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 significantly to hope for and it is much more prudent to aim for a modest objective, classification up to homotopy equivalance.

The algebraic tools utilised in topology contain a variety of cohomology theories, homology theories, homotopy groups, and groups of maps. These in turn have necessitated the development of far more complex algebraic tools such as derived functors and spectral sequences the machinery mostly derived from homological algebra is effective if rather daunting.

The value of algebraic topology is measured by the extent to which it answers questions which arise a lot more geometrically. These classical topics were, just before the development of the full algebraic tools, frequently treated as questions involving numbers. For example, the dimension of a real vector space is a topological invariant say, Rn is not homeomorphic to Rm if n &gt m, but this was not easy to prove for m &gt 1. Likewise, several 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-identified examples. Other questions of a geometric nature which can be addressed with an appeal to the algebra contain 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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