Lens Spaces

Introduction

The lens spaces are the quotient manifolds $L(p,q) = S^3/(\mathbb{Z}/p)$, where the cyclic group acts on the unit sphere of $\mathbb{C}^2$ through the diagonal matrices $\mathrm{diag}(\zeta,\zeta^q)$ with $\zeta$ a primitive $p$-th root of unity. They are the simplest closed three-manifolds after the sphere: they have cyclic fundamental group, they admit a Heegaard splitting of genus one, they are the boundaries of the four-dimensional disc plumbings whose intersection forms are the cyclic lattices, and they are the links of the quotient singularities of the plane. They carry the spherical geometry, and they exhaust the closed three-manifolds of constant positive curvature with cyclic fundamental group.

The lens spaces occupy a special place in topology as the first manifolds for which the classification by homeomorphism and by homotopy type were shown to differ, and for which the invariant that detects the difference — the Reidemeister torsion — was found. Reidemeister and Franz classified them up to homeomorphism by the arithmetic of the parameter $q$, Whitehead classified them up to homotopy type by a congruence involving squares, and the gap between the two classifications is the classical example of a homotopy-theoretic invariant that is not a homeomorphism invariant. The lens spaces are also the standard example in the theory of the $s$-cobordism: two lens spaces can be homotopy equivalent, $h$-cobordant and not diffeomorphic, so that the Whitehead torsion of a homotopy equivalence is the essential ingredient in the classification of the cobordisms of dimension at least five.

The boundaries of the article. The three-manifold topology involved — Heegaard splittings, Dehn surgery, the Seifert fibrations, the prime and torus decompositions, the classification of the spherical space forms and the geometrisation theorem — is not treated here, and the surgery and cobordism frame, the $s$-cobordism theorem and the Whitehead torsion are standard; the lens spaces are treated here as the class of examples in which the arithmetic of the classification is visible. The knots whose surgeries produce lens spaces are the standard ones. The Floer-theoretic $d$-invariants and correction terms that constrain the rational homology cobordisms of lens spaces are standard. The fundamental group, the homology, the Reidemeister torsion and the chain complexes are the algebraic-topological input of Algebraic Topology, being written by another agent; the present article states the classification and identifies the invariant, without re-deriving the algebra. The linear algebra of the cyclic lattices and the plumbing forms, and the signature and the Rochlin invariant, are those of Symplectic Forms and Poisson Brackets and of Characteristic Classes respectively, both written by other agents. The quotient singularities and the weighted projective constructions belong to algebraic geometry, and the analytic theory of the eigenvalues of the Laplacian — the spectral geometry of the lens spaces, the eta invariant and the multiplicities — is part of the spectral theory of Part III, where the measure and the limit are available. No physics is invoked.

Constructions of the Lens Spaces

Definition. Let $p\geq1$ and let $q$ be coprime to $p$. The lens space $L(p,q)$ is the quotient of the unit sphere

$$ S^3 = \{(z_0,z_1)\in\mathbb{C}^2 : |z_0|^2+|z_1|^2 = 1\} $$

by the free action of the cyclic group $\mathbb{Z}/p = \langle g\rangle$ generated by

$$ g\cdot(z_0,z_1) = (\zeta z_0, \zeta^q z_1), \qquad \zeta = e^{2\pi i/p} . $$

The action is free because the fixed-point set of $g^k$ in $\mathbb{C}^2$ is the coordinate axes when $\zeta^k\neq1$, and the axes are removed by the sphere condition; so the quotient is a closed three-manifold with fundamental group $\pi_1(L(p,q)) = \mathbb{Z}/p$. For $p = 1$ the space is $S^3$.

Proposition (equivalent constructions). The lens spaces admit the following descriptions, and the identifications are natural:

(a) Heegaard genus one. Cutting $S^3$ along the torus $\{|z_0|^2=|z_1|^2=1/2\}$ gives two solid tori, and the quotient acts on the torus as a translation; so $L(p,q) = V_0\cup_\phi V_1$ is the union of two solid tori glued by a homeomorphism $\phi$ of their boundaries. Every closed oriented three-manifold with a genus-one Heegaard splitting is a lens space, $S^3 = L(1,1)$, or $S^2\times S^1$, and the last is the only one of the three that is not a lens space.

(b) Dehn surgery on the unknot. $L(p,q) = S^3_{p/q}(U)$ is the manifold obtained by Dehn surgery on the unknot with coefficient $p/q$; the meridian of the solid torus glued back is the curve of slope $p/q$ on the boundary torus. Since the unknot has a genus-one Heegaard splitting, this is the same as (a).

(c) Seifert fibration. $L(p,q)$ is a Seifert fibred space over $S^2$ with at most two exceptional fibres; the Seifert invariants are recovered from the continued fraction of $p/q$. In particular the lens spaces are the Seifert fibred spaces with finite cyclic fundamental group and the spherical geometry.

(d) Plumbing of disc bundles. Write the continued fraction

$$ -\frac{p}{q} = a_1 - \cfrac{1}{a_2 - \cfrac{1}{\cdots - \cfrac{1}{a_k}}}, \qquad a_i\leq-2 , $$

and let $P(a_1,\ldots,a_k)$ be the four-dimensional plumbing of $k$ disc bundles over $S^2$ with Euler numbers $a_1,\ldots,a_k$ along a chain. Then $\partial P(a_1,\ldots,a_k) = L(p,q)$. The intersection form of the plumbing is the tridiagonal form with the $a_i$ on the diagonal and $1$ off the diagonal, and $|\det| = p$.

Proof sketch. The Heegaard description is the quotient of the standard genus-one splitting, and the surgery description follows because surgery on the unknot replaces one of the solid tori by the solid torus glued with slope $p/q$, which is exactly the quotient operation. The Seifert fibration is the quotient of the Hopf fibration by the cyclic action. For the plumbing, the boundary of the chain is computed by the gluing formula for the disc bundles, and the continued fraction is the algebra of the resulting torus automorphism; the determinant of the tridiagonal matrix is $p$ by the recurrence $D_n = a_n D_{n-1} - D_{n-2}$.

Example. For $q = p-1$, so that $L(p,p-1)$ is homeomorphic to $L(p,1)$, the continued fraction is $-p/(p-1) = \underbrace{[-2,-2,\ldots,-2]}_{p-1}$ with $p-1$ entries, and the chain of $p-1$ disc bundles with Euler number $-2$ has boundary $L(p,p-1)$; the determinants of the chains of $1,2,3,\ldots$ vertices are $2,3,4,\ldots$, computed from the recurrence with $a_i=-2$, so the chain of $p-1$ vertices has $|\det| = p$, in agreement with the general statement. For $p=2$ this is $L(2,1) = \mathbb{RP}^3$, the boundary of the disc bundle over $S^2$ with Euler number $-2$, which is the disc bundle of the line bundle $\mathcal{O}(-2)$; the disc bundle with Euler number $+2$ is the tangent disc bundle of $S^2$, whose boundary is the unit tangent bundle, and that too is $\mathbb{RP}^3$.

Example. The lens spaces occur as the links of the quotient singularities: the quotient $\mathbb{C}^2/(\mathbb{Z}/p)$ with the action $(z_0,z_1)\mapsto(\zeta z_0,\zeta^q z_1)$ is the cone on $L(p,q)$, so the lens spaces are the boundaries of the neighbourhoods of the cyclic quotient singularities, and they are the simplest family of Brieskorn-type links.

The Classification of the Lens Spaces

Theorem (homeomorphism classification; Reidemeister, Franz, de Rham). Let $p,q,q'$ be integers with $\gcd(q,p)=\gcd(q',p)=1$. Then $L(p,q)$ and $L(p,q')$ are homeomorphic if and only if

$$ p = p' \quad\text{and}\quad q' \equiv \pm q^{\pm1} \pmod p , $$

that is, if and only if $q' \equiv q$, $-q$, $q^{-1}$ or $-q^{-1}$ modulo $p$.

Proof sketch. The necessity is proved by the Reidemeister torsion: computing the torsion of the acyclic complex of the universal cover in the abelian representation sending the generator of $\mathbb{Z}/p$ to a primitive $p$-th root of unity $\zeta$ gives an element

$$ \tau(L(p,q)) \in \mathbb{Q}(\zeta)/\{\pm\zeta^{k}\}, $$

which is a homeomorphism invariant and which agrees for $L(p,q)$ and $L(p,q')$ exactly when $q' \equiv \pm q^{\pm1}\pmod p$. The sufficiency is by exhibiting the explicit homeomorphisms: the changes $q\mapsto -q$ and $q\mapsto q^{-1}$ are realised by the reflections of the sphere.

Theorem (homotopy classification; Whitehead). $L(p,q)$ and $L(p,q')$ are homotopy equivalent if and only if

$$ q q' \equiv \pm n^2 \pmod p $$

for some integer $n$. In particular the homeomorphism classification is strictly finer than the homotopy classification: $L(7,1)$ and $L(7,2)$ are homotopy equivalent, since $1\cdot2\equiv 3^2\pmod 7$, but they are not homeomorphic, since $2\not\equiv\pm1\pmod 7$.

Proof sketch. The homotopy classification is proved by the theory of the Moore spaces and the obstructions to the homotopy equivalences: the lens spaces are the Moore spaces $M(\mathbb{Z}/p,1)$ with their attaching data, and the obstruction to realising a homotopy equivalence lies in the group of units and squares modulo $p$. The construction of the homotopy equivalence for the pairs satisfying the congruence is by Whitehead's theorem on the simple homotopy types of the Moore spaces.

Theorem (the gap; Reidemeister, Franz, Milnor). The Reidemeister torsion is the complete homeomorphism invariant of the lens spaces with a fixed fundamental group, and it is not a homotopy invariant: the torsion distinguishes exactly the homeomorphism classes within a homotopy class. Moreover the lens spaces furnish a pair of homotopy equivalent, $h$-cobordant manifolds that are not diffeomorphic, and the obstruction to the diffeomorphism is the Whitehead torsion of the homotopy equivalence, an element of the Whitehead group of $\mathbb{Z}/p$. This is the classical example that shows the necessity of the torsion in the $s$-cobordism theorem; since the cobordism is four-dimensional, the example also marks the dimension hypothesis of the $h$-cobordism theorem, which holds in dimension at least five.

Proof sketch. The torsion is computed from the chain complex of the universal cover with the abelian coefficient system, and its behaviour under a homotopy equivalence is controlled by the Whitehead torsion, which need not vanish; the $h$-cobordism between the lens spaces is constructed from the difference of the attaching maps, and its nontriviality is read off from the torsion. The details of the $s$-cobordism and the Whitehead group belong.

Example. The smallest example of the phenomenon is the pair $L(7,1)$ and $L(7,2)$: they have the same fundamental group $\mathbb{Z}/7$, the same homology and the same homotopy type, they are $h$-cobordant, and they are not homeomorphic. The homeomorphism classes of lens spaces with fundamental group $\mathbb{Z}/p$ are the orbits of the units modulo $p$ under $q\mapsto \pm q^{\pm1}$, an orbit having four elements unless $q^2\equiv\pm1\pmod p$, when it has two; the orbits of size two are those of $q=\pm1$ and, for $p\equiv1\pmod4$, those of the two solutions of $q^2\equiv-1$, so that for an odd prime $p$ the number of classes is $1+(\varphi(p)-2)/4$ when $p\equiv3\pmod4$ and $2+(\varphi(p)-4)/4$ when $p\equiv1\pmod4$. For $p=7$ the classes are two, represented by $q=1$ and $q=2$, and the pair $L(7,1)$, $L(7,2)$ realises the failure of the homotopy classification; for $p=13$ the classes are four, represented by $q=1,2,3,5$.

Invariants of the Lens Spaces

Proposition (algebraic invariants). Let $L = L(p,q)$ with $p\geq2$. Then:

(a) $\pi_1(L) = \mathbb{Z}/p$, and the universal cover of $L$ is $S^3$, so $L$ is a spherical space form and its higher homotopy groups are those of the sphere;

(b) the homology is $H_0(L;\mathbb{Z}) = \mathbb{Z}$, $H_1(L;\mathbb{Z}) = \mathbb{Z}/p$, $H_2(L;\mathbb{Z}) = 0$, $H_3(L;\mathbb{Z}) = \mathbb{Z}$;

(c) the Euler characteristic vanishes, the manifold is orientable and its Heegaard genus is one;

(d) the linking form on the torsion subgroup $H_1(L) = \mathbb{Z}/p$ is the nonsingular form $\lambda : \mathbb{Z}/p\times\mathbb{Z}/p\to\mathbb{Q}/\mathbb{Z}$ determined by the value $\lambda(1,1) = q^*/p$ with $qq^*\equiv1\pmod p$, up to the sign convention fixed by the orientation; it is a homotopy invariant, and its isomorphism classes are strictly coarser than the homeomorphism classes.

Proof. The fundamental group is computed from the free action; the homology from the cell structure of the quotient or from the Poincaré duality $H_2\cong H^1 = \mathrm{Hom}(H_1,\mathbb{Z}) = 0$; the linking form is computed from the intersection form of the plumbing of (d) above, restricted to the torsion of the boundary.

Theorem (the linking form and the classification). Two lens spaces with the same fundamental group are homotopy equivalent if and only if their linking forms are isomorphic, equivalently if and only if $qq'\equiv\pm n^2\pmod p$ for some integer $n$; the isomorphism classes of the linking forms on $\mathbb{Z}/p$ are the classes of the units modulo the subgroup $\{\pm u^2\}$, and this is strictly coarser than the homeomorphism classification, which is by the classes modulo $\{\pm q^{\pm1}\}$. Consequently the arithmetic of the linking form is the arithmetic of the homotopy classification, and it is the Reidemeister torsion that performs the finer homeomorphism classification.

Proof sketch. The linking form is the nonsingular pairing on the torsion of $H_1$ induced by the intersection form of a bounding four-manifold; the plumbing provides such a four-manifold, and the form is computed from the matrix of the plumbing. The invariance of the form under homotopy equivalence and its completeness for the homotopy classification reduce to the corresponding statements for the Whitehead torsion, which is the finer invariant performing the homeomorphism classification.

Proposition (geometry and spectrum). The lens spaces carry the spherical geometry: $L(p,q)$ is the quotient of the round $S^3$ by a finite group of isometries, so it has constant sectional curvature $+1$, and it is the unique closed three-manifold (up to isometry) with this curvature and the given action. The eigenvalues of the Laplace operator on $L(p,q)$ are computed explicitly from the representation theory of $S^3$ and the decomposition of the spherical harmonics under the cyclic action; the multiplicities are the sums of the dimensions of the representations of $\mathbb{Z}/p$ occurring in each spherical harmonic.

Proof sketch. The spherical structure is the quotient structure; the spectral computation is the Peter–Weyl decomposition of $L^2(S^3)$ into the representations of $S^3$, restricted to the cyclic subgroup. The analysis of the spectrum and the heat kernel is part of the spectral theory of Part III.

Spherical Space Forms and Free Actions

Theorem (classification of the spherical space forms). Every closed three-manifold of constant curvature $+1$ is a quotient $S^3/\Gamma$ of the round sphere by a finite group $\Gamma$ acting freely by isometries; conversely every such quotient is a spherical space form. The finite groups acting freely and isometrically on $S^3$ are classified — they are the cyclic groups, the binary dihedral, the binary tetrahedral, the binary octahedral and the binary icosahedral groups — and the corresponding quotients are classified up to isometry and up to homeomorphism. The cyclic case gives exactly the lens spaces, and the other cases give the spherical space forms with non-cyclic fundamental groups, of which the Poincaré homology sphere $\Sigma(2,3,5) = S^3/I^*$ is the most famous.

Proof sketch. The action of a finite group of isometries on $S^3$ is a free action on the sphere; the classification of the finite subgroups of $SO(4)$ acting freely is obtained from the classification of the finite subgroups of $SO(3)$ and the two-fold covers, together with the analysis of the fixed-point sets; the list is that of Hopf and of Threlfall and Seifert.

Remark. The classification of the free actions of finite groups on the sphere is one of the earliest applications of the three-dimensional topology, and it distinguishes the lens spaces among the spherical space forms by their cyclic fundamental groups. Since the geometrisation theorem turns every closed three-manifold with finite fundamental group into a spherical space form, the class of the spherical space forms is exactly the class of the closed three-manifolds of finite fundamental group, and the Poincaré conjecture is the statement that the trivial group gives the sphere.

Four-Dimensional and Floer-Theoretic Aspects

Theorem (the plumbing and the bounding four-manifolds). Let $L = L(p,q)$ and let $P(a_1,\ldots,a_k)$ be the plumbing of the continued fraction of $-p/q$ as above. Then $\partial P = L$, the intersection form of $P$ is the form with the $a_i$ on the diagonal and $1$ off the diagonal, its determinant is $\pm p$, and its signature and Rochlin invariant are computed from the plumbing matrix by the standard formulae of Characteristic Classes. The lens space $L$ bounds a smooth rational homology four-ball if and only if the plumbing lattice $(\mathbb{Z}^k,Q)$ embeds as a sublattice of finite index in the standard unimodular lattice of the same rank, which for a negative definite plumbing is the lattice $\langle-1\rangle^k$; since the index is then $\sqrt{|\det Q|} = \sqrt{p}$, a necessary condition is that $p$ be a perfect square, and the sharp criterion on the continued fraction of $-p/q$ is Lisca's theorem. So not every lens space bounds a rational homology ball, and the obstruction is detected by the Floer-theoretic correction terms.

Proof sketch. The boundary computation is the plumbing formula; the intersection form is the tridiagonal matrix, and the existence of a rational homology ball is equivalent to the embedding of the plumbing lattice into a unimodular lattice of the same rank with index $\sqrt{p}$, which is a criterion on the continued fraction. The classification of the lens spaces that bound rational homology balls is the theorem of Lisca and is proved by the lattice-theoretic criterion together with the Floer-theoretic $d$-invariants, which obstruct the rational homology cobordisms.

Theorem (Lisca; Ozsváth–Szabó; the rational homology cobordism of lens spaces). A lens space $L(p,q)$ bounds a smooth rational homology four-ball if and only if the continued fraction of $p/q$ has the special form determined by Lisca; the criterion is proved by the classification of the negative definite plumbings and the Floer-theoretic correction terms of lens spaces, which are computable in closed form. Consequently the question of which lens spaces bound smooth rational homology balls is completely answered, while the classification of the lens spaces up to integral homology cobordism, and the question of which rational homology spheres bound contractible four-manifolds, remain open and are standard problems of the homology cobordism theory.

Proof sketch. The Floer correction terms $d(L(p,q))$ obstruct the rational homology cobordisms by the monotonicity of the invariant under the cobordism; the $d$-invariants of the lens spaces are computed explicitly from the continued fraction by the recursion of Ozsváth–Szabó. The lattice-theoretic classification gives the converse.

Remark. The lens spaces enter the theory of Dehn surgery as the building blocks: every closed oriented three-manifold is a surgery on a link, and the surgeries on a knot that produce lens spaces are the subject of a separate theory, with the Berge conjecture asserting that the knots in $S^3$ with lens space surgeries are exactly the knots in the Berge list. The lens spaces are also the rational homology spheres of minimal complexity, and their $d$-invariants are the standard example in which the Floer-theoretic invariants are computed in closed form and used as the input to the classification of the rational homology cobordisms. The integral homology cobordism classification of the lens spaces is not known, and it is equivalent to a statement about the outcomes of the surgeries on the links that bound.

Summary

The lens space $L(p,q)$ is the quotient of the round three-sphere by the free cyclic action $(z_0,z_1)\mapsto(\zeta z_0,\zeta^q z_1)$. It is equivalently the union of two solid tori glued along a slope, the Dehn surgery on the unknot with coefficient $p/q$, the Seifert fibred space over $S^2$ with at most two exceptional fibres, and the boundary of the disc plumbing determined by the continued fraction of $-p/q$; its fundamental group is $\mathbb{Z}/p$, its second homology vanishes, its Heegaard genus is one, and it carries the spherical geometry.

The classification is arithmetic. Up to homeomorphism $L(p,q)\cong L(p,q')$ if and only if $q'\equiv\pm q^{\pm1}\pmod p$, with the Reidemeister torsion as the complete invariant; up to homotopy equivalence $L(p,q)\simeq L(p,q')$ if and only if $q q'\equiv\pm n^2\pmod p$. The gap is the classical example of a homotopy invariant that is not a homeomorphism invariant: $L(7,1)$ and $L(7,2)$ are homotopy equivalent, $h$-cobordant and not homeomorphic, and the obstruction is the Whitehead torsion, so that the $s$-cobordism theorem needs the torsion hypothesis and the example marks the dimension hypothesis of the $h$-cobordism theorem. The linking form is a homotopy invariant and not a homeomorphism invariant, its classes being the units modulo $\pm u^2$ while the homeomorphism classes are the units modulo $\pm q^{\pm1}$. The lens spaces exhaust the spherical space forms with cyclic fundamental group; the general spherical space forms are the quotients of $S^3$ by the finite groups of the Hopf–Threlfall–Seifert list, including the Poincaré sphere. In dimension four, the lens spaces bound the plumbings of their continued fractions, and the rational homology balls they bound are classified by Lisca, with the Floer-theoretic correction terms giving the computable obstruction and the perfect-square condition on $p$ as the first restriction.

Summary of Notation

Symbol	Meaning
$L(p,q)$	Lens space $S^3/(\mathbb{Z}/p)$, action $\mathrm{diag}(\zeta,\zeta^q)$
$\zeta$, $p$, $q$	Primitive $p$-th root of unity; $\gcd(q,p)=1$
$V_0,V_1$, $\phi$	Solid tori and gluing homeomorphism of the genus-one Heegaard splitting
$S^3_{p/q}(U)$	Dehn surgery on the unknot producing $L(p,q)$
$a_1,\ldots,a_k$	Continued fraction coefficients of $-p/q$, $a_i\leq-2$; plumbing Euler numbers
$P(a_1,\ldots,a_k)$	Four-dimensional plumbing of disc bundles over $S^2$ with boundary $L(p,q)$
$\tau(L(p,q))$	Reidemeister torsion; complete homeomorphism invariant within fixed $\pi_1$
Linking form $\lambda$	Nonsingular form on $H_1=\mathbb{Z}/p$; $\lambda(1,1)=q^/p$, $qq^\equiv1$; a homotopy invariant
$\varphi(p)$	Euler totient; the homeomorphism classes with $\pi_1=\mathbb{Z}/p$ are the orbits of the units under $q\mapsto\pm q^{\pm1}$
$p$ perfect square	Necessary condition for $L(p,q)$ to bound a rational homology four-ball
$d(L(p,q))$	Heegaard Floer correction term; obstructs rational homology balls
$\Sigma(2,3,5)$	Poincaré homology sphere $S^3/I^*$ among the non-lens spherical space forms