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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. Although algebraic topology primarily uses algebra to study topological ...
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 .
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
In mathematics, more specifically algebraic topology, a pair (,) is shorthand for an inclusion of topological spaces:.Sometimes is assumed to be a cofibration.A morphism from (,) to (′, ′) is given by two maps : ′ and : ′ such that ′ =.
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space [note 1] is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer . A connected topological space X is called an Eilenberg–MacLane space of type K ( G , n ) {\displaystyle K(G,n)} , if it has n -th homotopy group π n ...
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