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Topology

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Moderate

Two Tori With Common Circle

TOPO-VQGXKQ

Let $X$ be the space obtained from two tori $S^1\times S^1$ by identifying a circle $S^1\times\{x_0\}$ in one torus with the corresponding circle $S^1\times\{x_0\}$ in another.

Which of the following is a valid presentation of $\pi_1(X)$?

A

$\langle a,b,c\rangle$

B

$\langle a,b,c\mid aba^{-1}b^{-1}\rangle$

C

$\langle a,b,c\mid aba^{-1}b^{-1},aca^{-1}c^{-1}\rangle$

D

$\langle a,b,c\mid aba^{-1}b^{-1},aca^{-1}c^{-1},bcb^{-1}c^{-1}\rangle$

E

$\langle a,b,c\mid abca^{-1}b^{-1}c^{-1}\rangle$