Calculation of the Cobordism Groups

Abstract

In Chapter 12 we gave the structure of various cobordism groups Ω ^G_>* ≅ MG_*(S°) = π*(MG) without giving any proofs. As it turns out, the determination of the homotopy groups π*(MG) is a problem to which the Adams spectral sequence is particularly well adapted for the reason that the homology H*(MG;Z _p ) turns out to have a relatively simple structure as A*-comodule for G = O, U, SO and SU. In this chapter we shall use the Adams spectral sequence to compute Ω^o, Ω.U and Ω^SO We shall also prove the theorem of Stong and Hattori which says that π*(MU) → P(K*(MU)) is an isomorphism.