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Topology

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Moderate

Identify The Covering Space

TOPO-CGZUVJ

Let $X=S^1\vee S^1$ be a graph with one vertex and two directed edges $a,b$. We know by the Seifert--van Kampen theorem that $\pi_1(X)=\langle a,b\rangle$.

Which of the following covering spaces is the connected covering space of $X$ corresponding to the subgroup $\langle a^3,b^2,aba^2b,a^2bab\rangle\lhd\pi_1(X)$?

A

A positively oriented equilateral triangle formed by three $b$'s whose vertices split a positively oriented circumscribed circle into three $a$'s.

B

A negatively oriented equilateral triangle of $b$'s whose vertices split a positively oriented circumscribed circle into three $a$'s.

C

Two concentric circles, both positively oriented and formed by three $a$'s, but with $b$'s going from each circle to the other at corresponding vertices.

D

Two concentric circles of opposite orientation formed by three $a$'s but with $b$'s going from each circle to the other at corresponding vertices.

E

A union of a doubly infinite sequence of circles each tangent to its two neighbors, alternating between a positively oriented circle formed by two $a$'s and a positively oriented circle formed by two $b$'s.