The restricted holonomy group is a normal subgroup

Question

I've been studying theory of connections, and in order to prove that the restricted holonomy group is a normal subgroup of the holonomy group, the book says: 'If $\tau$ and $\mu$ are two loops at $x$ and if $\mu$ is homotopic to zero, the composite curve $\tau \cdot \mu \cdot \tau^{-1}$ is homotopic to zero.' The problem is that it gives no further explanation. Why is that true?

Answer 1

Let $(X,x)$ be a pointed space (in your case $X$ is a manifold $M$), one defines the fundamental group of $X$ at $x$ denoted by $\pi_1(X,x)$ as follows.

First, consider the set of continuous loops at x, $\lbrace\tau:[0,1]\to X\, |\, \tau(0)=\tau(1)=x\rbrace$, and define a homotopy $H$ from $\tau$ to $\mu$ as a continuous map $H:[0,1]\times [0,1]\to X$ such that for all $t\in [0,1]$, $H(0,t) = \tau(t)$, $H(1,t) = \mu(t)$, and $H(t,0)=H(t,1)=x$.
This defines an equivalence relation on the set of loops at x, ie the relation, $\tau \sim \mu$ if and only if there exists a homotopy from $\tau$ to $\mu$, is an equivalence relation. Let $[\tau]$ denote the equivalence class of a loop $\tau$.

Second, take $0= [c_x]$, where $c_x$ is the constant loop at $x$; and $[\tau]^{-1} = [\tau^{-1}]$, where $\tau^{-1}$ denotes the inverse loop of $\tau$, ie $\tau^{-1}(t) = \tau(1-t)$; and $[\mu].[\tau] = [\mu.\tau]$, where $\mu.\tau$ denotes the concatenation of the loop $\tau$ followed by the loop $\mu$, namely if $0\leq t\leq 1/2$ then $(\mu.\tau)(t) = \tau(2t)$, and if $1/2\leq t\leq 1$ then $(\mu.\tau)(t) = \mu(2t-1)$.
This defines a group structure on the set of equivalence classes. One denotes the corresponding group by $\pi_1(X,x)$.

Now one has $[\tau.\mu.\tau^{-1}] = [\tau].[\mu].[\tau]^{-1}$, in particular if $\mu$ is homotopic to $0$ (ie $[\mu] = [c_x] = 0$) then $[\tau.\mu.\tau^{-1}] = 0$.
Note that when you write something like $\tau.\mu.\tau^{-1}$ you should use parentheses, since the concatenation of loops (and more generally of paths) is not associative. However, the composition becomes associative after passage to equivalence classes (this is the reason why one has a group law as claimed above).