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An H-space consists of a topological space X, together with an element e of X and a continuous map μ : X × X → X, such that μ(e, e) = e and the maps x ↦ μ(x, e) and x ↦ μ(e, x) are both homotopic to the identity map through maps sending e to e. [2]
Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage. Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1– 9.
The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows: the graded set of groups form a graded R-module; this can be given the structure of a graded R-algebra using the cup product; the Bockstein homomorphism β gives a differential.
Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc. May JP (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. Archived (PDF) from the original on 2022-10-09
Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0, MR 1867354 "Cohomology" , Encyclopedia of Mathematics , EMS Press , 2001 [1994] . May, J. Peter (1999), A Concise Course in Algebraic Topology (PDF) , University of Chicago Press , ISBN 0-226-51182-0 , MR 1702278
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946a, 1946b), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.
The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces X and Y. This is extraordinarily ambitious: in particular, when X is S n {\displaystyle S^{n}} , these maps form the n th homotopy group of Y .
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 978-0-19-856793-6