Index theorem example

Question

I understand characteristic classes, elliptic operators, vector bundles and the final statement of index theorem. I am desperately trying to work out an example of index theorem however almost all the refernces I have looked up are too dense for me. I have looked up Liviu I. Nicolaescu, spin geometry by Lawson, Atiyah-Singer Index Theorem by Shanahan and scouring the internet in general. The most common example I saw were Gauss Bonnet theorem, Hirzebruch signature theorem, Riemann Roch all require understanding of clifford algebra, twisted dirac operator etc which I do not know yet. Is there anything more accessible or am I just being over ambitious.

Answer 1

Index theory lies at the intersection of several areas of mathematics, including differential geometry, algebraic topology, functional analysis and operator theory. As such, it does require a lot of background knowledge to work out even easy examples. One fairly accessible example comes from the theory of Toeplitz operators. For each complex-valued continuous function $f\in C(\mathbb{T})$ on the circle, one associates an operator $T_f\in\mathbb{B}(\ell^2(\mathbb{N}))$, the Toeplitz operator with symbol $f$. The associated index theorem states that $T_f$ is a Fredholm operator precisely when $0$ is not in the range of $f$, and in this case the Fredholm index of $T_f$ equals the winding number of $f$ (considered as a loop around $0\in\mathbb{C}$), up to a choice of sign.

While this result is a corollary of the Atiyah-Singer index theorem, can be proven on its own with just a little understanding of operator theory and homotopy theory. The details of the proof can be found in multiple sources, for example, one can consult Chapter 3 of Gerard Murphy's book "$C^*$-Algebras and Operator Theory", or Chapter 7 of Ron Douglas's book "Banach Algebra Techniques in Operator Theory".