In this problem we will derive one half of the classification of lens spaces up to homotopy equivalence, by showing that if \(L_m(\ell_1, \dots, \ell_n) \simeq L_m(\ell_1', \dots, \ell_n')\) then \(\ell_1 \cdots \ell_n \equiv \pm \ell_1' \cdots \ell_n' k^n \pmod{m}\) for some integer \(k\). The converse is Exercise 29 for §4.2.
Let \(L = L_m(\ell_1, \dots, \ell_n)\) and let \(\mathbb{Z}_m^*\) be the multiplicative group of invertible elements of \(\mathbb{Z}_m\). Define \(t \in \mathbb{Z}_m^*\) by the equation \(xy^{n-1} = tz\) where \(x\) is a generator of \(H^1(L; \mathbb{Z}_m)\), \(y = \beta(x)\), and \(z \in H^{2n-1}(L; \mathbb{Z}_m)\) is the image of a generator of \(H^{2n-1}(L; \mathbb{Z})\). Show that the image \(\tau(L)\) of \(t\) in the quotient group \(\mathbb{Z}_m^* / \pm(\mathbb{Z}_m^*)^n\) depends only on the homotopy type of \(L\).
Given nonzero integers \(k_1, \dots, k_n\), define a map \(\tilde{f}: S^{2n-1} \to S^{2n-1}\) sending the unit vector \((r_1 e^{i\theta_1}, \dots, r_n e^{i\theta_n})\) in \(\mathbb{C}^n\) to \((r_1 e^{ik_1\theta_1}, \dots, r_n e^{ik_n\theta_n})\). Show:
\(\tilde{f}\) has degree \(k_1 \cdots k_n\).
\(\tilde{f}\) induces a quotient map \(f: L \to L'\) for \(L' = L_m(\ell_1', \dots, \ell_n')\) provided that \(k_j \ell_j \equiv \ell_j' \pmod{m}\) for each \(j\).
\(f\) induces an isomorphism on \(\pi_1\), hence on \(H^1(-; \mathbb{Z}_m)\).
\(f\) has degree \(k_1 \cdots k_n\), i.e., \(f_*\) is multiplication by \(k_1 \cdots k_n\) on \(H_{2n-1}(-; \mathbb{Z})\).
Using the \(f\) in (b), show that \(\tau(L) = k_1 \cdots k_n \tau(L')\).
Deduce that if \(L_m(\ell_1, \dots, \ell_n) \simeq L_m(\ell_1', \dots, \ell_n')\), then \(\ell_1 \cdots \ell_n \equiv \pm \ell_1' \cdots \ell_n' k^n \pmod{m}\) for some integer \(k\).
Solution
Write \[ L=L_m(\ell_1,\dots,\ell_n). \] Let \(x\in H^1(L;\mathbb Z_m)\) be a generator, put \(y=\beta(x)\), and let \(z\in H^{2n-1}(L;\mathbb Z_m)\) be the reduction of an integral orientation generator. Define \(t\in\mathbb Z_m^*\) by \[ xy^{n-1}=tz. \]
1. The class\[ \tau(L)=[t]\in\mathbb Z_m^*/\bigl(\pm(\mathbb Z_m^*)^n\bigr) \]depends only on the homotopy type of \(L\).
Let \(h:L\to L'\) be a homotopy equivalence. For suitable units \(a\in\mathbb Z_m^*\) and a sign \(\varepsilon=\pm1\), \[ h^*(x')=a x, \qquad h^*(y')=a y \] by naturality of the Bockstein, and \[ h^*(z')=\varepsilon z \] because \(h\) has degree \(\pm1\) on integral top cohomology. Pulling back \[ x'(y')^{n-1}=t'z' \] gives \[ a^nxy^{n-1}=\varepsilon t'z, \] so \[ a^nt=\varepsilon t'. \] Thus \(t\) and \(t'\) differ by a sign and an \(n\)th power of a unit, exactly the ambiguity quotiented out in \(\tau\).
2. For nonzero integers \(k_1,\dots,k_n\), the map\[ \widetilde f(r_1e^{i\theta_1},\dots,r_ne^{i\theta_n}) =(r_1e^{ik_1\theta_1},\dots,r_ne^{ik_n\theta_n}) \]has the stated properties.
For a regular value with every coordinate nonzero, the \(j\)th angular coordinate has \(|k_j|\) inverse choices, and each local sign contributes \(\operatorname{sgn}(k_j)\). Hence the sum of local degrees is \[ \deg\widetilde f=k_1\cdots k_n. \]
Let the generator of the \(\mathbb Z_m\) action defining \(L\) multiply the \(j\)th coordinate by \[ e^{2\pi i\ell_j/m}. \] Then \(\widetilde f\) carries this action to multiplication in the \(j\)th target coordinate by \[ e^{2\pi i k_j\ell_j/m}. \] Thus if \[ k_j\ell_j\equiv\ell_j'\pmod m \] for all \(j\), \(\widetilde f\) is equivariant for the identity automorphism of \(\mathbb Z_m\) and descends to \[ f:L\to L'. \] The induced map on \(\pi_1\cong\mathbb Z_m\) is therefore the identity, hence an isomorphism, and so is the induced map on \(H^1(-;\mathbb Z_m)\).
Finally the covering projections have degree \(m\) and fit into the commutative square with \(\widetilde f\) and \(f\). Multiplicativity of degree gives \[ m\deg f=(\deg\widetilde f)m, \] so \[ \deg f=k_1\cdots k_n. \]
3. For the map in part 2,\[ \tau(L)=k_1\cdots k_n\,\tau(L'). \]
Since \(f_*\) is the identity on \(\pi_1\), choose generators with \[ f^*(x')=x, \qquad f^*(y')=y. \] On top cohomology, \[ f^*(z')=(k_1\cdots k_n)z. \] Pulling back \(x'(y')^{n-1}=t'z'\) therefore gives \[ xy^{n-1}=t'(k_1\cdots k_n)z, \] so \[ t=(k_1\cdots k_n)t'. \]
4. The stated necessary condition for homotopy equivalence follows.
Let \[ L_0=L_m(1,\dots,1). \] Choose \(k_j\) with \[ k_j\ell_j\equiv1\pmod m. \] Part 3 gives \[ \tau(L)=\left(\prod_j\ell_j\right)^{-1}\tau(L_0). \] Similarly, \[ \tau(L')=\left(\prod_j\ell_j'\right)^{-1}\tau(L_0). \] If \(L\simeq L'\), part 1 says these two classes differ by a sign and an \(n\)th power of a unit. Cancelling the unit \(\tau(L_0)\) and inverting the resulting congruence yields \[ \boxed{\ell_1\cdots\ell_n \equiv\pm\ell_1'\cdots\ell_n' k^n\pmod m} \] for some unit \(k\) modulo \(m\), represented by an integer \(k\).