Algebraic topology

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Algebraic topology is a branch of mathematics that uses abstract algebra to understand topological spaces, and topology to understand abstract algebra.

The primary method of algebraic topology is to identify some kind of relation between a group (or a ring or other structure) and a topological space. Once such a relation is established, known results in one field can yield results in the other field.

Important results in this field include the Poincare conjecture, the unclassifiable nature of four-dimensional manifolds, and the proof that subgroups of free groups are necessarily free themselves.

The Homotopy Groups

One of the most elementary tools in algebraic topology is the fundamental group, which is symbolized by \pi_{1}(X) \ for a given topological space X \ . Some details must be ironed out: for example, the investigation of loops which are very similar gives no insight into the structure of the space, so we work instead with sets of loops called homotopy classes.

The formal definition is as follows: Let p \ be a fixed point in X \ . Consider the space of all curves \gamma:[0,1]\rightarrow X \ which begin and end at p \ . We consider two such curves to be (homotopically) equivalent if we can continuously deform the first curve into the second; that is to say, we consider \gamma_1 \ to be equivalent to \gamma_2 \ if there exists a mapping H:[0,1]\times[0,1]\rightarrow X \ with H(t,0) = \gamma_1(t) \ and H(t,1) = \gamma_2(t) \ . We can define an group structure on the resulting equivalence class of curves by declaring \gamma_1\cdot\gamma_2 \ to be the curve \gamma_2 \ followed by \gamma_1 \ , and rescaled so that the resulting curve still has domain [0,1]. This group is called the fundamental group of X \ .

Higher homotopy groups can be defined, and are written as \pi_n(X) \ . These are defined in just the same way, except we consider maps \gamma:[0,1]^n \rightarrow X \ instead.

A very important example is the topological space \mathbb{S}^1 \ , the circle. It can be shown that the fundamental group of \mathbb{S}^1 \ is isomorphic to the additive group of integers \mathbb{Z}. Other examples include the torus, \pi_1(\mathbb{T}^n) \ , which is the free abelian group on n \ generators, the sphere

\pi_m(\mathbb{S}^n) = \begin{cases} \left\{{e}\right\}, & \mbox{if }  m< n  \\ \mathbb{Z},  & \mbox{if } m=n \end{cases} \

where the former is the group with one element.

The Homology Groups

Homology groups, while lacking the simple definition of the homotopy groups, tend to be more studied and more useful for classification of topological spaces. There are several approaches to the study of homology: simplicial, which forms groups out of chain complexes, and de Rham, which forms groups from differential forms.

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