Use the preceding problem to show that a closed orientable surface \(M_g\) of genus \(g\) has a connected normal covering space with deck transformation group isomorphic to \(\mathbb{Z}^n\) (the product of \(n\) copies of \(\mathbb{Z}\)) if \(n \leq 2g\). For \(n = 3\) and \(g \geq 3\), describe such a covering space explicitly as a subspace of \(\mathbb{R}^3\) with translations of \(\mathbb{R}^3\) as deck transformations. Show that such a covering space in \(\mathbb{R}^3\) exists if there is an embedding of \(M_g\) in the 3-torus \(T^3 = S^1 \times S^1 \times S^1\) such that the induced map \(\pi_1(M_g) \to \pi_1(T^3)\) is surjective.
Solution
1. The abelianization of the genus-\(g\) surface group is\[ H_1(M_g;\mathbb Z) \cong \pi_1(M_g)_{\mathrm{ab}} \cong \mathbb Z^{2g}. \]
Using the standard presentation \[ \pi_1(M_g) = \left\langle a_1,b_1,\dots,a_g,b_g \ \middle|\ [a_1,b_1]\cdots[a_g,b_g]=1 \right\rangle, \] abelianization kills every commutator, so the single defining relation becomes trivial. The \(2g\) generators therefore freely generate the abelianization.
2. For every \(n\le2g\) there is a surjective homomorphism\[ \phi:\pi_1(M_g)\twoheadrightarrow\mathbb Z^n. \]
Compose the abelianization map \[ \pi_1(M_g)\twoheadrightarrow\mathbb Z^{2g} \] with projection onto any \(n\) coordinate factors.
3. Let\[ H=\ker\phi. \]The connected covering corresponding to \(H\) is normal and has deck group\[ \mathbb Z^n. \]
The kernel \(H\) is normal. For the associated connected normal cover \[ p:\widetilde M_g\to M_g, \] the deck group is \[ \pi_1(M_g)/H \cong \operatorname{im}\phi \cong\mathbb Z^n. \] This proves the first assertion for every \(n\le2g\).
4. Suppose now that\[ i:M_g\hookrightarrow T^3 \]is an embedding such that\[ i_*:\pi_1(M_g)\twoheadrightarrow\pi_1(T^3)\cong\mathbb Z^3 \]is surjective.Then the inverse image\[ \widetilde M_g:=\pi^{-1}(i(M_g))\subset\mathbb R^3, \]where\[ \pi:\mathbb R^3\to T^3=\mathbb R^3/\mathbb Z^3, \]is connected.
The restriction \[ \pi|_{\widetilde M_g}:\widetilde M_g\to M_g \] is the pullback of the universal cover of \(T^3\). The components of this pullback correspond to cosets of the subgroup \[ i_*\pi_1(M_g) \] in \[ \pi_1(T^3)=\mathbb Z^3. \] Since \(i_*\) is surjective, there is one coset, hence one component.
5. The restriction\[ \widetilde M_g\to M_g \]is a connected normal covering with deck group \(\mathbb Z^3\), acting by integer translations of \(\mathbb R^3\).
The universal cover \[ \mathbb R^3\to T^3 \] has deck group \(\mathbb Z^3\), acting by \[ x\longmapsto x+v, \qquad v\in\mathbb Z^3. \] Since \(\widetilde M_g\) is the full inverse image of \(M_g\), it is invariant under every integer translation. By 4 it is connected, so these translations are precisely the deck transformations of the restricted cover. Equivalently, the corresponding subgroup is \[ \ker i_*. \]
6. For \(g=3\), such an embedding \(M_3\hookrightarrow T^3\) is obtained as the boundary of a regular neighborhood of the wedge of the three coordinate circles in \(T^3\).
Let \[ \Gamma = (S^1\times\{1\}\times\{1\}) \cup (\{1\}\times S^1\times\{1\}) \cup (\{1\}\times\{1\}\times S^1). \] This is a wedge of three circles. A sufficiently small regular neighborhood \(N(\Gamma)\) is a genus-three handlebody, so \[ \partial N(\Gamma)\cong M_3. \]
The inclusion \[ \partial N(\Gamma)\hookrightarrow N(\Gamma) \] induces a surjection on fundamental groups: the three standard handlebody generators can be represented by loops on the boundary. The deformation retraction \[ N(\Gamma)\simeq\Gamma \] followed by inclusion into \(T^3\) sends these three generators to the three coordinate generators of \[ \pi_1(T^3)=\mathbb Z^3. \] Hence the composite \[ \pi_1(M_3)\to\pi_1(T^3) \] is surjective.
7. The resulting cover in \(\mathbb R^3\) is the boundary of a regular neighborhood of the standard cubic lattice graph.
The full inverse image of \(\Gamma\) under \[ \mathbb R^3\to T^3 \] is the graph with vertex set \(\mathbb Z^3\) and edges joining nearest-neighbor lattice points in the three coordinate directions. Pulling back the regular neighborhood \(N(\Gamma)\) gives a \(\mathbb Z^3\)-periodic regular neighborhood of this cubic lattice graph. Therefore \[ \widetilde M_3 = \partial\widetilde{N(\Gamma)} \subset\mathbb R^3 \] is an explicit connected periodic surface, and integer translations are its deck transformations.
8. For every \(g>3\), add \(g-3\) handles to \(M_3\) inside a small ball in \(T^3\) disjoint from the three loops that realize the coordinate generators.The resulting embedded surface is \(M_g\) and still induces a surjection on \(\pi_1\).
Attaching a local surface handle in a ball increases the genus by one. Because the modification is disjoint from the three loops on the original surface whose images generate \(\pi_1(T^3)\), those loops persist on the new surface and still map to the three coordinate generators. Thus surjectivity is preserved.
In the lifted picture in \(\mathbb R^3\), the same operation adds \(g-3\) handles in every fundamental cube, periodically under the \(\mathbb Z^3\) translation action.
9. Hence for \(n=3\) and every \(g\ge3\) there is an explicit connected normal cover\[ \widetilde M_g\subset\mathbb R^3 \]with deck group \(\mathbb Z^3\) acting by translations.
Combine <1>4–<1>8.