Index Theory

Lecture Notes taken by Nilufer Koldan In this lecture I will describe one of the most significant achievements of the second half of the XX century – the Atiyah-Singer index theorem. I will also discuss some more recent developments in the area as well as some open problems. 1. The definition of the index 1.1. The finite-dimensional case. Let A : V 1 → V 2 be a linear map between finite dimensional vector spaces. Define Coker(A) = V 2 / Im A. Then dim Ker(A) − dim Coker(A) = dim(V 1) − dim(V 2). (1.1) Example: A : V → V ⇒ dim(Ker(A)) − dim(Coker(A)) = 0 Exercise: Let A(t) be a continuous family of matrices then ∀t 0 ∃ > 0 : ∀t, |t − t 0 | < such that dim Ker A(t) ≤ dim Ker A(t 0). Hint: If dim(Ker A(t 0)) = 0 then A(t 0)v ≥ δv. Example: A = 1 0 0 0 and A(t) = 1 0 0 t In other words, the dimension of the kernel is a semi-continuous function of t. Similarly, the dimension of the cokernel is a semi-continuous function. But the equation (1.1) implies that the difference of those two functions is constant. 1.2. The infinite-dimensional case. Let now V be an infinite dimensional vector space and let A : V → V be a linear operator. Then A is not necessarily invertible even if Ker A = {0}. Definition 1. A linear operator A : V 1 → V 2 is called Fredholm if dim(Ker(A)) < ∞ and dim(Coker(A)) < ∞. The following definition was first suggested by Fritz Noether Definition 2. The index Ind A of a Fredholm operator A : V 1 → V 2 is defined by the formula Ind A := dim(Ker(A)) − dim(Coker(A)) (1.2) This definition is interesting because of the following theorems, which show the " stability " of the index