# Algebraic Topology

Algebraic topology is a branch of mathematics which 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 problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

### Other articles related to "topology, algebraic, algebraic topology":

Timeline Of Category Theory And Related Mathematics - 1971–1980
... which became the standard reference in category theory 1971 Horst Herrlich-Oswald Wyler Categorical topology The study of topological categories of structured sets (g ... General categorical topology study and uses structured sets in a topological category as general topology study and uses topological spaces ... Algebraic categorical topology tries to apply the machinery of algebraic topology for topological spaces to structured sets in a topological category ...
Algebraic Topology (object)
... In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e ... This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C) ... Another topology, the geometric topology (also called the Chabauty topology), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are not images of points in ...
Areas Of Mathematics - Major Divisions of Mathematics - Geometry and Topology
... geometry The study of geometry using calculus, and is very closely related to differential topology ... See also the glossary of differential geometry and topology ... Algebraic geometry Given a polynomial of two real variables, then the points on a plane where that function is zero will form a curve ...
Degree Of A Continuous Mapping - Definitions of The Degree - Between Manifolds - Algebraic Topology
... A continuous map f X→Y induces a homomorphism f* from Hm(X) to Hm(Y) ... Let, resp ...
Ronald Brown (mathematician) - Editing and Writing
... His mathematical research interests range from algebraic topology and groupoids, to homology theory, category theory, mathematical biology, mathematical ... Among his several books and standard topology and algebraic topology textbooks are Elements of Modern Topology (1968), Low-Dimensional Topology (1979, co-edited with T.L ... Thickstun), Topology a geometric account of general topology, homotopy types, and the fundamental groupoid (1998), Topology and Groupoids (2006) and Nonabelian Algebraic Topology Filtered ...

### Famous quotes containing the word algebraic:

I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
Henry David Thoreau (1817–1862)