Algebraic Topology is that branch of Mathematics which uses the algebra to study topological problems. The goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situtations this is too much to hope for and it is more prudent to aim for a modest goal, classification up to homotopy equivalance.
The algebraic tools used in topology include various cohomology theories, homology theories, homotopy groups, and groups of maps. These in turn have necessitated the development of more complex algebraic tools such as derived functors and spectral sequences; the machinery mostly derived from homological algebra is powerful if rather daunting.
The value of algebraic topology is measured by the extent to which it answers questions which arise more geometrically. These classical topics were, before the development of the full algebraic tools, often treated as questions involving numbers. For example, the dimension of a real vector space is a topological invariant say, R^n is not homeomorphic to R^m if n > m, but this was not easy to prove for m > 1. Likewise, many questions about maps between n-dimensional spaces could be resolved with an appeal to the degree of the map, an extension of the winding number from complex analysis, due to Brouwer– the Fundamental Theorem of Algebra and the Hairy Ball Theorem are two well-known examples. Other questions of a geometric nature which can be addressed with an appeal to the algebra include fixed-point theorems. For example, Lefschetz’s theorem guarantees a fixed point if a certain calculated number is nonzero; Smith theory studies the regularity of fixed points under self-homeomorphisms of finite order.
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