(Michigan Jan ’10) Let \(S^1=\{z\in\mathbb C\mid |z|=1\}\). Let \(X\) be obtained from the space \(S^1\times[0,1]\) by identifying every point \((z,0)\) with the point \((z^4,1)\), with the quotient topology. Compute the fundamental group and homology of \(X\).
Solution
Let \(a\) denote the loop around the \(S^1\)-fiber and \(t\) the loop obtained by moving once in the \([0,1]\)-direction and using the endpoint identification. The identification has degree \(4\) on the circle, so van Kampen gives the Baumslag–Solitar presentation \[ \pi_1(X)\cong\langle a,t\mid tat^{-1}=a^4\rangle. \]
For homology, use the Wang exact sequence for the mapping torus of the degree-\(4\) map \(f:S^1\to S^1\): \[ \cdots\to H_n(S^1)\xrightarrow{1-f_*}H_n(S^1)\to H_n(X) \to H_{n-1}(S^1)\xrightarrow{1-f_*}H_{n-1}(S^1)\to\cdots. \] On \(H_1(S^1)=\mathbb Z\), \(1-f_*\) is multiplication by \(1-4=-3\), while on \(H_0(S^1)=\mathbb Z\) it is \(0\). Therefore \[ H_2(X)=\ker(\mathbb Z\xrightarrow{-3}\mathbb Z)=0, \] and the degree-one part gives \[ 0\to\mathbb Z/3\to H_1(X)\to\mathbb Z\to0. \] Since \(\mathbb Z\) is free, this splits, so \[ H_1(X)\cong\mathbb Z\oplus\mathbb Z/3. \] Also \(H_0(X)=\mathbb Z\), and \(H_n(X)=0\) for \(n\ge2\). Equivalently, \(H_1\) is the abelianization of the displayed fundamental-group presentation: the relation becomes \(3a=0\), while \(t\) remains infinite cyclic.