Homotopy group

In mathematics, the first homotopy group of a pathwise-connected topological space $\cal{M}$, denoted by $\pi_1(\cal{M})$, is about the classification of all the loops in $\cal{M}$:

1. The relations between loops that can be continuously deformed into one another is an equivalence relation.
2. Then all loops in $\cal{M}$ can be partitioned into disjoint equivalent classes.
3. Finally, all equivalent classes of loops form a group. 

Note:

• Let $\alpha$ and $\beta$ be two loops in $\cal{M}$, the group multiplication between the equivalent classes $[\alpha]$ and $[\beta]$ is defined as $[\alpha]\cdot[\beta]=[\alpha * \beta]$, where $\alpha * \beta$ is a concatenated loop that we first traverse through the loop $\alpha$ and then through the loop $\beta$.
• In general, the n-th homotopy group, $\pi_n(\cal{M})$ is about the classification of the n-dimensional sphere $S^n$ in $\cal{M}$.

Examples:

• $\pi_1\left(S^1\right)=\mathbb{Z}$ suggests that we can label each equivalent class of loops in $S^1$ by an integer (winding number). Intuitively, if we wind a line along $S^1$ first by $n$ rounds and then by $m$ rounds, what ends up is a loop with $n+m$ rounds.
• $\pi_1\left(SO(3)\right)=\mathbb{Z}_2$ suggests that there are only two equivalent class of loops in SO(3): one can be continuously contracted to a single point while the other cannot. However, any loop traversed twice can be contracted to point.

Rigorously, a path connecting two points $p$ and $q$ in the topological space $\cal{M}$ is a continuous function $\alpha(s): [0, 1]\rightarrow \cal{M}$ such that $\alpha(0)=p$ and $\alpha(1)=p$; Furthermore, a loop in $\cal{M}$ is simply a path $\alpha$ such that $\alpha(0)=p=\alpha(1)$.

With the boundary condition $\alpha(0)=\alpha(1)$, we can formally extend the domain of definition of the function $\alpha$ from the interval $s\in [0, 1]$ to $\mathbb{R}$, with the requirement that the function be periodic with the period 1: $\alpha(s)=\alpha(s+1)$. Note that $\alpha(s)=\alpha(s+1)$ for $s\in \mathbb{R}$ is exactly a function defined on $S^1$, a loop $\alpha$ in $\cal{M}$ can then be viewed as a map: $\alpha:\,S^1\rightarrow \cal{M}$. As a result, the homotopy group $\pi_n(\cal{M})$ is also about the classification of all the maps from $S^n$ to $\cal{M}$.

In physics applications, we usually consider a field $\psi(x)$ defined on some $d$-dimensional region. If $\psi$ is in some Lie group $G$ and $\psi(x)$ takes the same value on all region boundaries, we can use the homotopy group $\pi_d(G)$ to classify all field configurations of $\psi(x)$.