Index theory

Summary

Index theory studies the Fredholm index of elliptic operators on manifolds. Its central result is the Atiyah–Singer index theorem. It says that the analytic index of an elliptic operator on a closed manifold equals its topological index. The topological index is an integer built from characteristic classes. The theory links analysis, topology and geometry. Classical results such as the Chern–Gauss–Bonnet theorem, the Hirzebruch signature theorem and the Hirzebruch–Riemann–Roch theorem are special cases.