Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^\*$-algebras is an equivalence of categories. In other words, all topological properties of such a topological space $X$ are encoded in the algebraic properties (and of course the norm) of $C(X)$.


I have the following related questions:


1. How does one recognize that $X$ is simply connected from the algebraic properties of $C(X)$?
2. How can the fundamental group $\pi\_1(X)$ be defined in terms of the algebraic properties of $C(X)$?
3. If $\tilde{X}$ is a universal cover of $X$, what is the algebraic relation between $C(\tilde{X})$ and $C(X)$?



Fundamental group under Gelfand duality
It seems that this question has been investigated in the literature, where the fundamental group is approached through the notion of a regular covering. For instance, the following reference establishes an equivalence between the categories of regular coverings $\tilde{X} \to X$ and Galois extensions of $C(X)$, meaning that the algebraic analog of a universal cover is a universal Galois extension and the fundamental group is the group associated to that extension:



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> *Høst-Madsen, Anders*, [**Separable algebras and covering spaces**](http://dx.doi.org/10.7146/math.scand.a-14306), Math. Scand. 87, No. 2, 211-239 (2000). [ZBL1002.54010](https://zbmath.org/?q=an:1002.54010). [MR1795745](https://mathscinet.ams.org/mathscinet-getitem?mr=1795745).
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More literature on this topic is cited therein.


