Let \(S\) be an oriented surface and let \(f:S\to S\) be a continuous map homotopic to the identity that has no fixed points. Find all the possible values for the genus of \(S\). Prove your answer is true.
Solution
1. If \(S\) may have boundary, genus \(0\) occurs.
Take the annulus \[ A=S^1\times[0,1]. \] It is an orientable surface of genus \(0\). For \(0\le t\le1\), define \[ F_t(z,s)=\left(e^{\pi i t}z,s\right). \] Then \(F_0=\operatorname{id}_A\), while \[ F_1(z,s)=(-z,s) \] has no fixed point. Steps <1>2–<1>4 treat a compact connected orientable surface without boundary, for which the genus is exactly \(1\).
2. Assume that \(S\) is a closed connected orientable surface of genus \(g\).Then\[ L(f)=2-2g. \]
For a closed connected orientable surface of genus \(g\), \[ H_0(S;\QQ)\cong\QQ, \qquad H_1(S;\QQ)\cong\QQ^{2g}, \qquad H_2(S;\QQ)\cong\QQ. \] Because \(f\simeq\operatorname{id}_S\), homotopy invariance of homology gives \[ f_*=\operatorname{id} \] on every homology group. Hence the Lefschetz number is \[ \begin{aligned} L(f) &=\sum_{q=0}^2(-1)^q \operatorname{tr}\left(f_*:H_q(S;\QQ)\to H_q(S;\QQ)\right)\\ &=1-2g+1\\ &=2-2g. \end{aligned} \]
3. Since \(f\) has no fixed points, \(g=1\).
The Lefschetz fixed point theorem says that if \[ L(f)\ne0, \] then \(f\) has a fixed point. Taking the contrapositive and using the hypothesis that \(f\) has no fixed points gives \[ L(f)=0. \] By 2, \[ 2-2g=0, \] so \[ g=1. \] Thus the torus is the only possible closed orientable surface.
4. Genus \(1\) actually occurs.
Write \[ T^2=S^1\times S^1. \] For \(0\le t\le1\), define \[ G_t(z,w)=\left(e^{\pi i t}z,w\right). \] Then \(G_0=\operatorname{id}_{T^2}\), whereas \[ G_1(z,w)=(-z,w). \] The equation \(G_1(z,w)=(z,w)\) would imply \(-z=z\), impossible for \(z\in S^1\). Hence \(G_1\) is fixed-point-free and homotopic to the identity. Therefore, for closed connected orientable surfaces, \[ \boxed{g=1} \] is both necessary and sufficient.
5. If \(S\) is a compact connected orientable surface that may have boundary, then the possible genera are \(0\) and \(1\).
Let \(S=\Sigma_{g,b}\) be compact, connected, orientable, of genus \(g\) with \(b\) boundary components. The same Lefschetz argument gives \[ 0=L(f)=\chi(S)=2-2g-b. \] For integers \(g,b\ge0\), the only solutions are \[ (g,b)=(1,0) \qquad\text{and}\qquad (g,b)=(0,2). \] These are the torus and annulus respectively, and 1 and 4 give fixed-point-free maps homotopic to the identity on both. Thus allowing compact surfaces with boundary changes the set of possible genera from \(\{1\}\) to \[ \{0,1\}. \] For noncompact \(S\) the Lefschetz fixed-point theorem does not apply: the translation \(x\mapsto x+v\) of \(\RR^2\) with \(v\ne0\) is homotopic to the identity through \(x\mapsto x+tv\) and has no fixed point, and \(\RR^2\) has genus \(0\).