Algebraic Topology · Problem 3E.2

Cohomology — Allen Hatcher

Mathematics — Topology · lens-space-homotopy-equivalence-classification

Problem

In this problem we will derive one half of the classification of lens spaces up to homotopy equivalence, by showing that if Lm(ℓ1,⋯ ,ℓn)≃Lm(ℓ1′,⋯ ,ℓn′)L_m(\ell_1, \cdots, \ell_n) \simeq L_m(\ell'_1, \cdots, \ell'_n) then ℓ1⋯ℓn≡±ℓ1′⋯ℓn′kn mod m\ell_1 \cdots \ell_n \equiv \pm \ell'_1 \cdots \ell'_n k^n \bmod m for some integer kk. The converse is Exercise 29 for \S4.2. (a) Let L=Lm(ℓ1,⋯ ,ℓn)L = L_m(\ell_1, \cdots, \ell_n) and let Zm∗\mathbb{Z}_m^* be the multiplicative group of invertible elements of Zm\mathbb{Z}_m. Define t∈Zm∗t \in \mathbb{Z}_m^* by the equation xyn−1=tzxy^{n-1} = tz where xx is a generator of H1(L;Zm)H^1(L; \mathbb{Z}_m), y=β(x)y = \beta(x), and z∈H2n−1(L;Zm)z \in H^{2n-1}(L; \mathbb{Z}_m) is the image of a generator of H2n−1(L;Z)H^{2n-1}(L; \mathbb{Z}). Show that the image τ(L)\tau(L) of tt in the quotient group Zm∗/±(Zm∗)n\mathbb{Z}_m^*/\pm(\mathbb{Z}_m^*)^n depends only on the homotopy type of LL. (b) Given nonzero integers k1,⋯ ,knk_1, \cdots, k_n, define a map f~:S2n−1→S2n−1\tilde{f}: S^{2n-1} \to S^{2n-1} sending the unit vector (r1eiθ1,⋯ ,rneiθn)(r_1 e^{i\theta_1}, \cdots, r_n e^{i\theta_n}) in Cn\mathbb{C}^n to (r1eik1θ1,⋯ ,rneiknθn)(r_1 e^{ik_1\theta_1}, \cdots, r_n e^{ik_n\theta_n}). Show: ((i)) f~\tilde{f} has degree k1⋯knk_1 \cdots k_n. ((ii)) f~\tilde{f} induces a quotient map f:L→L′f: L \to L' for L′=Lm(ℓ1′,⋯ ,ℓn′)L' = L_m(\ell'_1, \cdots, \ell'_n) provided that kjℓj≡ℓj′ mod mk_j \ell_j \equiv \ell'_j \bmod m for each jj. ((iii)) ff induces an isomorphism on π1\pi_1, hence on H1(−;Zm)H^1(-; \mathbb{Z}_m). ((iv)) ff has degree k1⋯knk_1 \cdots k_n, i.e., f∗f_* is multiplication by k1⋯knk_1 \cdots k_n on H2n−1(−;Z)H_{2n-1}(-; \mathbb{Z}). (c) Using the ff in (b), show that τ(L)=k1⋯knτ(L′)\tau(L) = k_1 \cdots k_n \tau(L'). (d) Deduce that if Lm(ℓ1,⋯ ,ℓn)≃Lm(ℓ1′,⋯ ,ℓn′)L_m(\ell_1, \cdots, \ell_n) \simeq L_m(\ell'_1, \cdots, \ell'_n), then ℓ1⋯ℓn≡±ℓ1′⋯ℓn′kn mod m\ell_1 \cdots \ell_n \equiv \pm \ell'_1 \cdots \ell'_n k^n \bmod m for some integer kk.
Intuitive explanation

Lens spaces are quotients of odd-dimensional spheres by finite group actions, and their homotopy type is constrained by subtle invariants. The key insight is to construct a cohomological invariant τ(L)\tau(L) that lives in the quotient group Zm∗/±(Zm∗)n\mathbb{Z}_m^*/\pm(\mathbb{Z}_m^*)^n—this invariant captures the "twisted product" of the parameters in a way that's independent of homotopy equivalence. By building explicit maps between lens spaces and tracking how τ\tau transforms under these maps, you can force a relationship between the parameter sequences of any two homotopy-equivalent lens spaces.

Intuitive explanation

Lens spaces are quotients of odd-dimensional spheres by finite group actions, and their homotopy type is constrained by subtle invariants. The key insight is to construct a cohomological invariant τ(L)\tau(L) that lives in the quotient group Zm∗/±(Zm∗)n\mathbb{Z}_m^*/\pm(\mathbb{Z}_m^*)^n—this invariant captures the "twisted product" of the parameters in a way that's independent of homotopy equivalence. By building explicit maps between lens spaces and tracking how τ\tau transforms under these maps, you can force a relationship between the parameter sequences of any two homotopy-equivalent lens spaces.

Related theory

Browse the problem library