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From what I understand, the fundamental group of a topological space $X$ with base point $x_0$ is the set of all equivalence classes of continuous paths in $X$ that start and end at $x_0$. Formally speaking:

Let $X$ be a topological space and let $x_{0}$ be an element of $X$. Consider the set $S$ of continuous loops with base point $x_{0}$:

$$S=\{f:[0,1]\rightarrow X:f(0)=f(1)=x_{0}\}.$$

The fundamental group of $X$ with base point $x_{0}$, denoted $\pi_{1}(X,x_{0})$, is $S$ modulo homotopy $h$ $(S\bmod h)$, or simply $S/h$. I'm not going to write out the group operation here, because it doesn't relate to my question.

I'm wondering: when the definition says "$S$ modulo homotopy $h$", what does this mean without defining the homotopy $h$? Does it mean some homotopy $h$? Any homotopy $h$?

I mean we can speak of the set of integers modulo $5$, which is the set of congruence classes of $5$ (i.e., $\{[0]_{5},[1]_{5},[2]_{5},[3]_{5},[4]_{5} \}$. So the set of integers modulo $n$ would be $\{[0]_{n},[1]_{n},...,[n-1]_{n}\}$. But without saying what $n$ is, what good is it using that in a definition of something else? For example, $3$ is congruent to $7$ modulo $4$, so $7 \in [3]_4$. But if $n=8$, then $3$ and $7$ form two different classes. I guess what I'm getting at is: without specifying the homotopy $h$, how can we define what the set of loops modulo $h$ is?

I intuitively understand the fundamental group as being the set of all loops and their deformations that start and end at $x_0$, but are not homotopic to each other. Or, we could say the set of all equivalence classes of loops that start and end at $x_0$, under the relation homotopy. So we don't include $f_2$ in the fundamental group if $f_1$ and $f_2$ are homotopic. But for one: I'm not sure if either of my intuitive understandings are right, and two: I can't put the intuitive definition together with the formal one. Particularly, this stems from the use of the word "modulo".

If anyone can help me with understanding what the fundamental group is (more generally, a homotopy group), I would appreciate it.

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  • $\begingroup$ It's not "modulo a homotopy," but modulo the equivalence relation defined by all homotopies between elements of $S$. $\endgroup$ – Thomas Andrews Dec 26 '14 at 2:45
  • $\begingroup$ So am I correct when I say that the fundamental group is just the set containing the equivalence classes of loops, where two loops are in the same equivalence class if they are homotopic (i.e., they are equivalent under the relation homotopy)? EDIT: This is a comment for all answers. $\endgroup$ – Sultan of Swing Dec 26 '14 at 2:50
  • $\begingroup$ Yes, that's it exactly. In general, we take a set $X$ "modulo an equivalence relation $R$ on $X$." In the case of integers modulo $m$, the equivalence relation is $x\equiv y$ if $m\mid x-y$. So $m$ here is shorthand for this congruence equivalence relation. (Although I suspect the word "modulo" was originally used for the integer case, and only generalized later.) $\endgroup$ – Thomas Andrews Dec 26 '14 at 2:53
  • $\begingroup$ I suppose that was the nature of my confusion, is the use of the word modulo. I was trying to generalize the case of integers modulo $m$. $\endgroup$ – Sultan of Swing Dec 26 '14 at 3:00
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The notion is that of a set "mod" a relation: we take equivalence classes under the relation.

In this case, the relation is "two loops being homotopic, holding basepoints fixed". (There's no single homotopy $h$, by the way -- we speak of "classes of curves, mod basepoint-fixing homotopies".)

For the circle, any path that misses some point turns out to be homotopic to a constant curve (one that never moves away from the basepoint). Any curve that "wraps" around the circle counterclockwise is different from that (i.e., not homotopic to any loop in that class). Any curve that wraps counterclockwise twice is different from those. In the end, the number of times the curve wraps around the circle gives an isomorphism to $\mathbb Z$, so we say that the fundamental group of the circle is $\mathbb Z$.

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Since homotopy is an equivalence relation between loops, one can speak of ‘the equivalence class of a loop’ modulo or under this relation, just as for any other equivalence relation. One speaks then of ‘S modulo homotopy’, which means that two loops in $S$ (with the same base point $x_0$) are in the same class (homotopically equivalent) if there exists a homotopy $h$ such that &c.

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