How do I prove the existence of inverse of the fundamental group $ \pi ( X)$.

Question

I know that the definition of the inverse path $a^{-1}$ of a path $a$ is defined as $$a^{-1} (t) = a (|| a || - t ), 0 \leq t \leq ||a||.$$ And I am guessing that the identity is the constant loop $C(t) = p$ for all $ 0 \leq t \leq || a ||$. But still I do not know how to show the existence of the inverse. Could anyone help me please?

Answer 1

Let's start from the beginning and you can walk through each step along the way to make sure you're comfortable:

The elements of the fundamental group of $X$ at a fixed basepoint $p$ are paths $a : [0,1]\to X$ with $a(0)=a(1)=p$, quotiented out by the equivalence relation that $a = b$ if there is a homotopy $\kappa : a\to b$ that fixes the basepoint $p$; so $\kappa : [0,1]\times [0,1]\to X$ with $\kappa(t,0)= a(t)$ and $\kappa(t,1) = b(t)$ for all $t$, as well as $\kappa(0,s) = \kappa(1,s) = p$ for all $s \in [0,1]$.

Define the composition of $a \cdot b$ to be the function $(a\cdot b)(t) = a(2t), 0\leq t\leq 1/2$ ; $b(2t-1)$, $ 1/2\leq t \leq 1$.

Now the first thing you should do is prove that this is even a well-defined function on equivalence classes; that is, if $a$ and $a'$ are two paths that are homotopic, and $b, b'$ are two paths that are homotopic, you should prove that $a\cdot b$ and $a'\cdot b'$ are homotopic.

The next thing is to prove that this is associative up to homotopy; that is, in order for this to be a associative function on the set of homotopy equivalence classes, prove that $(a\cdot b)\cdot c = a\cdot (b\cdot c)$. You can do this by rescaling the paths, i.e. reparametrizing the curves by altering how much time it takes to move along each segment.

You are right that the identity is the constant loop $C(t)=p$ for all $t$. To prove that this is the identity you should verify that for any loop $a$, $a\cdot C = C\cdot a = a$ up to homotopy; again you can do this by reparametrizing the curves and showing that this constitutes a continuous deformation.

Now you have just written down a proposed definition of the inverse. So, first ask yourself: is this a loop with basepoint p? If yes, move on. Then you want to prove that $a\cdot a^{-1} = C(t)=p$, the constant loop, which means you want to demonstrate a homotopy of $a\cdot a^{-1}$ onto the constant path. Remember that $(a^{-1}\cdot a)(1/2) = p$, but it is no longer "attached" to $p$ - you can freely deform the curve away from $p$ at this point. Try writing down a retraction that pulls back $a\cdot a^{-1}$ along itself, entirely within the original path.