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An Introduction to Algebraic Topology (1988), Springer-Verlag; ISBN 0-387-96678-1 An Introduction to the Theory of Groups (1995), Springer-Verlag; ISBN 0-387-94285-8 A First Course in Abstract Algebra (2000), Prentice Hall; ISBN 0-13-011584-3
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .
Download QR code; Print/export Download as PDF; ... A., Algebraic Topology, Cambridge University Press (2002) ISBN ...
isbn 978-0-387-94087-8. Hatcher, Allen (2002), Algebraic Topology , Cambridge: Cambridge University Press, ISBN 0-521-79540-0 . Jean-Pierre Marquis (2006) "A path to Epistemology of Mathematics: Homotopy theory", pages 239 to 260 in The Architecture of Modern Mathematics , J. Ferreiros & J.J. Gray , editors, Oxford University Press ISBN 978-0 ...
The Betti numbers of the manifold are the rank of the free part of the homology group, and in the special case of surfaces, the torsion part of the homology group only occurs for non-orientable cycles. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology". [26]
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]
He was known for his book on non-Euclidean geometry (1st edition, 1974; 4th edition, 2008) [3] [4] and his book on algebraic topology (1st edition, 1967, published with the title Lectures on Algebraic Topology; revised edition published, with John R. Harper as co-author, in 1981 with the title Algebraic Topology: A First Course). [5] [6] [7]
However, in 2012 Bourbaki resumed publication of the Éléments with a revised and expanded edition of the eighth chapter of Algebra, the first of new books on algebraic topology (covering also material that had originally been planned as the eleventh chapter of the group's book on general topology) [11] and the two volumes of significantly ...