Index Theory

Index Theory

The  aim of this  course is to introduce  topological K -theory,  formulate  the  index theorem,  sketch the K -theoretic  proof and  exhibit  its power by deriving celebrated  results such the Gauss-Bonnet theorem and Hirzebruch’s signature  theorem.
An outline  of the  course is as follows.  We begin  by defining vector  bundles  and  generalising standard  constructions such  as  direct  sum,  tensor  product  and  exterior  powers,  from  vector spaces to vector  bundles.  We explain how to define K -theory  by considering  formal differences of vector bundles.  After establishing  some formal properties  of K-theory,  we turn  to the theory of characteristic classes and Chern-Weil  theory.  As a first application we present an easy proof, due to Adams  and  Atiyah,  of the  Hopf invariant  one theorem  using certain  operations  on K-theory.    One  easy  consequence  of this  theorem  is that there  are  no real  division  algebras  in dimensions  other  than  1, 2, 4 and 8.
Next we introduce  elliptic differential  operators  (or more generally,  elliptic complexes) on com- pact  manifolds  and  explain  how their  symbol  determines  a class in compactly  supported K-theory.    We  define the  notion  of analytic  index  and  topological  index  for elliptic  operators. The Atiyah-Singer index theorem  is the statement that these two index maps are equal.  After explaining  the  basic idea behind  the  proof and  deriving  a cohomological  formula,  we consider several applications of the index theorem.


Referências:

A. Hatcher, Algebraic Topology.
A. Hatcher, Vector Bundles and K-theory.
P. Shanahan, The Atiyah-Singer Index Theorem, Lecture Notes in Mathematics, 638, Springer, Berlin, 1978.
H. B. Lawson and M.-L. Michelson, Spin Geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton, 1989.