Bott Periodicity and Thorn Isomorphism

Abstract

This chapter begins with a special kind of Fredholm operator called Toeplitz operator. A family of such operators over a space X leads us to Bott periodicity theorem in the following form

$$K\left( X \right) \otimes K\left( {{S^2}} \right) \cong K\left( {X \times {S^2}} \right),or\tilde K\left( X \right) \otimes \tilde K\left( {{S^2}} \right) \cong \tilde K\left( {X \wedge {S^2}} \right)$$

. Then we use Bott periodicity theorem to prove Thorn Isomorphism theorem for a complex vector bundle over X. This is approach is different from that considered in Atiyah and Bott [6]. The formulation of the proof of the Bott periodicity theorem presented here fits into the axiomatic treatment as given in Atiyah [5].