Lectures on Algebraic Topology : Reprint of the 1972 Edition

Originally published as volume 200 in the series: Grundlehren der mathematischen Wissenschaften This is essentially a book on singular homology and cohomology with special emphasis on products and manifolds. It does not treat homotopy theory except for some basic notions, some examples, and some applica­ tions of (co-)homology to homotopy. Nor does it deal with general(-ised) homology, but many formulations and arguments on singular homology are so chosen that they also apply to general homology. Because of these absences I have also omitted spectral sequences, their main applications in topology being to homotopy and general (co-)homology theory. Cech­ cohomology is treated in a simple ad hoc fashion for locally compact subsets of manifolds; a short systematic treatment for arbitrary spaces, emphasizing the universal property of the Cech-procedure, is contained in an appendix. The book grew out of a one-year's course on algebraic topology, and it can serve as a text for such a course. For a shorter basic course, say of half a year, one might use chapters II, III, IV (§§ 1-4), V (§§ 1-5, 7, 8), VI (§§ 3, 7, 9, 11, 12). As prerequisites the student should know the elementary parts of general topology, abelian group theory, and the language of categories - although our chapter I provides a little help with the latter two. For pedagogical reasons, I have treated integral homology only up to chapter VI; if a reader or teacher prefers to have general coefficients from the beginning he needs to make only minor adaptions. Springer-Verlag began publishing books in higher mathematics in 1920, when the series Grundlehren der mathematischen Wissenschaften, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A few years later, a new series Ergebnisse der Mathematik und Ihrer Grenzgebiete, survey reports of recent mathematical research, was added. Of over 400 books published in these series, many have become recognized classics and remain standard references for their subject. Springer is reissueing a selected few of these highly successful books in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. Front Matter....Pages I-XI Preliminaries on Categories, Abelian Groups and Homotopy....Pages 1-15 Homology of Complexes....Pages 16-28 Singular Homology....Pages 29-53 Applications to Euclidean Space....Pages 54-84 Cellular Decomposition and Cellular Homology....Pages 85-122 Functors of Complexes....Pages 123-185 Products....Pages 186-246 Manifolds....Pages 247-367 Back Matter....Pages 368-377 Springer is reissuing a selected few highly successful books in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. Springer-Verlag began publishing books in higher mathematics in 1920. This is a reprint of the Second Edition.