I am trying to construct the universal cover of a wedge sum of two tori. My question is: Can I treat $T^2 ∨ T^2$ as a special case of a two-holed torus with an octagon as its representation? Would appreciate any hints. Thanks
Universal cover of wedge sum $T^2 ∨ T^2$
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algebraic-topology
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0You cannot. The wedge point of $T^2 \vee T^2$ doesn't appear on the two-holed torus. – 2017-02-21
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1Have you seen the universal cover of $S^1\vee S^1$? Your cover space is somewhat similar, but with planes instead of lines. – 2017-02-21
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First we look at the universal cover of $S^1\vee S^1$, for analogy. It starts with a copy of $\Bbb R$ for one circle. At each integer point (which represent the wedge point), we wedge a copy of $\Bbb R$ representing the second circle. At each integer point of those lines, we attach a copy of $\Bbb R$, and so on.
For $T^2\vee T^2$, we start with a copy of $\Bbb R^2$ for the first torus. At each grid point (which represent the wedge point), we wedge a copy of $\Bbb R^2$, representing the second torus. At each grid point of each of those, we attach a copy of $\Bbb R^2$. At each grid point of each of those, we attach a copy of $\Bbb R^2$, and so on.