TITLE: Index of a family of operators
QUESTION [11 upvotes]: In the usual setting of the Atiyah-Singer index theorem the situation is as follows: we have a closed smooth manifold $M$ without boundary and $D$ is some elliptic differential operator acting on sections of some vector bundle over $M$. In this situation $D$ turns out to be Fredholm and one can compute its index which is an integer. However I read that AS index theorem has various generalisations: one of them is about a whole family $(D_x)_{x \in X}$ of elliptic operators parametrized by some topological space. I read that in such a situation one can define the index of this family as an element of $K^0(X)$. 

Why the index of such a family is not just an integer valued function? Why it is defined as an element of $K$-theory?

REPLY [6 votes]: This is just an addendum to Sebastian Goette's excellent answer:
In fact, you can retrieve the integer-valued function that you mention from the $K$-theory class: Let $\iota_x \colon \{pt\} \to X$ be the inclusion that sends $pt$ to $x \in X$. Let $ind(D) \in K^0(X)$ be the $K$-theory class representing the family index of $(D_x)_{x \in X}$. The function you mention will be 
$$
X \to K^0(pt) \cong \mathbb{Z} \quad ; \quad 
x \mapsto \iota_x^*(ind(D))
$$
But this loses a lot of information. In fact, it will just recover the virtual dimension of the fiber over $x$ of the vector bundle Sebastian Goette is talking about in his answer. The index class $ind(D) \in K^0(X)$ also contains global information about "how non-trivial" the vector bundle is. The $K$-theory class $ind(D) \in K^0(X)$ is not a bug, it is a feature!