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Algebraic topology

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Algebraic topology is the study of topology using methods from abstract algebra. In general, given a topological space, we can associate various algebraic objects, such as groups and rings.

Fundamental Groups

Perhaps the simplest object of study in algebraic topology is the fundamental group. Let X be a path-connected topological space, and let x\in X be any point. Now consider all possible "loops" on X that start and end at x, i.e. all continuous functions f:[0,1]\to X with f(0)=f(1)=x. Call this collection L. Now define an equivalence relation \sim on L by saying that p\sim q if there is a continuous function g:[0,1]\times[0,1]\to X with g(a,0)=p(a), g(a,1)=q(a), and g(0,b)=g(1,b)=x. We call g a homotopy. Now define \pi_1(X)=L/\sim. That is, we equate any two elements of L which are equivalent under \sim.

Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map 1:[0,1]\to X given by 1(a)=x for all a\in[0,1]. The inverse of a map h is the map h^{-1} given by h^{-1}(a)=h(1-a). We can compose maps as follows: g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases} One can check that this is indeed well-defined.

Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is {\mathbb{Z}}, which is abelian.

More generally, if X is an h-space, then \pi_1(X) is abelian, for there is a second multiplication on \pi_1(X) given by (\alpha\beta)(t) = \alpha(t)\beta(t), which is "compatible" with the concatenation in the following respect:

We say that two binary operations \circ, \cdot on a set S are compatible if, for every a,b,c,d \in S, (a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).

If \circ,\cdot share the same unit e (such that a \cdot e = e \cdot a = a \circ e = e \circ a = a) then \cdot = \circ and both are abelian.

Higher Homotopy Groups

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Homology and Cohomology

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