24 - New ideas in algebraic topology

Summary

The final piece is an excellent survey article by Novikov. As with many other survey articles, the reader's first object in reading it should be to gain a general understanding of what is going on rather than a grasp of the technical details behind each sentence.

Introduction

In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems. The K-functor, which arose in algebraic geometry in the well-known work of Grothendieck, has been successfully applied by Atiyah and Hirzebruch to differential topology and has led quickly to the solution of a number of delicate problems.

Among the results obtained strictly with the help of K-theory the work of Atiyah and Singer on the problem of the index of elliptic operators and of Adams on vector fields on spheres and the Whitehead J-homomorphism are outstanding. More or less influenced by the K-functor other functors have appeared, with importance for topology – the J-functor, bordism theories and Milnor's microbundle k-functor. These have thrown new light on old results and have led to some new ones.