The previous post on kernels and cokernels mentioned that for a linear operator T: V → W, the index of T is defined as the difference between the dimension of its kernel and the dimension of its cokernel:
index T = dim ker T − dim coker T.
The index was first called the Fredholm index, because of it came up in Fredholm’s investigation of integral equations. (More on this work in the next post.)
Robustness
The index of a linear operator is robust in the following sense. If V and W are Banach spaces and T: V → W is a continuous linear operator, then there is an open set around T in the space of continuous operators from V to W on which the index is constant. In other words, small changes to T don’t change its index.
Small changes to T may alter the dimension of the kernel or the dimension of the cokernel, but they don’t alter their difference.
Relation to Fredholm alternative
The next post discusses the Fredholm alternative theorem. It says that if K is a compact linear operator on a Hilbert space and I is the identity operator, then the Fredholm index of I − K is zero. The post will explain how this relates to solving linear (integral) equations.
Analogy to Euler characteristic
We can make an exact sequence with the spaces V and W and the kernel and cokernel of T as follows:
0 → ker T → V → W → coker T → 0
All this means is that the image of one map is the kernel of the next.
We can take the alternating sum of the dimensions of the spaces in this sequence:
dim ker T − dim V + dim W − dim coker T.
If V and W have the same finite dimension, then this alternating sum equals the index of T.
The Euler characteristic is also an alternating sum. For a simplex, the Euler characteristic is defined by
V − E + F
where V is the number of vertices, E the number of edges, and F the number of faces. We can extend this to higher dimensions as the number of zero-dimensional object (vertices), minus the number of one-dimensional objects (edges), plus the number of two-dimensional objects, minus the number of three dimensional objects, etc.
A more sophisticated definition of Euler characteristic is the alternating sum of the dimensions of cohomology spaces. These also form an exact sequence.
The Atiyah-Singer index theorem says that for elliptic operators on manifolds, two kinds of index are equal: the analytical index and the topological index. The analytical index is essentially the Fredholm index. The topological index is derived from topological information about the manifold.
This is analogous to the Gauss-Bonnet theorem that says you can find the Euler characteristic, a topological invariant, by integrating Gauss curvature, an analytic calculation.
Other posts in this series
This is the middle post in a series of three. The first was on kernels and cokernels, and the next is on the Fredholm alternative.