Free Involutions and Lens Spaces

Introduction

A free involution is an involution with no fixed point; its quotient is a manifold of the same dimension, the orbit map is a two-sheeted covering, and the involution is recovered from the covering. This article treats the free involutions of spheres and of manifolds, and the lens spaces, which are the quotients of odd-dimensional spheres by free cyclic actions and are the standard family in which the free quotients are fully computable.

The free case is where the equivariant theory simplifies — the fixed-set part of the surgery obstruction vanishes — and where it nevertheless shows the subtleties of the topological category: the quotient of a free involution on $S^n$ is a manifold homotopy equivalent to $\mathbb{RP}^n$, and for $n\geq5$ there are quotients that are not $\mathbb{RP}^n$, the "fake" projective spaces. The lens spaces exhibit the same phenomenon arithmetically: the homeomorphism classification (Reidemeister) is strictly finer than the homotopy classification (Whitehead), and the classical example is the pair $L(7;1)$ and $L(7;2)$, homotopy equivalent and not homeomorphic.

The article assumes the covering space theory and the fundamental group, the classification of surfaces and of manifolds, the homology and the Euler characteristic, and the fixed-set theory of Involutions on Manifolds and Equivariant Surgery. It uses the elementary theory of the groups $\mathbb{Z}/m$ and the Smith normal form of a matrix over $\mathbb{Z}/m$.

The boundaries of the article. The general involution of a manifold and its equivariant surgery are Involutions on Manifolds and Equivariant Surgery; the classification of the involutions of a surface is The Classification of Involutions on Surfaces; the periodic maps of a prime order are Periodic Maps and the Smith Theory; the classification of three-manifolds by the geometrisation is Low-Dimensional Topology, and the lens spaces as three-manifolds are Lens Spaces. The smooth structures and the four-dimensional classifications are Part III and Cobordism and Surgery Theory. No analysis is used.

Free Involutions and Coverings

Definition. An involution $T$ of a space $X$ is free if it has no fixed point. The quotient $Q = X/T$ is then a space and the orbit map $\pi : X\to Q$ is a two-sheeted covering.

Proposition (the elementary invariants of a free involution). Let $T$ be a free involution of a closed $n$-manifold $X$ with quotient $Q$. Then $Q$ is a closed $n$-manifold, the orbit map is a covering, and $$ \chi(Q) = \tfrac12\chi(X), \qquad \pi_1(Q) = \pi_1(X)/\text{(the subgroup generated by the class of a loop }x\ast Tx\text{)}, $$ the quotient of the fundamental group by the image of the covering; the involution $T$ is orientation-preserving if and only if the covering $X\to Q$ is the orientation double cover of a non-orientable $Q$, and $T$ is orientation-reversing if and only if $X$ is connected, $Q$ is non-orientable and the covering is the orientation double cover.

Proof. The quotient of a manifold by a free finite action is a manifold; the Euler characteristic halves because the covering is two-sheeted; the fundamental group is the quotient by the image of $\pi_1(X)$ in $\pi_1(Q)$ under the transfer. The orientation statement is the definition of the orientation double cover: the covering $X\to Q$ is the orientation double cover exactly when $X$ is orientable and $Q$ is not, and its deck transformation reverses the orientation of $X$; if $Q$ is orientable, the deck transformation preserves an orientation.

Proposition (classification of free involutions as coverings). Free involutions of a fixed space $X$ are classified up to conjugacy by the two-sheeted coverings of the appropriate quotient: the data are the quotient manifold $Q$ and the covering $X\to Q$, and two free involutions of $X$ are conjugate if and only if the corresponding coverings are isomorphic over $X$.

Proof. Given the involution, the quotient and the covering are determined; given the covering with $X$ connected, the deck transformation is the unique nontrivial automorphism, and it is the involution. The passage in both directions preserves conjugacy by construction.

Free Involutions on Spheres

Theorem (the quotient of a free involution on a sphere). Let $T$ be a free involution of $S^n$. Then the quotient $Q = S^n/T$ is a closed $n$-manifold with $\pi_1(Q) = \mathbb{Z}/2$, double covered by $S^n$, and homotopy equivalent to $\mathbb{RP}^n$. Moreover $Q$ is orientable exactly when $T$ preserves the orientation and non-orientable exactly when $T$ reverses it; the antipodal map has degree $(-1)^{n+1}$, so it preserves the orientation for $n$ odd and reverses it for $n$ even, and correspondingly $\mathbb{RP}^n$ is orientable exactly for $n$ odd.

Proof. The universal cover of $Q$ is $S^n$, because the cover $S^n\to Q$ is simply connected and the total space is simply connected; hence $\pi_1(Q)$ has order two; the homotopy equivalence with $\mathbb{RP}^n$ follows from the comparison of the quotients of $S^n$ by a free involution, both having a cell decomposition with one cell in each dimension $0,\dots,n$. For the orientability, an orientation of $Q$ pulls back to a $T$-invariant orientation of the connected $S^n$, and conversely a $T$-invariant orientation descends; hence $Q$ is orientable exactly when $T$ preserves some orientation of $S^n$, that is, exactly when $T$ is orientation-preserving. The antipodal map is the composition of the $n+1$ sign changes, of degree $(-1)^{n+1}$, and the orientation of $\mathbb{RP}^n$ is detected by the degree of the deck transformation, which gives the stated parity.

Theorem (uniqueness and non-uniqueness). For $n = 1,2,3$ the free involution of $S^n$ is unique up to conjugacy, with quotient $\mathbb{RP}^n$; for $n\geq5$ there are free involutions whose quotient is not homeomorphic to $\mathbb{RP}^n$, classified up to conjugacy by the Reidemeister torsion of the quotient, an invariant in the Whitehead group of $\mathbb{Z}/2$ together with the normal data; the free involutions with quotient $\mathbb{RP}^n$ are the standard ones.

Proof sketch. For $n = 3$ the quotient is a closed three-manifold with $\pi_1 = \mathbb{Z}/2$, hence by the geometrisation a quotient of $S^3$ by a free action of $\mathbb{Z}/2$ with quotient $\mathbb{RP}^3$, unique; the cases $n=1,2$ are elementary. For $n\geq5$ the quotient is a homotopy $\mathbb{RP}^n$ and the surgery theory of Cobordism and Surgery Theory classifies such manifolds by their normal data and their torsion; the non-standard quotients exist and are detected by the Reidemeister torsion, which is a simple-homotopy invariant. The full classification is the theorem of López de Medrano and of Browder–Livesay for the free case, and the low-dimensional uniqueness follows from the geometrisation in Low-Dimensional Topology.

Remark (the "fake" projective spaces). The non-standard quotients of the free involutions on $S^n$ for $n\geq5$ are the "fake projective spaces": manifolds homotopy equivalent to $\mathbb{RP}^n$ and not homeomorphic to it. They are the free analogues of the fake lens spaces and are classified by the same simple-homotopy and surgery data; their existence shows that the free involution is not determined by the homotopy type of its quotient, exactly as the lens spaces below show for the cyclic actions.

Lens Spaces

Definition. For $m\geq2$, $n\geq1$ and integers $q_1,\dots,q_n$ coprime to $m$, the lens space is $$ L(m;q_1,\dots,q_n) = S^{2n-1}/\mathbb{Z}/m, \qquad \lambda\cdot(z_1,\dots,z_n) = (\lambda^{q_1}z_1,\dots,\lambda^{q_n}z_n), $$ the quotient of the unit sphere of $\mathbb{C}^n$ by the free action of $\mathbb{Z}/m$; the quotient is a closed orientable $(2n-1)$-manifold with $\pi_1 = \mathbb{Z}/m$ and a cell decomposition with one cell in each dimension $0,1,3,\dots,2n-1$.

Proposition (lens spaces as free quotients and the involution). The lens space $L(m;q_1,\dots,q_n)$ is the quotient of $S^{2n-1}$ by the free action of $\mathbb{Z}/m$; for $m$ even the subgroup of order two of $\mathbb{Z}/m$ acts freely on $S^{2n-1}$ and the quotient map factors, $$ S^{2n-1}\longrightarrow L(m;q_1,\dots,q_n)\longrightarrow L(m/2;q_1,\dots,q_n), $$ so that the lens space of even order double covers the lens space of half the order, with the free involution $(z_1,\dots,z_n)\mapsto(-z_1,\dots,-z_n)$ as the covering transformation. The quotient of $S^{2n-1}$ by the subgroup of order two is $L(2;1,\dots,1) = \mathbb{RP}^{2n-1}$.

Proof. The action is free because the $q_i$ are coprime to $m$ and at least one coordinate is nonzero; the subgroup of order two is $\{1,-1\}$ acting by the global sign, whose quotient is the projectivisation; the factorisation of the quotient map is the factorisation of the group action through the quotient group.

Theorem (classification of lens spaces). Two lens spaces are homeomorphic if and only if their parameters are related as in the Reidemeister classification, and homotopy equivalent if and only if $$ q_1\cdots q_n \;\equiv\; \pm k^{\,n}\,q'_1\cdots q'_n \pmod m $$ for some $k$ coprime to $m$ (Whitehead). In dimension three ($n=2$) the homeomorphism classification is $$ L(m;q_1,q_2)\cong L(m;q'_1,q'_2) \quad\Longleftrightarrow\quad (q'_1,q'_2)\equiv (\varepsilon k q_1^{\pm1},\,\varepsilon k q_2^{\pm1}) \pmod m $$ up to the order of the two entries, with $\varepsilon = \pm1$ and $k$ coprime to $m$; the higher-dimensional case allows a permutation of the entries and the corresponding unit factors. Homotopy equivalence is strictly weaker than homeomorphism: the classical example is the pair $L(7;1)\simeq L(7;2)$, which is homotopy equivalent but not homeomorphic.

Proof sketch. The Reidemeister classification computes the classification of the free actions up to conjugation in the homeomorphism group, equivalently the classification of the quotients; the Franz–de Rham torsion and the linking form give the invariants, and the signs and the inverses come from the orientation and from $\mathbb{Z}/m$-changes of generator. Whitehead's theorem computes the homotopy classification by the Reidemeister torsion and the linking form, which see only the product of the parameters and the power $k^n$. For the example, the homeomorphism condition for $L(7;1)$ and $L(7;2)$ would require $2\equiv\pm1^{\pm1}\pmod7$, which is false, while the homotopy condition $1\cdot 2\equiv\pm k^2\cdot1\pmod 7$ has the solution $k^2 = 4$; hence the pair. The computations of the torsion and the invariants are the bookkeeping of Lens Spaces.

Corollary (amphichiral lens spaces). A three-dimensional lens space $L(m;q)$ is homeomorphic to its mirror image if and only if $$ q^2\equiv -1 \pmod m . $$ The lens space $L(5;2)$ is amphichiral, since $2^2 = 4\equiv-1\pmod5$; the lens space $L(7;2)$ is chiral, since $4\not\equiv-1\pmod7$. The chirality of a lens space is thus an arithmetic condition on the parameter, and the example is the three-dimensional companion of the knot-theoretic chirality of Amphichiral Knots and the Orientation-Reversing Involution.

Proof. The mirror of $L(m;q_1,q_2) = L(m;1,q)$ is $L(m;1,-q)$. By the homeomorphism classification the two are homeomorphic exactly when there are $\varepsilon = \pm1$ and a unit $k$ with $(1,-q)\equiv(\varepsilon k\cdot 1^{\pm1},\ \varepsilon k q^{\pm1})$ up to order; the first coordinate gives $\varepsilon k\equiv1$, so the second gives $-q\equiv q^{\pm1}$, that is either $2q\equiv0$ or $q^2\equiv-1\pmod m$. For $m$ odd the first alternative forces $q\equiv0$, which is impossible, leaving $q^2\equiv-1\pmod m$; the computations $2^2\equiv-1\pmod5$ and $2^2\not\equiv-1\pmod7$ give the two examples.

Remark (the free involutions on the lens spaces). A lens space of even order carries the free involution given by the subgroup of order two, with quotient the lens space of half the order; in particular every even-order lens space is a two-sheeted cover of a lens space, and the free involution is the covering transformation. A lens space of odd order has no free involution arising from a subgroup of $\mathbb{Z}/m$; a free involution on it, if it exists, is an additional structure classified by the same covering data, and in dimension three the free involutions on the lens spaces are classified by the covering theory of Low-Dimensional Topology.

Examples

Example (the projective spaces). The antipodal map of $S^n$ is the standard free involution, with quotient $\mathbb{RP}^n = L(2;1,\dots,1)$ in odd dimensions; the quotient is the model for the free-involution data, and every free involution on $S^n$ has a quotient homotopy equivalent to it.

Example (the lens space $L(5;2)$ and its mirror). The condition $q^2\equiv-1\pmod5$ holds, so the lens space is amphichiral; the mirror is $L(5;-2)=L(5;3)$, and the classification gives $3\equiv-2\pmod5$, so the mirror is homeomorphic to the original. The example is one of the smallest amphichiral lens spaces.

Example (the pair $L(7;1)$, $L(7;2)$). The two lens spaces are homotopy equivalent by Whitehead's condition and not homeomorphic by the Reidemeister classification; they are the standard example warning that the homotopy type of the quotient does not determine the free involution. The pair is also the standard example of the failure of the "homotopy equivalence implies homeomorphism" for three-manifolds, and its place in the classification is Low-Dimensional Topology.

Example (the free involution on $S^3$). The quotient of a free involution on $S^3$ is a closed three-manifold with $\pi_1 = \mathbb{Z}/2$ and universal cover $S^3$, hence $\mathbb{RP}^3 = L(2;1,1)$ by the geometrisation; the free involution on $S^3$ is unique up to conjugacy and is the antipodal map.

Example (the free involution on a lens space of even order). The lens space $L(6;1,1)$ double covers $L(3;1,1)$ by the involution $(z_1,z_2)\mapsto(-z_1,-z_2)$, and the covering is the free involution of the even-order lens space. The example shows the general factorisation $L(2k;\dots)\to L(k;\dots)$ and is the source of the free involutions on the lens spaces.

Summary

A free involution has no fixed point, its quotient is a manifold of the same dimension, the orbit map is a two-sheeted covering, and the involution is classified by the covering: the Euler characteristic halves, the fundamental group is the quotient by the image of the cover, and the involution is orientation-reversing exactly when the covering is the orientation double cover of a non-orientable quotient. A free involution on $S^n$ has a quotient with $\pi_1 = \mathbb{Z}/2$ homotopy equivalent to $\mathbb{RP}^n$; the free involution is unique for $n=1,2,3$ and for $n\geq5$ there are non-standard "fake projective" quotients classified by the Reidemeister torsion. The lens spaces $L(m;q_1,\dots,q_n)$ are the quotients of the odd spheres by free cyclic actions, with one cell in each dimension $0,1,3,\dots$; a lens space of even order double covers the lens space of half the order with the free involution given by the subgroup of order two. The homeomorphism classification (Reidemeister, with the signs and inverses of the parameters) is strictly finer than the homotopy classification (Whitehead, by the product and the power $k^n$), the standard example being $L(7;1)\simeq L(7;2)$ not homeomorphic; and a three-dimensional lens space $L(m;q)$ is amphichiral exactly when $q^2\equiv-1\pmod m$.

Summary of Notation

Symbol	Meaning
free involution $T$	an involution with no fixed point; quotient $X/T$ and two-sheeted cover $X\to X/T$
$Q = X/T$	the quotient; $\chi(Q) = \chi(X)/2$
$\mathbb{RP}^n$	the standard quotient of $S^n$ by the antipodal free involution
fake projective space	a quotient of a free involution on $S^n$, $n\geq5$, not homeomorphic to $\mathbb{RP}^n$
Reidemeister torsion	the simple-homotopy invariant classifying the free involutions for $n\geq5$ and the lens spaces
$L(m;q_1,\dots,q_n)$	the lens space $S^{2n-1}/(\mathbb{Z}/m)$, with $\lambda\cdot z = (\lambda^{q_1}z_1,\dots)$
$L(2k;q)\to L(k;q)$	the double cover by the order-two subgroup; the free involution on an even-order lens space
$L(7;1)\simeq L(7;2)$	homotopy equivalent by $q_1q_2\equiv\pm k^n q'_1q'_2$, not homeomorphic
$q_1\cdots q_n\equiv\pm k^n q'_1\cdots q'_n$	Whitehead's homotopy classification condition
$L(m;q)\cong$ mirror $\iff q^2\equiv-1\pmod m$	the amphichirality condition for a three-dimensional lens space