# Index of an Operator

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?

I would say that the basic reason why the index of an operator is an interesting number is that one may care about the dimension of the vector space of smooth solutions of the corresponding differential equation. The index theorem serves to describe this number in terms of the symbol of the operator (in one formulation, using cohomology, in another, using $K$-theory). From my point of view, observations about the homotopy type of the space of Fredholm operators have a place in explaining why the theorem is true, not why it is interesting or useful.