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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.
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory.In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X 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
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces The main article for this category is Algebraic topology . Contents
The Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks.
In mathematics, especially in algebraic topology, the homotopy limit and colimit [1] pg 52 are variants of the notions of limit and colimit extended to the homotopy category (). The main idea is this: if we have a diagram:
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]
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 .