Lagmental Vicfred

Loop Concatenation Defines the Fundamental Group by Vicfred

tags: mathematics, fundamental groups, covering spaces, van Kampen theorem

Based loops multiply by traversing one path and then the other with a reparameterization. I will separate the object being defined from the consequence being claimed.

Notation

The fundamental group \(\pi_1(X,x_0)\) records based loops modulo based homotopy. A covering map \(p:\widetilde X\to X\) turns loop classes into endpoint data upstairs.

$$ \alpha,\beta:[0,1]\to X,\qquad\alpha(0)=\alpha(1)=\beta(0)=\beta(1)=x_0 $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ [\alpha]\,[\beta]=[\alpha*\beta] $$

Stress the formula

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ (\alpha*\beta)(t)=\begin{cases}\alpha(2t),&0\le t\le\tfrac12,\\\beta(2t-1),&\tfrac12\le t\le1.\end{cases} $$

Interpretation

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad \alpha,\beta:[0,1]\to X,\qquad\alpha(0)=\alpha(1)=\beta(0)=\beta(1)=x_0,\\[5pt] \mathsf{C}\;&:\quad [\alpha]\,[\beta]=[\alpha*\beta]. \end{aligned} $$

Limit of the argument

Basepoints matter for literal homomorphisms. Changing basepoint produces an isomorphism only after choosing a path, and the choice is visible up to conjugation.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] [\alpha]\,[\beta]=[\alpha*\beta] \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Sun 01 March 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.