Understanding the notion of the fundamental group

Question

Can anyone explain what we mean by homotopy classes and for how can for example the fundamental group of the real number line be 0? And what exactly does the fundamental group consist of?

Answer 1

If you want a full, rigorous, Mathematical description of what is meant by the fundamental group you will need to watch some lectures or read some notes. But I guess an informal explanation would help with understanding intuitively what it all means.

Two continuous functions, ${f : X\to Y}$ and ${g:X\to Y}$ are said to be homotopic, written as ${f\simeq g}$, if there is some nice continuous way to "deform" $f$ into $g$. It turns out that the ${\simeq}$ symbol actually gives us an equivalence relation. That means that given some continuous function $f$, we can "slot it into" exactly one class of functions that are all homotopic to each other basically.

Now - a loop in a topological space is a nice continuous path that starts and ends at the same point. Loops are continuous functions, and so these have their own "homotopy classes". In other words - every loop can be fit nicely into exactly one class of loops, and all loops within their classes can be "deformed" nicely into one another.

With this in mind, the fundamental group is essentially the set of these "loop homotopy classes", where we define some nice group operation on these classes of loops (we do the "path product").

The reason the fundamental group of the real line, ${\pi_1(\mathbb{R})}$, is $0$ ($0$ in this context means "the group with just the identity element") is because any loop in ${\mathbb{R}}$ can be deformed continuously into any other loop in ${\mathbb{R}}$ (it's not difficult to see this). So there is only one "loop" class, and it must be the identity element (since groups need at the minimum the identity element).