While I’m still continuing to work through Gallian, I recently stumbled across an interesting mention of group theory in the textbook (The Shape of Space, by Jeffrey Weeks) for my geometry and topology class this year. It mentioned that a number of techniques and tools can be used to determine if surfaces are in fact the same surface or not. This is not always immediately apparent, because surfaces can have different representations that can look completely different upon inspection. For example, the hexagonal 3-torus and the regular 3-torus are in fact topologically the same, even though they appear to be very different from their definitions.
One of the tools that can be used is the fundamental group, which can provide information about the properties of a surface. While there’s certainly no way that I could understand the formal definition of the fundamental group right now, given that it requires knowledge of formal definitions in topology (which our textbook does not include—instead, we’ve been taking an intuitive approach to the subject), I was able to find an intuitive definition of the fundamental group of a topological space on Wikipedia that actually does make sense to me. This intuitive definition reads:
Start with a space (e.g. a surface), and some point in it, and all the loops both starting and ending at this point — paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group for that particular space.
In short, the fundamental group of a space is the set of the equivalence classes of closed loops (starting and ending at some given point) in the space under the equivalence relation where two loops are equivalent if one can be continuously deformed into the other. The operation of the group is basically the concatenation of the two paths: trace one path and then trace the other. The resulting path is still a closed loop because it starts and ends at the same point and is continuous because the first two loops were both continuous and closed. Thus, the fundamental group is indeed a group.
Some examples: the fundamental group of the sphere is the trivial group (the group containing only the identity and nothing else) because any closed path can be continuously deformed into any other closed path. The fundamental group of the cylinder is isomorphic to the group of the integers under addition, where each equivalence class of loops is mapped to an integer based on how many times it wraps around the cylinder. If a loop wraps around twice, then it can’t be continuously deformed into a loop that wraps around only once. But any two loops that wrap around the same number of times are equivalent under this equivalence relation.
It can be sort of tricky to figure out a space’s fundamental group intuitively—the Klein bottle is one that pushes the limits of this intuitive definition—but I thought that it was cool that group theory was referenced in a book on intuitive topology, highlighting how group theory is at once both a field of study in and of itself as well as a tool that is leveraged in many other areas of mathematics.