Lagmental Vicfred

The Lifting Criterion Is a Subgroup Condition by Vicfred

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

A map lifts to a covering space exactly when its induced fundamental group lands in the covering subgroup. A small computation will anchor the general statement before the abstraction takes over.

The data

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.

$$ p:(\widetilde X,\tilde x_0)\to(X,x_0),\qquad f:(Y,y_0)\to(X,x_0) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \exists\tilde f:Y\to\widetilde X,\ p\tilde f=f\Longleftrightarrow f_\ast\pi_1(Y,y_0)\subseteq p_\ast\pi_1(\widetilde X,\tilde x_0) $$

Derivation

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \begin{array}{ccc}&\widetilde X&\\[-2pt]\tilde f\nearrow&&\downarrow{\scriptstyle p}\\Y&\xrightarrow{\ f\ }&X\end{array} $$

Invariant content

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad p:(\widetilde X,\tilde x_0)\to(X,x_0),\qquad f:(Y,y_0)\to(X,x_0),\\[5pt] \mathsf{C}\;&:\quad \exists\tilde f:Y\to\widetilde X,\ p\tilde f=f\Longleftrightarrow f_\ast\pi_1(Y,y_0)\subseteq p_\ast\pi_1(\widetilde X,\tilde x_0). \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists\tilde f:Y\to\widetilde X,\ p\tilde f=f\Longleftrightarrow f_\ast\pi_1(Y,y_0)\subseteq p_\ast\pi_1(\widetilde X,\tilde x_0) \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Thu 02 May 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.