Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to...

One of the best ways to learn topology is to read the classics. This translation of the 2005 Russian edition include the original methods and constructions from the originals of the 1950s and 1960s. The extended entries include Pontrjagin's article on smooth manifolds and their application in homotopy theory; Thoms' work on global properties of differential manifolds; Novikov's paper on homotropy properties of Tom complexes; papers by Smale on the generalized Poincare's conjecture in dimensions greater than four and the structure of manifolds; Quillen's article on the formal group laws of...

This is the second of a three-volume set collecting the original and now-classic works in topology written during the 1950s-1960s. The original methods and constructions from these works are properly documented for the first time in this book. No existing book covers the beautiful ensemble of methods created in topology starting from approximately 1950, that is, from Serre's celebrated “singular homologies of fiber spaces.”

Presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds. This book unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory.