The notion of a fundamental group: How strong is it?

Abstract: This post will demonstrate the strength of the notion of a fundamental group by indicating several well-known problems whose solutions can be found much easier thanks to the fundamental group. 

1. Compute the (co)homology groups

   The (co)homology groups of an arbitrary topological space X are defined complicatedly via the groups generated by all continuous maps from the standard simplices to X. Meanwhile, fundamental groups allow us to restrict ourselves to the space X alone. By Hurewicz theorem, which was named after the Polish mathematician Witold Hurewicz, in certain cases, we can compute the (co)homology groups via the fundamental groups, which simplify the set of mathematical objects that we need to consider.

   Example: Consider the unorientable surface of genus g. Its fundamental group has the following representation: <c(1), c(2), ..., c(g) | c(1)^2c(2)^2...c(g)^2>. By Hurewicz theorem, its first homology group is the abelization of this group. In other words, the first homology group has the following representation: <c(1), c(2), ..., c(g) | c(1)^2c(2)^2...c(g)^2, c(i)c(j) = c(j)c(i) for all 1 <= i < j <= g> (i.e. obtained from the representation of the fundamental group by adding the commutativity of all pairs of generators). It is the quotient group of Z^g obtained by identifying (2, 2, ..., 2) with the identity element, which is isomorphic to Z^(n - 1) x Z_2. It is easier to see this if we consider the basis {(1, 0, 0, ..., 0), (0, 1, 0, ..., 0), ..., (0, 0, ..., 0, 1, 0), (1, 1, ..., 1, 1)} instead of the standard basis. 

2. Proof of Nielsen - Schreier's theorem

   Nielsen - Schreier's theorem is named after the Danish mathematician Jakob Nielsen (1890 - 1959) and the Jewish - Austrian mathematician Otto Schreier (1901 - 1929). It is stated as follows: any subgroup of a free group is free. In fact, there are other ways to prove this theorem without using the fundamental group. Fundamental group helps us to tackle the geometric picture of the free group instead of considering everything group-theoretically or combinatorially. 

   The topological idea of the group can be briefly explained as follows. First, a free group can be considered as the fundamental group of a wedge sum of circles such that the cardinality of the set of circles equals the rank of the free group. As a result, each of its subgroup is also the fundamental group of a covering space of this wedge sum, which is also a graph. By contracting the "remaining" edges, the graph can be converted to a wedge sum of circles that has the same fundamental group. The fundamental group of a wedge sum of circles is a free group, which yields the theorem. 

   The idea of contraction is illustrated in the figure below, which appears in the book "A course on geometric group theory" (Brian H. Bowditch). To be more exact, in the finite case, if the connected graph has V vertices and E edges, then its fundamental group is the free group of rank E - V + 1. For example, in the following image, since there are five edges "outside" the spanning tree, the fundamental group is F_5. 

3. Proof of Brouwer's fixed point theorem

   Brouwer's fixed point theorem is perhaps a "classical" and "well-known" theorem, which was named after the Dutch mathematician Luitzen Egbertus Jan "Bertus" Brouwer (1881 - 1966). It states that any continuous mapping f from the disk D^2 to itself has a fixed point. To prove this theorem, for each continuous mapping f, consider the mapping g defined as follows. For each x, consider the ray whose origin is x and goes through f(x). The point g(x) is the intersection point of the ray and the boundary of the disk D^2. It is obvious that g is continuous, and its restriction on the boundary of D^2 is the identity map, thus it is a retraction. While these might seem to be elementary analytic arguments, things become different when the fundamental group appears. 

   First, note that the fundamental group of D^2 is the trivial group and the fundamental group of its boundary is the additive group of integers. If there is a retraction r from D^2 to its boundary, consider the natural embedding i from D^2's boundary to D^2 that send each point to itself. The composition of r and i is the identity mapping, thus its induced homomorphism must send each element of the fundamental group of D^2's boundary to itself. Since the domain of the induced homomorphism of r has only an element, while the fundamental group of D^2's boundary has infinitely many elements, this can't be the case. This leads to a contradiction, which yields the Brouwer's fixed point theorem.