This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

Algebraic K-theory describes a branch of algebra that centers about two functors. K₀ and K₁, which assign to each associative ring ∧ an abelian group K₀∧ or K₁∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K₂, also from associative rings to abelian groups. Just as functors K₀ and K₁ are important to geometric topologists, K₂ is now considered to have similar topological applications. The exposition includes, besides K-theory, a...