Undergrad presenter: Finzi, Leonardo A
Title: The Galois Correspondence of Covering Spaces
Faculty Advisor: Carl Wang-Erickson
Description: In field theory, Galois theory allows us to translate the problem of finding intermediate field extensions into the problem of finding subgroups of a group. Analogously, given certain "nice" conditions on a topological space $X$, we can translate between problems involving covering spaces of $X$ and problems involving groups. In particular, we correspond to any covering spaces of $X$, a subgroup of the fundamental group $\pi_1(X, x)$, for some fixed $x \in X$. As in the "usual" Galois theory, the power of this result lies in the fact that it allows us to algebraically determine the pointed covering spaces of $X$, which is generally difficult to do by topological inspection. This presentation presents a proof for this correspondence by showing that the category of orbits of $\pi_1(X,x)$ is equivalent to the category of the covers of $X$.
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704 Thackeray