Algebraic and Differential Topology presents in a clear, concise, and detailed manner the fundamentals of homology theory. It first defines the concept of a complex and its Betti groups, then discusses the topolgoical invariance of a Betti group. The book next presents various applications of homology theory, such as mapping of polyhedrons onto other polyhedrons as well as onto themselves. The third volume in L.S. Pontryagin's Selected Works, this book provides as many insights into algebraic topology for today's mathematician as it did when the author was making his initial endeavors into this field.

R. V. Gamkrelidze

FOUNDATIONS OF ALGEBRAIC TOPOLOGY, CHAPTER I: COMPLEXES AND THEIR HOMOLOGY GROUPS, CHAPTER II: INVARIANCE OF HOMOLOGY GROUPS CHAPTER III. CONTINUOUS MAPPINGS AND FIXED POINTS, SMOOTH MANIFOLDS AND THEIR APPLICATIONS IN HOMOTOPY THEORY CHAPTER I: SMOOTH MANIFOLDS AND THEIR SMOOTH MAPPINGS CHAPTER II: FRAMED MANIFOLD, CHAPTER III: THE H OPF INVARANT ..CHAPTER IV: CLASSIFICATION OF THE APPINGS OF (« + 1)- A N D (« + 2)-DIMENSIONAL SPHERES INTO AN «-DIMENSIONALSPHERE