let be a
-manifold, and let
be a twice differentiable function. The Hessian
would be
Since this matrix is symmetric, as a result of the Spectral Theorem it has mutually orthonormal eigenvectors.
A critical point of a Morse function
is that where the derivative is
. A non-degenerate critical point is that where the derivative is
, but the Hessian is non-singular.
The index of a Morse function at a non-degenerate critical point would be the number of negative eigenvalues of the Hessian at that point.
Now we come to a very interesting lemma of Morse Theory. Say is a function, of which
is a non-degenerate critical point. Then there is a neighbourhood
of
in
such that
can be written as
, where
is the index of the critical point. For instance, if
, then
is a non-degenrate critical point with one negative eigenvalue. Hence, there exists a neighbourhood of
where
.
Now we move on to homotopy types. Let and
be two topological spaces. We say that
and
are of the same homotopy type if there exist functions
and
such that
and
. For example,
is homotopy equivalent to
.
Theorem: If is a smooth real-valued function on a manifold
, and if
and
are points such that
and
has no critical points, then
is diffeomorphic to
.
Experimenting with functions from to
should convince you of this fact. It is important to note that there not being a critical point between
and
is not a necessary condition for
to be diffeomorphic to
. Experimenting further should convince you of this too.
Now we move on to a rather interesting theorem. If is a smooth real valued function on a manifold
, and
a non-degenerate critical point with index
, if
is compact with exactly one critical point, then for all
sufficiently small,
, with an
-cell attached appropriately, is of the same homotopy type as
.
Take for instance. Clearly,
is a critical point, and the hessian here is
; hence no negative eigenvalues. The theorem above says that a point (attaching a
-simplex to an empty set as
for
is empty) is homotopic to the bounded
-manifold
.
Here, it is instructional to note that two manifolds of the same homotopy type will have the same Euler characteristic. The number of -simplices may not remain the same, however. For instance, the unit disc is homotopic to the unit square. One way to think about his is that whenever you’re adding vertices, you’re also adding faces and edges; and somehow the alternating sum always remains constant.
It is also important to note that if is a Morse function, then the Euler characteristic of
is the same as the Euler characteristic of
. They’re obviously the same, and hence have the same Euler characteristic.
If you’ve convinced yourself of the fact above, we will move on to some interesting formulations.
Let be the number of critical points of
with index
. Then
. In other words, I am saying that the number of
-simplices in
is equal to the number of critical points of Morse index
. How come that is true? Assume that
is the lowest critical value (we’re assuming that it exists). Now keep hopping one at a time until you reach the greatest critical value. After doing this, we now have a manifold
, which has the same Euler characteristic as
attached with certain simplices at critical points; a quotient space of the disjoint union of
and some simplices. Let us call the latter
. Using the standard formula for Euler characteristic, we see that
. This is the Poincare-Hopf conjecture!