I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the isomorphism between the homotopy classes of maps from a compact topological space to the space of Fredholm operators and the first K-group of the topological space given by the index map, but is there a simpler example of why we want to study the index?



Index of an Operator
Usually one really wants to know the dimension of the kernel of an operator rather than its index. The problem is that the dimension of the kernel is not a continuous function of the operator, so is very hard to compute in terms of topological data. The key point about the index is that it is a continuous function of the operator, which makes it a lot easier to compute. And it's a modification of the kernel, so with some luck one can compute the kernel from it. 


