A NOTE ON ARITHMETIC TOPOLOGY

NIRANJAN RAMACHANDRAN

Abstract. Mazur, Manin and others have indicated striking analogies between number fields and 3-manifolds. In this note, we prove the number-theoretic counterpart of the statement integral homology 3-spheres are homotopy 3-spheres. From this, we deduce that the arithmetical analog of the Poincaré conjecture is false.

0. Introduction

B. Mazur and Yu. I. Manin (see [7], [9]) have indicated deep analogies between number fields and 3-manifolds. This analogy has been baptized Arithmetic Topology by A. Reznikov [13], [14]. Let us briefly review this analogy (cf. the dictionary by Reznikov and M. Kapranov in §14 of [14]). We shall write A for Arithmetic and T for topology. For any abelian group N, we shall write its torsion subgroup as tors(N) and its torsion-free quotient by N/tors.

T1. A closed connected 3-manifold M. A1. X = Spec O, the ring of algebraic integers of a number field L. T2. S^{3}, the 3-dimensional sphere. A2. Spec ℤ, the ring of integers. T3. tors(H_{1}(M; ℤ)). A3. P ic (X), the ideal class group of O. T4. H_{1}(M; ℤ)/tors. A4. O^{∗}/tors; here O^{∗} denotes the units of O. T5. The fundamental group π_{1}(M) of M. A5. The Galois group G_{L}^{ur} of the maximal extension of L (contained in a fixed algebraic closure L̄ of L) which is unramified at all primes (both finite and infinite); this is not always equal to the étale π_{1}(X). T6. The group π_{0}(Diff(M)) of connected components of the group Diff(M) of diffeomorphisms of M. A6. The group K_{1}(X) of X.

Since M is a closed manifold, there is no essential difference between cohomology, cohomology with compact support, and homology of M (Poincaré duality). The singular (co)homology groups of M vanish beyond dimension three. The scheme X is analogous to a 3-dimensional manifold since X is a smooth scheme and its étale cohomology groups H^{i}(X; 𝔾_{m}) are non-zero only for i ≤ 3. ^{1}

Date: 20 April 2000.

Let us recall that a closed orientable connected 3-manifold is said to be an integral homology 3-sphere if H_{1}(M; ℤ) = 0, a homotopy 3-sphere if π_{1}(M) = 0, and a rational homology 3-sphere if H_{1}(M; ℚ) = 0.

The purpose of this note is to prove the following ^{2}

Theorem 1. Every integral homology 3-sphere (in number theory) is a homotopy 3-sphere.

This is in utter contrast to the situation in topology. First, the topological analog of Theorem 1 is false (p. 244-246 of [15]). Second, there are infinitely many (mutually non-homeomorphic) integral homology 3-spheres. The Poincaré conjecture [12] states that every homotopy 3-sphere is homeomorphic to S^{3} (Consequently, there should be only one homotopy 3-sphere, up to homeomorphism).

From Theorem 1, we shall deduce that there are several mutually distinct integral homology 3-spheres (in number theory) and that the arithmetic analog of the Poincaré conjecture is false (Corollary 9).

1. Etale cohomology of number fields

The etale cohomology H^{∗}(X; 𝔾_{m}) of X with coefficients in the sheaf 𝔾_{m} is the guiding light of arithmetic topology. Good references for this are [1], [6] and [8] (II §2). Let r_{1} and 2r_{2} be the number of real and complex embeddings of L. We shall denote these infinite primes of L by v. The completion of L with respect to a prime v is denoted L_{v}. The groups H^{∗}(X; 𝔾_{m}) can be calculated explicitly (Proposition 2.1 of [8]):

(1)  H^{0}(X; 𝔾_{m}) = O^{∗},  H^{1}(X; 𝔾_{m}) = P ic (X)
(2)  H^{2}(X; 𝔾_{m}) = \begin{cases}0 & r_{1} = 0 \\ (ℤ/2ℤ)^{r_{1}−1} & r_{1} > 0 \end{cases}
(3)  H^{3}(X; 𝔾_{m}) = ℚ/ℤ
(4)  H^{n}(X; 𝔾_{m}) = ⨁_{v real} H^{n}(L_{v}; 𝔾_{m}) (n ≥ 4).

In the above, note that, for v real, L_{v} = ℝ and H^{∗}(L_{v}; 𝔾_{m}) = H^{∗}(G; 𝔾_{m}) where G = Gal(ℂ/ℝ) ≅ ℤ/2ℤ. We recall the identities

H^{2m}(G; 𝔾_{m}) = ℤ/2ℤ, H^{2m−1}(G; 𝔾_{m}) = 0 (for m > 0). (5)

With the étale cohomology in hand, we can now turn to our main question.

1Strictly speaking, this is true only up to 2-torsion (see §1).

2A. Reznikov recently informed me that he and M. Kapranov had also discovered this result.

2. When is X an integral homology 3-sphere?

As remarked earlier, we shall use cohomology (!) to define an integral homology sphere in arithmetic:

Definition 2. We say that X is an integral homology 3-sphere if it satisfies (a) H^{i}(X; 𝔾_{m}) = 0 for i ≠ 0, 3 ^{3}.

(b) H^{0}(X; 𝔾_{m}) is torsion.

Theorem 3. X is an integral homology 3-sphere if and only if L is an imaginary quadratic field of class number one.

Proof. By Dirichlet’s theorem, H^{0}(X; 𝔾_{m}) = O^{∗} is a finitely generated abelian group of rank r_{1} +r_{2} −1. Condition (b) says that r_{1} +r_{2} = 1. So we find that L must be an imaginary quadratic field or ℚ. Let us turn now to (a). When i = 1, it means that the ideal class group P ic (X) of O must be trivial, i.e., L has class number one. If L has a real embedding (r_{1} > 0), then H^{2m}(X; 𝔾_{m}) will be nonzero for all m > 1. Therefore, (a) says that L is a totally complex field of class number one. We have proved our claim.

The imaginary quadratic fields of class number one have been the subject of intense study. The conjecture of Gauss (proved by Baker, Heegner and Stark [2]) states that there are exactly nine of them. These are the fields ℚ(\sqrt{−d}) where d runs over the set \{1, 2, 3, 7, 11, 19, 43, 67, 163\}. Thus, there are exactly nine distinct integral homology 3-spheres in number theory.

Definition 4. X is a rational homology 3-sphere if it satisfies (a’) H^{∗}(X; 𝔾_{m}) ⊗ ℚ = 0.

Proposition 5. X is a rational homology 3-sphere exactly when L is an imaginary quadratic field or ℚ.

Proof. Since H^{i}(X; 𝔾_{m}) is torsion for i ≠ 0, the restriction comes from the requirement H^{0}(X; 𝔾_{m}) ⊗ ℚ = 0 (see the proof of Theorem 3).

Corollary 6. Spec ℤ is not an integral homology 3-sphere. It is, however, a rational homology 3-sphere.

3. Homotopy 3-spheres

We need an arithmetic analog of a homotopy 3-sphere in order to discuss a version of the Poincaré conjecture. We say that X is a homotopy 3-sphere if the group G_{L}^{ur} = 0 ;cf. A4. So our Theorem 1 translates to the

Theorem 7. Let L be an imaginary quadratic field of class number one. Then L has no nontrivial finite unramified extension.

Proof. It can be found in [19] Exercise 11.2, [20] §2. The ingredients are:

(i) the smallest perfect group has order 60 (it is the alternating group A_{5} of degree five). (ii) A. Odlyzko’s [10], [11] bounds on discriminants of number fields (see Theorem 11.20 and Table on p. 223 of [19]).

3See p. 137 of [9].

Corollary 8. In number theory, there are exactly ten rational homology 3spheres which are homotopy 3-spheres.

By Minkowski’s theorem, Spec ℤ is simply connected. So, it is also a homotopy 3-sphere.

Corollary 9. In number theory, there exist (at least) ten mutually nonisomorphic homotopy 3-spheres.

The previous corollary shows that the arithmetic analog of the Poincaré conjecture is false.

Remark 10. (i) Each of the integral homology 3-spheres has an involution with one fixed point. ^{4} The quotient by the involution is always Spec ℤ.

(ii) (B. Mazur) Every X comes equipped with a canonical projection π (the structure morphism) to Spec ℤ. This suggests considering enriched 3manifolds, namely, pairs (M, π) of a 3-manifold M and a map π : M → S^{3}. The Poincaré conjecture is false for enriched 3-manifolds: there exist many self-maps of S^{3}. This point shall be pursued further elsewhere.

4. Variants

Since X is not compact (it is an affine scheme), there is a difference between homology, cohomology, and cohomology with compact support of X. Here we consider the following two variants of cohomology which could be used in definitions 2 and 4.

4.1. Étale cohomology with compact support. The groups H_{c}^{∗}(X; 𝔾_{m}) are the right objects for duality theorems; they are defined in [8] (pp. 207-208). From Proposition 2.6 and Remark 2.8a of [8], we obtain

(a) H_{c}^{i}(X; 𝔾_{m}) = 0 for i ≠ 0, 1, 3,

(b) H_{c}^{3}(X; 𝔾_{m}) = ℚ/ℤ,

(c) the exact sequence

0 → H_{c}^{0}(X; 𝔾_{m}) → O^{∗} → ⊕_{v real}L_{v}^{∗}/(L_{v}^{∗})^{2} → H_{c}^{1}(X; 𝔾_{m}) → P ic (X) → 0.

Hence H_{c}^{0}(X; 𝔾_{m}) is the group of totally positive units of O and H_{c}^{1}(X; 𝔾_{m}) is the narrow class group of O.

We should view the étale fundamental group π_{1}(X) of the scheme X as the narrow fundamental group of X since its abelianization π_{1}^{ab}(X) is H_{c}^{1}(X; 𝔾_{m}). Utilizing H_{c}^{∗}(X) as our defining cohomology theory, we obtain only one new integral homology sphere, namely, Spec ℤ. Theorem 1, Proposition 5 and Corollary 9 remain true in this definition.

4Note that S^{3} is a 2-fold branched covering of S^{3} in an essentially unique manner [18].

4.2. Artin-Verdier theory. Consider the space X_{∞} = X(ℂ)/G where X(ℂ) is the set of points of X with values in ℂ and G = Gal(ℂ/ℝ) operates by complex conjugation. M. Artin and J.-L. Verdier [1] have defined an étale topology on the space X̄ = X ∐ X_{∞}. The étale fundamental group of X̄ is the group G_{L}^{ur} of A4.

They define 𝔾_{m} on X̄ to be j_{∗}𝔾_{m} where j : X ↪ X̄ denotes the open immersion. Theorem 2.4 of [1] shows that the étale cohomology groups H^{i}(X̄; 𝔾_{m}) are zero for i ≠ 0, 1, 3, and that H^{0}(X̄; 𝔾_{m}) = O^{∗}, H^{1}(X̄; 𝔾_{m}) = P ic (X) and H^{3}(X; 𝔾_{m}) = ℚ/ℤ.

In this variant as well, it turns out that we acquire only one new integral homology sphere, namely, Spec ℤ. Theorem 1, Proposition 5 and Corollary 9 remain true in this definition as well. Perhaps the Artin-Verdier definition is the most appropriate.

Remark 11. The three different cohomology theories agree for totally complex fields (r_{1} = 0). For these fields, it is also the case that the étale fundamental group π_{1}(X) is the same as G_{L}^{ur}.

Remark 12. We wish to contribute a few more points to the analogy.

T7. π_{0}(Diff(S^{3})) = ℤ/2ℤ (See [3]). A7. K_{1}(ℤ) = ℤ/2ℤ since K_{1}(O) = O^{∗}. T8. The ends of a (possibly non-compact) manifold. A8. X_{∞} (see 4.2).

Mazur and Manin have suggested that primes correspond to knots. A knot is given by a closed immersion S^{1} ↪ M and a finite prime is given by a closed immersion Spec 𝔽 ↪ X of a finite field 𝔽. T9. S^{1} is an Eilenberg-Mac Lane space K(ℤ, 1). A9. For any finite field 𝔽, the scheme Spec 𝔽 is an Eilenberg-Mac Lane space K(ℤ̂, 1) where ℤ̂ is the profinite completion of the group ℤ.

Let T denote the tubular neighbourhood of a knot K in M. Let O_{𝔭} (resp. L_{𝔭}) denote the local ring (resp. field) corresponding to a finite prime 𝔭 of X. As usual, ℕ𝔭 is the cardinality of the residue field 𝔽_{𝔭} = O/𝔭. T10. The surface S = ∂T (homotopy equivalent to T − K). A10. Spec L_{𝔭} = Spec O_{𝔭}− Spec 𝔽_{𝔭}. T11. π_{1}(S) has a presentation ⟨x, y|xyx^{−1} = y⟩. A11. The Galois group ^{5} of the maximal tamely ramified extension ([5], p.

173) of L_{𝔭} has a presentation ⟨x, y|xyx^{−1} = y^{ℕ𝔭}⟩.

Since there is no wild ramification in characteristic zero, it is the tame Galois group of L_{𝔭} that enters into the analogy. Should we consider the residue field of a knot to be 𝔽_{1} (the field with one element!) [17]? T12. Every closed 3-manifold M is (non-canonically) a branched covering of S^{3} [4]. A12. Every X is (canonically) a branched covering of Spec ℤ.

5This group has cohomological dimension two ([16] p.86).

To conclude, in number theory, there are nine legitimate ( = insensitive to the variation of definition) integral homology 3-spheres and ten homotopy 3-spheres. Even if Spec ℤ is adopted as the analog of S^{3}, the Poincaré conjecture is still false in number theory.

Acknowledgements. I have benefited greatly from discussions with A. Giacobbe, J. Gordon, I. Kofman, M. Marcolli, A. Sikora and L. Washington.

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Department of Mathematics, University of Maryland, College Park MD 20742-4015, USA

E-mail address: atma@math.umd.edu