Suppose \(X\) is a path-connected CW-complex with \(\pi_1(X) \cong \mathbb{Z}^2\) and \(\pi_{\geq 2}(X) = 0\). Show that \(X\) is homotopy-equivalent to \(S^1 \times S^1\). Use this to show that the fundamental group of a closed orientable surface \(\Sigma_g\) of genus \(g \geq 2\) cannot contain a subgroup isomorphic to \(\mathbb{Z}^2\).
Solution
1. Choose loops in \(X\) representing a basis of \(\pi_1(X)\cong\mathbb Z^2\). Since their commutator is null-homotopic, these loops extend across the standard \(2\)-cell of the torus to a map\[f:T^2\to X\]which induces an isomorphism on \(\pi_1\).
The standard CW presentation of the torus is one \(0\)-cell, two \(1\)-cells \(a,b\), and one \(2\)-cell attached by the commutator \(aba^{-1}b^{-1}\). The chosen loops commute in \(\pi_1(X)\), so the attaching loop maps null-homotopically and the map on the \(1\)-skeleton extends.
2. The map \(f\) is an isomorphism on every homotopy group.
It is an isomorphism on \(\pi_1\) by construction. Both \(T^2\) and \(X\) have vanishing \(\pi_k\) for \(k\ge2\): the torus has universal cover \(\mathbb R^2\), while this vanishing is assumed for \(X\).
3. Therefore\[\boxed{X\simeq T^2=S^1\times S^1.}\]
Both spaces are CW complexes, so Whitehead’s theorem applies to the weak homotopy equivalence in 2.
4. Now suppose \(\pi_1(\Sigma_g)\) contained a subgroup \(H\cong\mathbb Z^2\). Let \(p:\widetilde\Sigma\to\Sigma_g\) be the connected covering corresponding to \(H\).
Closed orientable surfaces are path connected, locally path connected, and semilocally simply connected, so the subgroup-covering correspondence applies.
5. The covering surface \(\widetilde\Sigma\) is aspherical and has \(\pi_1(\widetilde\Sigma)\cong\mathbb Z^2\), hence by <1>1–<1>3\[\widetilde\Sigma\simeq T^2.\]
The universal cover of \(\widetilde\Sigma\) is also the universal cover of \(\Sigma_g\), namely the plane, so all higher homotopy groups vanish.
6. The cover cannot have finite degree.
If it had degree \(d\), then \[\chi(\widetilde\Sigma)=d\chi(\Sigma_g)=d(2-2g)<0,\] whereas homotopy equivalence to \(T^2\) gives Euler characteristic \(0\).
7. The cover cannot have infinite degree either.
An infinite-sheeted connected cover of the compact surface \(\Sigma_g\) is noncompact. Every connected noncompact surface has \(H_2(-;\mathbb Z)=0\), while 5 gives \(H_2(\widetilde\Sigma;\mathbb Z)\cong H_2(T^2;\mathbb Z)\cong\mathbb Z\).
8. Hence\[\boxed{\pi_1(\Sigma_g)\text{ contains no subgroup isomorphic to }\mathbb Z^2\quad(g\ge2).}\]
The finite- and infinite-index cases in <1>6–<1>7 exhaust all possibilities.