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Albrecht Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1. Allen Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the author's homepage
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 ...
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 ′ =.
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
An example of a classifying space is that when G is cyclic of order two; then BG is real projective space of infinite dimension, corresponding to the observation that EG can be taken as the contractible space resulting from removing the origin in an infinite-dimensional Hilbert space, with G acting via v going to −v, and allowing for homotopy ...
For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.
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 mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations .