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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
Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring with any topological space. Every continuous map: determines a homomorphism from the cohomology ring of to that of ; this puts strong restrictions on the possible maps from to .
Differential graded algebra: the algebraic structure arising on the cochain level for the cup product; Poincaré duality: swaps some of these; Intersection theory: for a similar theory in algebraic geometry
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.
Let : be a Serre fibration of topological spaces, and let F be the (path-connected) fiber.The Serre cohomology spectral sequence is the following: , = (, ()) + (). Here, at least under standard simplifying conditions, the coefficient group in the -term is the q-th integral cohomology group of F, and the outer group is the singular cohomology of B with coefficients in that group.
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]
In algebraic topology, the pushforward of a continuous function : between two topological spaces is a homomorphism: () between the homology groups for . Homology is a functor which converts a topological space X {\displaystyle X} into a sequence of homology groups H n ( X ) {\displaystyle H_{n}\left(X\right)} .