Homology, Homotopy and Applications, vol.1(2), 2000, pp.325–344

THE BLOCH INVARIANT AS A CHARACTERISTIC CLASS IN B(SL_{2}(ℂ), 𝔗)

JOSÉ LUIS CISNEROS-MOLINA and JOHN D. S. JONES

(communicated by Jonathan Rosenberg)

Abstract Given an orientable complete hyperbolic 3-manifold of finite volume M we construct a canonical class α(M) in H_{3}(B(SL_{2}(ℂ), 𝔗)) with B(SL_{2}(ℂ), 𝔗) the SL_{2}(ℂ)-orbit space of the classifying space for a certain family of isotropy subgroups. We prove that α(M) coincides with the Bloch invariant β(M) of M defined by Neumann and Yang in [13], giving with this a simpler proof that the Bloch invariant is independent of an ideal triangulation of M. We also give a new proof of the fact that the Bloch invariant lies in the Bloch group B(ℂ).

1. Introduction

Given a compact oriented hyperbolic 3-manifold M endowed with a spin structure there is a representation ρ: π_{1}(M) → SL_{2}(ℂ) given by the spin structure. Such representation corresponds to a map Bρ: M → BSL_{2}(ℂ) where BSL_{2}(ℂ) is the classifying space of SL_{2}(ℂ) considered as a discrete group. There is a well-known invariant ⟨M⟩ of M in H_{3}(BSL_{2}(ℂ); ℤ) given by the image of the fundamental class of M under the homomorphism induced in homology by Bρ. We extend this invariant to an invariant α(M) when M is a complete oriented hyperbolic 3-manifold of finite volume (i.e. M is compact or with cusps) but in this case α(M) takes values in H_{3}(B(SL_{2}(ℂ), 𝔗); ℤ) where B(SL_{2}(ℂ), 𝔗) is the SL_{2}(ℂ)-orbit space of E(SL_{2}(ℂ), 𝔗), the classifying space for SL_{2}(ℂ)-actions with isotropy subgroups in certain subgroup-closed SL_{2}(ℂ)-family 𝔗. We show that there is a canonical map ψ: M_{+} → B(SL_{2}(ℂ), 𝔗) where M_{+} is the one-point compactification of M. The invariant α(M) is defined as the image of the fundamental class of M_{+} under the homomorphism ψ_{∗}: H_{3}(M_{+}; ℤ) → H_{3}(B(SL_{2}(ℂ), 𝔗); ℤ). The invariant α(M) extends ⟨M⟩ in the following sense: There is a canonical map k: BSL_{2}(ℂ) → B(SL_{2}(ℂ), 𝔗) and when M is compact M_{+} = M and ψ ≃ k ∘ Bρ. Therefore the homomorphism in homology induced by k sends ⟨M⟩ to α(M).

The first author was partially supported by proyecto CONACYT G36357-E. Received August 23, 2003, revised September 9, 2003; published on September 23, 2003. 2000 Mathematics Subject Classification: 57M27, 53R40, 19E99, 19D99 Key words and phrases: Hyperbolic 3-manifolds, Classifying space for families of isotropy subgroups, Bloch group, Bloch invariant, acyclic maps \begin{matrix}c \\ ◯ \end{matrix} 2000, José Luis Cisneros-Molina and John D. S. Jones. Permission to copy for private use granted.

By [17, Lemma 2.2] we have that H_{3}(B(SL_{2}(ℂ), 𝔗); ℤ) ≅ P(ℂ) where P(ℂ) is the pre-Bloch group. Using (degree one) ideal triangulations of M Neumann and Yang defined in [13] an element β(M) ∈ P(ℂ) called the Bloch invariant of M. They proved that β(M) is independent of the ideal triangulation, showing that the generator of certain relative homology group isomorphic to ℤ is mapped to β(M). We prove that α(M) coincides with the Bloch invariant β(M). This gives a simpler proof that the Bloch invariant is independent of the ideal triangulation.

A priori, we have that α(M) lies in P(ℂ) but in [13] Neumann and Yang proved that β(M) actually lies in certain subgroup B(ℂ) of P(ℂ) called the Bloch group. We also give a new proof of this fact. We do this by constructing a map R: M_{+} → BSL_{2}(ℂ)^{+} such that the canonical map ψ: M_{+} → B(SL_{2}(ℂ), 𝔗) factors through BSL_{2}(ℂ)^{+} as ψ ≃ h ∘ R, where h: BSL_{2}(ℂ)^{+} → B(SL_{2}(ℂ), 𝔗) is the map induced by the canonical map k: BSL_{2}(ℂ) → B(SL_{2}(ℂ), 𝔗) by the plusconstruction. Taking the homomorphisms induced in homology we have that ψ_{∗} factors through H_{3}(SL_{2}(ℂ); ℤ) and by the Bloch-Wigner exact sequence [8, Appendix A] (see Section 7) we have that the image of h_{∗} is precisely the Bloch group B(ℂ).

The paper is organised as follows. In Section 2 we recall a well-known theorem which gives the global geometry of the aforementioned hyperbolic 3-manifolds. In Section 3 we define the classifying spaces for families of isotropy subgroups and we list some of their main properties. In Section 4 we define the invariant α(M). In Section 5 we recall the definitions of the Bloch group B(ℂ) and the Bloch invariant and in Section 6 we prove that this invariant coincides with α(M). In Section 7 we describe the Bloch-Wigner exact sequence and finally, in Section 8 we construct the map R: M_{+} → BSL_{2}(ℂ)^{+} and prove that α(M) lies in the Bloch group B(ℂ).

2. Hyperbolic 3-manifolds

A complete oriented hyperbolic 3-manifold M is the quotient of the hyperbolic 3space H^{3} by a discrete, torsion-free subgroup Γ of orientation preserving isometries.

Identify the upper half space model H^{3} with the set of quaternions { z + tj | z ∈ ℂ, t > 0 }. The space H^{3} is bounded by the extended complex plane ℂ̂ = ℂ ∪ {∞} (Riemann sphere). The group of orientation preserving isometries of H^{3} is isomorphic to P SL_{2}(ℂ) = SL_{2}(ℂ)/\{±I\} and the action of ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ P SL_{2}(ℂ) in H^{3} is given by the linear fractional transformation

ϕ(w) = (aw + b)(cw + d)^{−1},  w = z + tj,  ad − bc = 1,

which is the extension to H^{3} of the complex linear fractional transformation on ℂ̂ given by ( \begin{matrix}a & b \\ c & d \end{matrix} ). This action is transitive and the isotropy group of the point (0, 1) ∈ H^{3} is P SU(2) ≅ SO(3), its maximal compact subgroup. Thus the correspondence g ↦ g(0, 1) for g ∈ P SL_{2}(ℂ) induces a diffeomorphism

P SL_{2}(ℂ)/SO(3) ≅ SL_{2}(ℂ)/SU(2) ⟶^{≅} H^{3}.  (1)

Any discrete, torsion-free subgroup Γ of P SL_{2}(ℂ) acts freely and properly discontinuously on H^{3}. Then, the quotient map

π: H^{3} → Γ\H^{3} = M,

is the universal cover and therefore π_{1}(M) = Γ. Note that M is a K(Γ, 1), i.e. M = BΓ, the classifying space of Γ.

Two hyperbolic 3-manifolds M = Γ\H^{3} and M′ = Γ′\H^{3} are isometric by an orientation-preserving isometry if and only if Γ and Γ′ are conjugate in P SL_{2}(ℂ) [14, Thm. 8.1.5]. To such an hyperbolic 3-manifold we can associate a representation ρ: π_{1}(M) = Γ → P SL_{2}(ℂ) which is canonical up to equivalence.

Using the diffeomorphism (1) we have that any complete, oriented, hyperbolic 3-manifold M is given by the double coset space

M = Γ\P SL_{2}(ℂ)/SO(3),

with Γ = π_{1}(M) a discrete torsion-free subgroup. The canonical representation Γ → P SL_{2}(ℂ) is given by the inclusion and it can be lifted to a representation ρ: Γ → SL_{2}(ℂ) [6, Prop. 3.1.1]. There is a one-to-one correspondence between such lifts and spin structures on M. Hence, any complete, oriented hyperbolic 3-manifold M equipped with a spin structure can be seen as the double coset space

M = Γ\SL_{2}(ℂ)/SU(2),

where we identify Γ with a subgroup of SL_{2}(ℂ) using the representation ρ: Γ → SL_{2}(ℂ) which corresponds to the spin structure.

Throughout this paper, M will denote a non-compact orientable complete hyperbolic 3-manifold of finite volume. The global geometry of such manifolds is given in the following theorem, see for instance [14, p. 647 Cor. 4 and Thm. 10.2.1].

Theorem 2.1. Let M be an orientable complete hyperbolic 3-manifold of finite volume. Then there is a compact 3-manifold-with-boundary M_{0} in M such that M − M_{0} is the disjoint union of a finite number of cusps. Each cusp of M is diffeomorphic to T^{2} × (0, ∞), where T^{2} denotes the 2-torus.

Remark 2.2. Let M be an oriented complete hyperbolic 3-manifold of finite volume with d cusps. Take a base point x_{0} in M and a point x_{i} in a torus section T_{i}^{2} of the end corresponding to the i-th cusp (i = 1, …, d). Let q_{i}(t) (0 ⩽ t ⩽ 1) be a path in M with q_{i}(0) = x_{0} and q_{i}(1) = x_{i}. Consider the homomorphism (q_{i})_{#} : π_{1}(T_{i}^{2}, x_{i}) → π_{1}(M, x_{0}) induced by q_{i} and put Γ_{i} = (q_{i})_{#}(π_{1}(T_{i}^{2}, x_{i})). The subgroups Γ_{i} are called the peripheral torus subgroups of Γ. The image of Γ_{i} under the representation ρ: Γ → SL_{2}(ℂ) given by the inclusion is a free abelian group of rank 2 of SL_{2}(ℂ). The subgroups Γ_{i} consist only of parabolic elements, that is, they fix no points of H^{3} and fix a unique point of the boundary ℂ̂ of H^{3}. All the elements in Γ_{i} have as a fixed point the corresponding cusp point [18, §4.5]. Since the action of SL_{2}(ℂ) in ℂ̂ is transitive and the conjugates of parabolic isometries are parabolic [14, p. 141] we can assume that the fixed point is the point at infinity ∞ which we denote by its homogeneous coordinates ∞ = [1, 0] and therefore the subgroups Γ_{i} are conjugate to a group of matrices of the form ( \begin{matrix}±1 & λ \\ 0 & ±1 \end{matrix} ), with λ ∈ ℂ.

3. Classifying spaces for families of isotropy subgroups

Let G be a group and let 𝔰𝔲𝔟(G) denote the set of all subgroups of G. A subset 𝔉 of 𝔰𝔲𝔟(G) is a subgroup-closed G-family if each subgroup of each element of 𝔉 belongs to 𝔉 and, moreover, 𝔉 is closed under taking conjugates by elements of G. Let {H_{i}}_{i∈I} be subgroups of G, we denote by 𝔉(H_{i}) the subgroup-closed G-family consisting of all the subgroups of the {H_{i}}_{i∈I} and all their conjugates by elements of G.

Let X be a G-space. For each subgroup H of G, we define the set X^{H} consisting of points of X fixed by all of H.

A classifying space for G-actions with isotropy subgroups in 𝔉, or a space of type E(G, 𝔉), is a G-space with the following properties: for each H ∈ 𝔉, X^{H} is contractible and for each H ∈ 𝔰𝔲𝔟(G) − 𝔉, X^{H} is empty.

The G-space E(G, 𝔉) has all its isotropy subgroups in 𝔉 and it has the following universal property. Given any G-space X with isotropy subgroups in 𝔉 there exist a G-map

X → E(G, 𝔉)

which is unique up to G-homotopy [19, Rmk. 7.2.1], [20, Thm. 6.6]. The classifying space E(G, 𝔉) is unique up to G-homotopy equivalence [7, Prop. 2.2], [19, Rmk. 7.2.2].

There are different ways of constructing the spaces E(G, 𝔉). We consider the following two:

1. There exist a countable system {H_{i}}_{i∈I} of subgroups H_{i} ∈ 𝔉 such that every group in 𝔉 is conjugate to an H_{i}, that is, 𝔉 = 𝔉(H_{i}). Let E_{i} = G/H_{i}∗G/H_{i}∗… be the join of a countable number of copies of G/H_{i}. Then E(G, 𝔉) = ∗_{i∈I}E_{i} is the join of the E_{i}, i ∈ I, with the Milnor topology [19, Prop. 7.2.1].

2. Let Δ be any G-set such that 𝔉 is precisely the set of subgroups of G which fix at least one point of Δ. Then E(G, 𝔉) = Δ∗Δ∗…, the join of a countably number of copies of Δ [7, Prop. 2.2].

Example 3.1. Consider SL_{2}(ℂ) as a discrete group. Since the action of SL_{2}(ℂ) on the universal bundle ESL_{2}(ℂ) is free and ESL_{2}(ℂ) is contractible we have that ESL_{2}(ℂ) is a space of type E(SL_{2}(ℂ), {1}).

Example 3.2. Consider the sphere S^{2} as the extended complex numbers ℂ̂. We have that SL_{2}(ℂ) acts triply transitively on S^{2}. Let T be the isotropy subgroup of ∞, T consists of the matrices of the form ( \begin{matrix}z & a \\ 0 & z^{−1} \end{matrix} ), with a, z ∈ ℂ and z ≠ 0. Hence we have a diffeomorphism given by

\begin{matrix}SL_{2}(ℂ)/T & ⟶^{≅} S^{2} \\ gT & ↦ g · ∞. \end{matrix}

Let J(S^{2}) = S^{2} ∗ S^{2} ∗ …, the join of a countable number of copies of S^{2}. By construction 1 of E(G, 𝔉) we have that J(S^{2}) is an E(SL_{2}(ℂ), 𝔉(T)).

Example 3.3. Let Γ be a discrete torsion-free subgroup of SL_{2}(ℂ). The action of Γ on the hyperbolic 3-space H^{3} is free and since H^{3} is contractible, it is a space of type E(Γ, {1}).

Let M = Γ\H^{3} be a non-compact orientable complete hyperbolic 3-manifold of finite volume. Let π: H^{3} → Γ\H^{3} = M be the universal cover of M. Let C be the set of cusp points, that is, the preimages of cusps of M in ∂H̄^{3} = ℂ̂, or, equivalently, fixed points of parabolic elements of Γ and let Ŷ = H^{3} ∪ C. The action of Γ on Ŷ is no longer free. The cusp points have as isotropy subgroups the peripheral torus subgroups Γ_{1}, …, Γ_{d} of Γ or their conjugates; therefore, by the definition of E(Γ, 𝔉), we have that Ŷ is a E(Γ, 𝔉(Γ_{1}, …, Γ_{d})).

We list some results about E(G, 𝔉):

Proposition 3.4 ([7, Cor. 2.3]). If 𝔉_{1} ⊆ 𝔉_{2} are subgroup-closed G-families, then there exist a G-map E(G, 𝔉_{1}) → E(G, 𝔉_{2}) unique up to G-homotopy.

Corollary 3.5. • There is a SL_{2}(ℂ)-map ℓ: ESL_{2}(ℂ) → J(S^{2}) unique up to

SL_{2}(ℂ)-homotopy.

• There is a Γ-map l: H^{3} → Ŷ unique up to Γ-homotopy.

Let H be a subgroup of G. For a G-space X, let res_{H}X be the H-space obtained by restricting the group action. If 𝔉 is a subgroup-closed G-family, let 𝔉/H = { L ∩ H | L ∈ 𝔉 } be the induced family of subgroups of H.

Proposition 3.6 ([19, Prop. 7.2.4]).

res_{H}E(G, 𝔉) = E(H, 𝔉/H).

Corollary 3.7. 1. res_{Γ}ESL_{2}(ℂ) is Γ-homotopy equivalent to H^{3}.

2. There is a Γ-map Ŷ → res_{Γ}J(S^{2}) unique up to Γ-homotopy.

Proof. Claim 1 is a direct consequence of Proposition 3.6. Also by Proposition 3.6 we have that res_{Γ}J(S^{2}) is an E(Γ, 𝔉(T)/Γ). By Example 3.3 Ŷ is an 𝔉(Γ_{1}, …, Γ_{d}) where Γ_{1}, …, Γ_{d} are the peripheral torus subgroups. We have that 𝔉(Γ_{1}, …, Γ_{d}) ⊂ 𝔉(T)/Γ and the result follows by Proposition 3.4.

Following this, we give a simplicial version of the map ℓ: ESL_{2}(ℂ) → J(S^{2}) of Corollary 3.5 which we use in following sections.

Example 3.8. Consider the universal bundle ESL_{2}(ℂ) as the geometric realisation of the simplicial complex where an n-simplex is an (n + 1)-tuple (g_{0}, …, g_{n}) with g_{i} ∈ SL_{2}(ℂ), and the i-th face (resp., degeneracy) of such a simplex is obtained by omitting (resp., repeating) g_{i}. The action of g ∈ SL_{2}(ℂ) gives the simplex (gg_{0}, …, gg_{n}). The infinite join J(S^{2}) is the geometric realisation of the simplicial set whose n-simplices are the n + 1-tuples (a_{0}, …, a_{n}) of elements of S^{2} = ℂ̂ and the i-th face (resp., degeneracy) of such a simplex is obtained by omitting (resp., repeating) a_{i}. The action of g ∈ SL_{2}(ℂ) gives the simplex (ga_{0}, …, ga_{n}). Hence the map ℓ: ESL_{2}(ℂ) → J(S^{2}) is the (geometric realisation of the) SL_{2}(ℂ)-equivariant simplicial map

(g_{0}, …, g_{n}) → (g_{0} · ∞, …, g_{n} · ∞).  (2)

Choosing a different basepoint instead of ∞ by Corollary 3.5 we get a SL_{2}(ℂ)- homotopic map.

4. The invariant α(M) of M

To simplify notation, if G is a group and X is a G-space, we denote the space of orbits G\X by X_{G}. Also for the rest of the paper, let 𝔗 = 𝔉(T) with T as in Example 3.2 and B(SL_{2}(ℂ), 𝔗) = E(SL_{2}(ℂ), 𝔗)_{SL_{2}(ℂ)}.

Let M be an orientable complete hyperbolic 3-manifold of finite volume, choosing a spin structure on M we have that M = Γ\H^{3} for some discrete torsion-free subgroup of SL_{2}(ℂ). Let Ŷ = H^{3}∪C with C the set of cusp points and let M̂ = Γ\Ŷ. If M is compact C = ∅ and M̂ = M, if M is non-compact we have that M̂ is the end-compactification of M which is the result of adding the cusps of M.

By Corollary 3.7-1, we have that M = Γ\H^{3} is homotopically equivalent to ESL_{2}(ℂ)_{Γ} = BΓ and by Corollary 3.7-2 there is a Γ-map

M̂ → J(S^{2})_{Γ} = E(SL_{2}(ℂ), 𝔗)_{Γ}

which is unique up to Γ-homotopy. We also have that ESL_{2}(ℂ)_{SL_{2}(ℂ)} = BSL_{2}(ℂ). Combining this with Corollary 3.5 and taking the quotients by SL_{2}(ℂ) and Γ we get the following commutative diagram

\xymatrix{M \ar[r]^{q̂} \ar[d]^{f} & M̂ \ar[d]^{ψ} \\ BSL_{2}(ℂ) \ar[r]^{k} & B(SL_{2}(ℂ), 𝔗)}  (3)

where q̂ (resp., k) is the map induced by the map l (resp. ℓ) of Corollary 3.5 on the sets of Γ (resp. SL_{2}(ℂ)) orbits; f = Bρ: BΓ → BSL_{2}(ℂ) is the map between classifying spaces which on fundamental groups induces the canonical representation ρ: Γ → SL_{2}(ℂ) of M, and ψ is given by the composition

ψ: M̂ → E(SL_{2}(ℂ), 𝔗)_{Γ} → B(SL_{2}(ℂ), 𝔗).  (4)

The map ψ induces an homomorphism

ψ_{∗}: H_{3}(M̂; ℤ) → H_{3}(B(SL_{2}(ℂ), 𝔗); ℤ).  (5)

Let [M̂] denote the fundamental class of M̂ in H_{3}(M̂; ℤ). We denote by α(M) the canonical class in H_{3}(B(SL_{2}(ℂ), 𝔗); ℤ) given by

α(M) = ψ_{∗}([M̂]).

5. The Bloch invariant

In the present section, following [13], we shall define the Bloch invariant of a non-compact orientable complete hyperbolic 3-manifold of finite volume which has an ideal triangulation.

The pre-Bloch group P(F) of a field F is the quotient of the free abelian group generated by the formal symbols [x], x ∈ F ∪ {∞} by the following relations

[0] = [1] = [∞] = 0
[x] − [y] + [y/x] − [(1 − x^{−1})/(1 − y^{−1})] + [(1 − x)/(1 − y)] = 0.

The Bloch group B(F) is the kernel of the map

\begin{matrix}P(F) & ⟶^{λ} F^{×} ∧ F^{×} \\ [x] & ↦ 2(x ∧ (1 − x)), \end{matrix}  (6)

where F^{×} is the multiplicative group of F.

Let H̄^{3} = H^{3} ∪ ℂ̂ be the standard compactification of the hyperbolic 3-space H^{3}. We shall refer to the elements of the boundary ℂ̂ of H^{3} as points at infinity.

An ideal tetrahedron (or ideal simplex) is a tetrahedron in H̄^{3} with all its vertices in ℂ̂. Let △(a_{0}, a_{1}, a_{2}, a_{3}) be an ideal tetrahedron with vertices a_{0}, a_{1}, a_{2}, a_{3}. The group P SL_{2}(ℂ) acts on ℂ̂ by fractional linear transformations and the action is triply transitive. Therefore, there exists an element g ∈ P SL_{2}(ℂ) such that

g · a_{0} = 0, g · a_{1} = ∞, g · a_{2} = 1, g · a_{3} = z

where z = [a_{0} : a_{1} : a_{2} : a_{3}] is the cross-ratio of the vertices given by

[a_{0} : a_{1} : a_{2} : a_{3}] = \frac{(a_{0} − a_{3})(a_{1} − a_{2})}{(a_{0} − a_{2})(a_{1} − a_{3})}.  (7)

Therefore the congruence classes of ideal tetrahedra are parameterised by z ∈ ℂ − {0, 1}. We extend the definition of the cross-ratio to [a_{0} : a_{1} : a_{2} : a_{3}] = 0 whenever a_{i} = a_{j} for some i ≠ j.

Let M be a non-compact orientable complete hyperbolic 3-manifold of finite volume. An ideal triangulation for M is a triangulation where all the tetrahedra △_{i} are ideal tetrahedra.

Remark 5.1. In [9] Epstein and Penner proved that these kind of manifolds can be decomposed into convex ideal polyhedra, but to the best of our knowledge it is still unknown whether this decomposition can be subdivided into an ideal triangulation (compare [13, §10]).

For this reason, from now on we shall assume that when M is a non-compact orientable complete hyperbolic 3-manifold of finite volume it has an ideal triangulation.

Let M be an hyperbolic 3-manifold and let △_{1}, …, △_{n} be the ideal tetrahedra of an ideal triangulation of M. Let z_{i} ∈ ℂ be the parameter of △_{i} for each i. These parameters define an element β(M) = ∑_{i=1}^{n}[z_{i}] in the pre-Bloch group. The element β(M) ∈ P(ℂ) is called the Bloch invariant of M.

Remark 5.2. Neumann and Yang defined the Bloch invariant using degree one ideal triangulations, in that way it is defined for all hyperbolic 3-manifolds of finite volume, even the compact ones, see [13, §2] for details.

In [13, Thm. 1.1] Neumann and Yang proved that β(M) depends only on M and not on the triangulation, and that it actually lies in the Bloch group B(ℂ). In the following section, we give a simpler proof of the independence of the Bloch invariant of the ideal triangulation proving that α(M) = β(M). In section 8 we give another proof that α(M) = β(M) ∈ B(ℂ).

6. The invariant α(M) is the Bloch invariant

In this section we prove that the characteristic class α(M) of an orientable complete hyperbolic 3-manifold of finite volume M equals its Bloch invariant β(M), in particular, this gives a new proof that the Bloch invariant is independent of the ideal triangulation, or from a different point of view, we can compute α(M) using an ideal triangulation of M.

For the rest of the paper we shall consider J(S^{2})_{SL_{2}(ℂ)} as the model for B(SL_{2}(ℂ), 𝔗).

Firstly, by [17, Lemma 2.2] (but using SL_{2}(ℂ) instead of GL_{2}(ℂ)) we have that H_{3}(B(SL_{2}(ℂ), 𝔗); ℤ) = H_{3}(J(S^{2})_{SL_{2}(ℂ)}; ℤ) ≅ P(ℂ), hence α(M) ∈ P(ℂ). The isomorphism is given by

(a_{0}, a_{1}, a_{2}, a_{3})_{SL_{2}(ℂ)} ↦ [a_{0} : a_{1} : a_{2} : a_{3}],  (8)

(a_{0}, a_{1}, a_{2}, a_{3})_{SL_{2}(ℂ)} denotes the SL_{2}(ℂ)-orbit of the 3-simplex (a_{0}, a_{1}, a_{2}, a_{3}) and [a_{0} : a_{1} : a_{2} : a_{3}] is the cross-ratio of the a_{i} ∈ ℂ̂, i = 0, 1, 2, 3.

Theorem 6.1. Let M be an orientable complete hyperbolic 3-manifold of finite volume. Then α(M) = β(M).

Proof. Let M be a non-compact orientable complete hyperbolic 3-manifold of finite volume. Let π: H^{3} → Γ\H^{3} = M be the universal cover of M. Then M lifts to an exact, convex, fundamental, ideal polyhedron P for Γ [14, Thm. 11.2.1]. An ideal triangulation of M gives a decomposition of P into a finite number of ideal tetrahedra. Since P = { gP | g ∈ Γ } is an exact tessellation of H^{3} [14, Thm. 6.7.1], this decomposition of P gives an ideal triangulation of H^{3}.

As in Example 3.3, let C be the set of cusp points and let Ŷ = H^{3} ∪ C, which is the result of adding the vertices of the ideal tetrahedra of the ideal triangulation of H^{3}. Hence we can consider Ŷ as a simplicial complex with vertices given by the cusp points.

To prove the theorem, we use the simplicial version of the Γ-map Ŷ → J(S^{2}) given in Corollary 3.7-2. Consider J(S^{2}) as the geometric realization of a simplicial complex as in Example 3.8. Considering Ŷ as the geometric realisation of its ideal triangulation and since its vertices are the cusp points C ⊂ ℂ̂ we have that the Γ-map Ŷ → J(S^{2}) is given by the (geometric realisation of the) Γ-equivariant simplicial map

Ŷ → J(S^{2})
(a_{0}, a_{1}, a_{2}, a_{3}) ↦ (a_{0}, a_{1}, a_{2}, a_{3}).

The composition (4) induces on 3-chains the map

C_{3}(M̂) → C_{3}(J(S^{2})_{SL_{2}(ℂ)})
(a_{0}, a_{1}, a_{2}, a_{3})_{Γ} ↦ (a_{0}, a_{1}, a_{2}, a_{3})_{SL_{2}(ℂ)} = (0, ∞, 1, z)_{SL_{2}(ℂ)}.

where (a_{0}, a_{1}, a_{2}, a_{3})_{Γ} (resp. (a_{0}, a_{1}, a_{2}, a_{3})_{SL_{2}(ℂ)}) denotes the Γ-orbit (resp. SL_{2}(ℂ)- orbit) of the 3-simplex (a_{0}, a_{1}, a_{2}, a_{3}) and z = [a_{0} : a_{1} : a_{2} : a_{3}].

If △_{i} = △(a_{0}^{i}, a_{1}^{i}, a_{2}^{i}, a_{3}^{i}), i = 1, …, n, are the ideal tetrahedra of an ideal triangulation of M and z_{i} = [a_{0}^{i} : a_{1}^{i} : a_{2}^{i} : a_{3}^{i}] ∈ ℂ is the parameter of △_{i} for each i, now is clear that under the homomorphism (5) the fundamental class [M̂] is sent to the Bloch invariant β(M)

H_{3}(M̂) → H_{3}(J(S^{2})_{Γ}) → P(ℂ)

[M̂] ↦ ∑_{i=1}^{n}(a_{0}^{i}, a_{1}^{i}, a_{2}^{i}, a_{3}^{i})_{Γ} ↦ ∑_{i=1}^{n}(a_{0}^{i}, a_{1}^{i}, a_{2}^{i}, a_{3}^{i})_{SL_{2}(ℂ))} = ∑_{i=1}^{n}[z_{i}].

For the case when M is compact, following [13, §2] choose a degree one ideal triangulation of M given by a geometric 3-cycle Y and a map f: Y − Y^{(0)} → M with |Y | = M. As in [13, §4], form the pull-back covering

\xymatrix{{ \atop Y − Y^{(0)}} \ar[r] \ar[d] & H^{3} \ar[d] \\ Y − Y^{(0)} \ar[r] & M}

and complete to get a simplicial complex Ŷ with a Γ-action (which is free except maybe at the vertices). By Corollary 3.7 there is a Γ-map Ŷ → J(S^{2}) and we can substitute H^{3} by ESL_{2}(ℂ). Taking the quotients by SL_{2}(ℂ) and Γ we get the following commutative diagram

\xymatrix{Y − Y^{(0)} \ar[r] \ar[d]^{f} & M \ar[d]^{ψ} \\ BSL_{2}(ℂ) \ar[r]^{k} & B(SL_{2}(ℂ), 𝔗)}

since Γ\Ŷ = Y = M. The rest of the proof is as for the non-compact case using this Ŷ.

Remark 6.2. Recall that by Theorem 2.1 M is homotopically equivalent to M_{0} therefore we can think that the map q̂: M_{0} → M̂ collapses each torus component of M_{0} to a point. On the other hand, consider the quotient map c: M̂ → M_{+} from the end-compactification to the one-point-compactification of M which consists in identifying all the cusps points of M̂ to a single point. Since the action of G on J(S^{2}) identifies all the 0-simplices, all the cusps points of M̂ are mapped to the unique 0-simplex of J(S^{2})_{SL_{2}(ℂ)} under the map ψ: M̂ → J(S^{2})_{SL_{2}(ℂ)}, thus the map ψ descends to a map M_{+} → J(S^{2})_{SL_{2}(ℂ)} which we shall also call ψ. Since the map c induces an isomorphism in H_{3} we also have that α(M) = ψ_{∗}([M_{+}]) = β(M) with [M_{+}] the fundamental class of M_{+}. Hence, diagram (3) is equivalent to the following with q = c ∘ q̂

\xymatrix{M_{0} \ar[r]^{q} \ar[d]^{f} & M_{+} \ar[d]^{ψ} \\ BSL_{2}(ℂ) \ar[r]^{k} & J(S^{2})_{SL_{2}(ℂ)}}  (9)

7. The Bloch-Wigner exact sequence

In the present section we describe the Bloch-Wigner exact sequence [8, Appendix A] which will help us in the next section to prove that α(M) = β(M) ∈ B(ℂ).

Recall that for a (discrete) group G its (Eilenberg–Mac Lane) cohomology groups H_{EM}^{n}(G, A) with coefficients in a G-module A is the homology of the complex

C^{n}(G, A) = { f: G^{n+1} → A | gf(g_{0}, …, g_{n}) = f(gg_{0}, …, gg_{n}) }

with coboundary δ: C^{n}(G, A) → C^{n+1}(G, A) given by

δ(f)(g_{0}, …, g_{n+1}) = ∑_{i=0}^{n+1}(−1)^{i}f(g_{0}, …, ĝ_{i}, …, g_{n+1}),

where ĝ_{i} indicates that g_{i} is omitted.

Lemma 7.1. Let G = SL_{2}(ℂ). Consider the function given by

τ̄: G × G × G × G → P(ℂ)
τ̄(g_{0}, g_{1}, g_{2}, g_{3}) = [[g_{0} · ∞ : g_{1} · ∞ : g_{2} · ∞ : g_{3} · ∞]], ∞ ∈ ℂ̂,

where [a_{0} : a_{1} : a_{2} : a_{3}] with a_{i} ∈ ℂ̂ is the cross-ratio given by (7) and a cross-ratio of non-distinct points is taken as zero. Then τ̄ represents a cohomology class in H_{EM}^{3}(SL_{2}(ℂ), P(ℂ)) which we denote by τ.

Proof. Clearly τ̄ is invariant under the left diagonal action of G. First we are going to show that it is a cocycle of the complex C^{•}(G, P(ℂ)). Since the action of SL_{2}(ℂ) in ℂ̂ is triply transitive we can assume that

(g_{0} · ∞, g_{1} · ∞, g_{2} · ∞, g_{3} · ∞, g_{4} · ∞) = (0, ∞, 1, x, y).

Thus, we have

δ(τ̄)(g_{0}, g_{1}, g_{2}, g_{3}, g_{4}) = [∞ : 1 : x : y] − [0 : 1 : x : y] + [0 : ∞ : x : y] − [0 : ∞ : 1 : y] + [0 : ∞ : 1 : x], and applying the definition of the cross-ratio we get

δ(τ̄)(g_{0}, g_{1}, g_{2}, g_{3}, g_{4}) = [\frac{1 − x}{1 − y}] − [\frac{1 − x^{−1}}{1 − y^{−1}} ] + [\frac{y}{x}] − [y] + [x],

which is the 5-term relation which defines P(ℂ) and therefore it is zero. Hence, τ represents a class in H_{EM}^{3}(SL_{2}(ℂ), P(ℂ)).

Evaluating the class τ of Lemma 7.1 on homology we get a homomorphism

\begin{matrix}H_{3}^{EM}(SL_{2}(ℂ), ℤ) ⟶^{τ} P(ℂ) \\ (g_{0}, g_{1}, g_{2}, g_{3})_{SL_{2}(ℂ)} ↦ [[g_{0} · ∞ : g_{1} · ∞ : g_{2} · ∞ : g_{3} · ∞]], \end{matrix}  (10)

also denoted by τ.

Consider the map k: BSL_{2}(ℂ) → J(S^{2})_{SL_{2}(ℂ)} of (3) which is the map induced by the SL_{2}(ℂ)-map ℓ: ESL_{2}(ℂ) → J(S^{2}) of Corollary 3.5 on the sets of SL_{2}(ℂ)- orbits.

Proposition 7.2. The homomorphism k_{∗}: H_{3}(BSL_{2}(ℂ), ℤ) → P(ℂ) induced by k in homology is the homomorphism τ in (10).

Proof. From (2) we have that the map k is given by (the geometric realisation of) the simplicial map

(g_{0}, …, g_{n})_{SL_{2}(ℂ)} → (g_{0} · ∞, …, g_{n} · ∞)_{SL_{2}(ℂ)},

which applied to 3-simplices and composed with the isomorphism (8) is precisely the map τ (10).

The Bloch–Wigner exact sequence [8, Appendix A] is given by

0 → ℚ/ℤ → H_{3}^{EM}(SL_{2}(ℂ), ℤ) ⟶^{τ} P(ℂ) ⟶^{λ} ∧_{ℤ}^{2}(ℂ^{×}) ⟶^{sym} K_{2}(ℂ) → 0,  (11)

where ℚ/ℤ = H_{3}(μ_{ℂ}; ℤ) with μ_{ℂ} ⊂ ℂ^{×} the group of roots of unity. The first map is induced by the inclusion of μ_{ℂ} into the diagonal of SL_{2}(ℂ). The homomorphism τ is the one given in (10). The homomorphism λ was defined in (6) and for u, v ∈ ℂ^{×}, sym(u ∧ v) = {u, v} ∈ K_{2}(ℂ) denotes the K_{2}-symbol.

8. The Bloch invariant lies in B(ℂ)

In the present section we prove that α(M) = β(M) ∈ B(ℂ) by factoring the canonical homomorphism ψ_{∗}: H_{3}(M_{+}; ℤ) → P(ℂ) through the homomorphism τ: H_{3}^{EM}(SL_{2}(ℂ), ℤ) → P(ℂ) of (11). We do this constructing a map R: M_{+} → BSL_{2}(ℂ)^{+} such that ψ ≃ h ∘ R, where h: BSL_{2}(ℂ)^{+} → J(S^{2})_{SL_{2}(ℂ)} is the map induced by the canonical map k: BSL_{2}(ℂ) → J(S^{2})_{SL_{2}(ℂ)} by the plus-construction. Therefore ψ_{∗} = τ ∘ R_{∗}.

8.1. Construction of the map R

Let G be a group and let H be a perfect normal subgroup. Consider Quillen’s plus construction relative to H. Let F be the homotopy fibre of

F → BG → BG^{+}  (12)

then, F has the following properties (see [16, (2.1)], [10])

1. F is connected and π_{1}(F) is a universal central extension of H.

2. F is acyclic, i.e., H̃_{•}(F) = 0.

3. π_{i}(F) = π_{i+1}(BG^{+}) for i ⩾ 2.

4. The sequence (12) is both a fibration sequence and a cofibration sequence (see [10, Thm. (2.5)], [4, Thm. (7.7)]).

Let M be a non-compact orientable complete hyperbolic 3-manifold of finite volume endowed with a spin structure. By Theorem 2.1 M is homotopically equivalent to M_{0}, and therefore, the representation ρ: Γ → SL_{2}(ℂ) corresponds to a map

f = Bρ: M_{0} → BSL_{2}(ℂ).

Recall that SL_{2}(ℂ) is the group of elementary matrices E_{2}(ℂ) of GL_{2}(ℂ). Since ℂ is a field, we have that SL_{2}(ℂ) = E_{2}(ℂ) is a perfect group. Applying the plusconstruction to the space BSL_{2}(ℂ) we get the (co-)fibration (12) for G = SL_{2}(ℂ)

F ⟶^{j} BSL_{2}(ℂ) ⟶^{p} BSL_{2}(ℂ)^{+}.  (13)

Let ∂M_{0} be the boundary of M_{0}. Note that M_{0}/∂M_{0} is homeomorphic to the one-point compactification M_{+} of M. Consider the following diagram where ι: ∂M_{0} → M_{0} is the inclusion and q: M_{0} → M_{+} is given by collapsing ∂M_{0} to a point as in (9).

\xymatrix{∂M_{0} \ar[r]^{ι} \ar@{.>}[d]_{g} & M_{0} \ar[r]^{q} \ar[d]^{f} & M_{+} \ar@{.>}[d]^{R} \\ F \ar[r]^{j} & BSL_{2}(ℂ) \ar[r]^{p} & BSL_{2}(ℂ)^{+}.}  (14)

Our aim is to extend the map p ∘ f to a map R: M_{+} → BSL_{2}(ℂ)^{+}, this is possible if and only if the map p ∘ f ∘ ι is nullhomotopic and every extension R corresponds to a nullhomotopy of p ∘ f ∘ ι. The homotopy class of the extension R only depends on f and the nullhomotopy of p ∘ f ∘ ι [11, Lemma 11.2’].

Lemma 8.1. There is a one-to-one correspondence between nullhomotopies H_{t}: p∘ f ∘ ι ∼ ∗ and pairs (g, H̃_{t}) with g: ∂M_{0} → F and H̃_{t} a homotopy H̃_{t}: f ∘ ι ∼ j ∘ g.

Proof. Let H_{t} be a nullhomotopy of p ∘ f ∘ ι, since p is a fibration we can lift H_{t} to a homotopy H̃_{t} such that H̃_{0} = f ∘ ι and p ∘ H̃_{t} = H_{t} for all t ∈ [0, 1]. Since H_{1} is the constant map, the image of H̃_{1} is contained in F. Let g = H̃_{1}, hence (g, H̃_{t}) is the pair corresponding to the nullhomotopy H_{t}.

Let (g, H̃_{t}) with g: ∂M_{0} → F and H̃_{t}: f ∘ ι ∼ j ∘ g. Since p is a fibration and j: F → BSL_{2}(ℂ) is the inclusion of its fibre, p ∘ H̃_{t} gives the desired nullhomotopy of p ∘ f ∘ ι.

By Lemma 8.1 to define an extension R is enough to give a map g: ∂M_{0} → F such that the left square of (14) commutes up to homotopy and to choose a homotopy H̃_{t}: f ∘ ι ∼ j ∘ g.

The boundary ∂M_{0} consists of a disjoint union of tori, we construct the map g: ∂M_{0} → F on each torus component separately. The restriction of f ∘ ι to a component T_{i}^{2} of the boundary gives a map

f_{i} = f ∘ ι|_{T_{i}^{2}} : S^{1} × S^{1} → BSL_{2}(ℂ).

This map induces in homotopy the homomorphism

ℤ × ℤ → SL_{2}(ℂ)
(m, n) ↦ A^{m}B^{n}

for some commuting matrices A, B ∈ SL_{2}(ℂ), which are the generators of the corresponding peripheral torus subgroup Γ_{i} of Γ. The matrices A and B corresponds to the homotopy classes of the loops given by the restrictions f_{i}|_{S^{1}×{∗}}: S^{1} ×{∗} → BSL_{2}(ℂ) and f_{i}|_{{∗}×S^{1}} : {∗} × S^{1} → BSL_{2}(ℂ).

We need to find maps g_{i}: T_{i}^{2} → F which make the following diagram commute up to homotopy

\xymatrix{T_{i}^{2} \ar@{.>}[r]^{g_{i}} \ar[d]_{ι} & F \ar[d]^{j} \\ M_{0} \ar[r]^{f} & BSL_{2}(ℂ).}  (15)

Since SL_{2}(ℂ) is a perfect group it has a universal central extension given by

1 → H_{2}(SL_{2}(ℂ)) → St_{2}(ℂ) ⟶^{φ} SL_{2}(ℂ) → 1,  (16)

where St_{2}(ℂ) is the Steinberg group. This universal central extension is induced by the homotopy exact sequence of the fibration (13) and we have that π_{1}(F) = St_{2}(ℂ) (property 1 of (12)).

Take loops a, b: S^{1} → F such that φ(a) = A and φ(b) = B. Then we can form the map

a ∨ b: S^{1} ∨ S^{1} → F.  (17)

The obstruction to extend this map to a map S^{1}×S^{1} → F is given by the Whitehead product [a, b] of a and b given by the commutator. Thus, we need to prove that [a, b] vanish for any lifts a, b ∈ St(ℂ) of A and B respectively, when A and B are the generators of a peripheral torus subgroup of Γ. For that, we need to recall a construction of elements in K_{2}(ℂ) = H_{2}(SL_{2}(ℂ)) given in [12, §8].

Consider the universal central extension given in (16). Given two matrices A, B ∈ SL_{2}(ℂ) which commute, choose representatives a, b ∈ St_{2}(ℂ) such that φ(a) = A and φ(b) = B. The commutator [a, b] = aba^{−1}b^{−1} is then an element of K_{2}(ℂ). We shall use Milnor’s notation

A ⋆ B = aba^{−1}b^{−1} ∈ K_{2}(ℂ)

for this commutator. The element A ⋆ B does not depend on the choice of representatives a and b.

As a direct consequence of identities satisfied by commutators the product ⋆ satisfies the following properties:

• Bimultiplicativity

(A_{1}A_{2}) ⋆ B = (A_{1} ⋆ B)(A_{2} ⋆ B)  (18)

• Invariance under inner automorphisms of SL_{2}(R)

(P AP^{−1}) ⋆ (P BP^{−1}) = A ⋆ B.  (19)

By Remark 2.2 the matrices A, B ∈ SL_{2}(ℂ) represent parabolic isometries of H^{3} and they are conjugate to matrices of the form

Ā = (\begin{matrix}λ & δ \\ 0 & λ \end{matrix}),  B̄ = (\begin{matrix}μ & ε \\ 0 & μ \end{matrix})

with λ = ±1, μ = ±1, and δ ≠ 0, ε ≠ 0. By property (19), to prove that A ⋆ B = 1 is enough to prove that Ā ⋆ B̄ = 1 for all the possible values of λ and μ.

Proposition 8.2. Let Ā and B̄ as above. Then Ā ⋆ B̄ = 1.

Proof. By the bimultiplicativity (18) of the ⋆-product we have that

Ā^{2} ⋆ B̄^{2} = (Ā ⋆ B̄^{2})(Ā ⋆ B̄^{2}) = (Ā ⋆ B̄)^{4}.

By a result of Bass and Tate [3, Prop. 1.2] K_{2}(ℂ) is a uniquely divisible abelian group, i.e., a vector space over ℚ. Therefore, if we prove that (Ā ⋆ B̄)^{4} = 1, it will imply that Ā ⋆ B̄ = 1. We have that

Ā^{2} = (\begin{matrix}1 & 2λδ \\ 0 & 1 \end{matrix}) = e_{12}^{2λ}  B̄^{2} = (\begin{matrix}1 & 2με \\ 0 & 1 \end{matrix}) = e_{12}^{2με}

where e_{ij}^{a} denotes the elementary matrix with its ij entry equal to a. Let x_{ij}^{a} be the corresponding generator of the Steinberg group, thus,

Ā^{2} ⋆ B̄^{2} = x_{12}^{2λδ}x_{12}^{2με}x_{12}^{−2λδ}x_{12}^{−2με} = x_{12}^{2(λδ+με)}x_{12}^{−2(λδ+με)} = 1.

We thank Michel Matthey for suggesting to use that K_{2}(ℂ) is uniquely divisible to simplify the computations. This proposition can also be proved without using this fact but one has to analyse the four cases separately (see [5]).

Therefore, by Proposition 8.2 the map (17) extends to a map S^{1} ×S^{1} → F which makes the diagram (15) commute up to homotopy since S^{1}×S^{1} is a classifying space B(ℤ×ℤ) and the maps f_{i} and j∘g_{i} induce the same homomorphism on fundamental groups.

Taking all the maps g_{i}: T_{i}^{2} → F together we get the desired map g: ∂M_{0} → F. Choose a homotopy H̃_{t}: f ∘ ι ∼ j ∘ g, since ι: ∂M_{0} → M_{0} is a cofibration we can lift H̃_{t} to a homotopy H̄_{t} such that H̄_{0} = f and H̄_{t} ∘ ι = H̃_{t} for all t ∈ [0, 1]. Let f_{1} = H̄_{1}, then we have the following strictly commutative diagram

\xymatrix{∂M_{0} \ar[r]^{ι} \ar[d]_{g} & M_{0} \ar[r]^{q} \ar[d]^{f_{1}} & M_{+} \ar[d]^{R_{H}} \\ F \ar[r]^{j} & BSL_{2}(ℂ) \ar[r]^{p} & BSL_{2}(ℂ)^{+},}  (20)

where

R_{H}: M_{+} → BSL_{2}(ℂ)^{+}  (21)

is the map defined by the nullhomotopy H corresponding to (g, H̃_{t}) by Lemma 8.1.

8.2. Factoring ψ through R

Our next task is to prove that τ ∘ R_{H}_{∗} = ψ_{∗}. For this, we need the following lemmata.

Lemma 8.3. π_{1}(J(S^{2})_{SL_{2}(ℂ)}) = 0.

Proof. It is a direct application of a result by M. A. Armstrong [1, Thm. 3], [2, Thm. 6.18] which says that π_{1}(J(S^{2})_{SL_{2}(ℂ)}) = SL_{2}(ℂ)/F where F is the normal subgroup of SL_{2}(ℂ) generated by those elements which leave fixed at least one point of J(S^{2}). Obviously the elements of SL_{2}(ℂ) which are matrices of the form ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ), with a ∈ ℂ and of the form ( \begin{matrix}1 & 0 \\ b & 1 \end{matrix} ), with b ∈ ℂ are in F, they fix (∞, …, ∞) and (0, …, 0) respectively. But these are elementary matrices and the whole SL_{2}(ℂ) is generated by elementary matrices (since SL_{2}(ℂ) = E_{2}(ℂ)), therefore F = SL_{2}(ℂ).

Lemma 8.4. π_{2}(J(S^{2})_{SL_{2}(ℂ)}) = 0, π_{3}(J(S^{2})_{SL_{2}(ℂ)}) = P(ℂ).

Proof. By the proof of [17, Lemma 2.2] we have that H_{1}(J(S^{2})_{SL_{2}(ℂ)}) = 0, H_{2}(J(S^{2})_{SL_{2}(ℂ)}) = 0 and H_{3}(J(S^{2})_{SL_{2}(ℂ)}; ℤ) ≅ P(ℂ). By Lemma 8.3 J(S^{2})_{SL_{2}(ℂ)} is simply connected and the lemma follows by Hurewicz Theorem.

Remember that a map between path connected spaces is acyclic if its homotopy fibre is an acyclic space. By the properties of (12) the homotopy fibration

F ⟶^{j} BSL_{2}(ℂ) ⟶^{p} BSL_{2}(ℂ)^{+}

is acyclic since F is acyclic. We have also that j is a cofibration and p is the cofibre of j.

By Corollary 3.5 there is a SL_{2}(ℂ)-map ℓ: ESL_{2}(ℂ) → J(S^{2}) unique up to SL_{2}(ℂ)-homotopy which induces a map k: BSL_{2}(ℂ) → J(S^{2})_{SL_{2}(ℂ)} between the SL_{2}(ℂ)-orbit spaces. Let π_{1}(p) and π_{1}(k) the homomorphisms on fundamental groups induced by p and k, by Proposition 8.3 we have that ker π_{1}(p) = ker π_{1}(k) = SL_{2}(ℂ) and by [10, Prop. (3.1)], [4, (5.2)] there exist a map h: BSL_{2}(ℂ)^{+} → J(S^{2})_{SL_{2}(ℂ)} unique up to homotopy which makes the following diagram commute

\xymatrix{ &  & J(S^{2})_{SL_{2}(ℂ)} \\ F \ar[r]^{j} & BSL_{2}(ℂ) \ar[ur]^{k} \ar[r]^{p} & BSL_{2}(ℂ)^{+}. \ar@{.>}[u]_{h}}  (22)

Remark 8.5. The map h is also given by the functoriality of the plus-construction applied to the map k, that is h = k^{+}, since J(S^{2})_{SL_{2}(ℂ)}^{+} = J(S^{2})_{SL_{2}(ℂ)}.

Proposition 8.6. The following diagram commutes (up to homotopy)

\xymatrix{M_{+} \ar[r]^{R_{H}} \ar[dr]_{ψ} & BSL_{2}(ℂ)^{+} \ar[d]^{h} \\  & J(S^{2})_{SL_{2}(ℂ)}}

where R_{H} is the map given in (21), ψ is the canonical map (9) and h is the map given in (22).

Proof. Firstly, we modify diagram (9) to make it compatible with diagram (20). Consider the homotopy H̄_{t}: f ∼ f_{1} that we use in (20) and a homotopy F̄_{t}: k ∼ h∘p. Define a homotopy F_{t}: k∘f ∼ h∘p∘f_{1} by F̄_{t}∘H̄_{t}. Since q: M_{0} → M_{+} is a cofibration we can extend F_{t} to a homotopy F̃_{t} such that F̃_{1} = ψ and F̃_{t}∘q = F_{t} for all t ∈ [0, 1]. Let ψ_{1} = F̃_{1}, then we get the following strictly commutative diagram

\xymatrix{M_{0} \ar[r]^{q} \ar[d]^{f_{1}} & M_{+} \ar[d]^{ψ_{1}} \\ BSL_{2}(ℂ) \ar[r]^{h∘p} & J(S^{2})_{SL_{2}(ℂ)}.}  (23)

Putting together diagrams (20) and (23) we get

\xymatrix{∂M_{0} \ar[r]^{ι} \ar[d]_{g} & M_{0} \ar[r]^{q} \ar[d]^{f_{1}} & M_{+} \ar[d]^{R_{H}} \ar[dr]^{ψ_{1}} \\ F \ar[r]^{j} & BSL_{2}(ℂ) \ar[r]^{p} & _{BSL_{2}(ℂ)_{+}} \ar[r]^{h} & J(S^{2})_{SL_{2}(ℂ)}}  (24)

From diagram (20) we have that

f_{1} ∘ ι = j ∘ g,  (25)
R_{H} ∘ q = p ∘ f_{1}.  (26)

From diagram (23) we have

ψ_{1} ∘ q = h ∘ p ∘ f_{1}.  (27)

We want to prove that the triangle in (24) commutes (up to homotopy), that is ψ_{1} ≃ h ∘ R_{H}. Substituting (26) in (27) we obtain

ψ_{1} ∘ q = h ∘ R_{H} ∘ q.

To simplify notation, put v_{o} = ψ_{1}, v_{1} = h ∘ R_{H}, G = SL_{2}(ℂ) and J = J(S^{2})_{SL_{2}(ℂ)}. Thus we have two maps v_{i}: M_{+} → J, i = 0, 1, such that v_{0} ∘q = v_{1} ∘q. We want to prove that v_{o} ≃ v_{1}.

Let F: BG × I → J given by F (z, t) = h ∘ p for all t ∈ [0, 1] then H = F ∘ (f_{1} × id): M_{0} × I → J is the constant homotopy between v_{o} ∘ q and v_{1} ∘ q since

H(x, 0) = F (f(x), 0) = h ∘ p ∘ f_{1}(x) = ψ_{1} ∘ q(x) = v_{0} ∘ q(x)
H(x, 1) = F (f(x), 1) = h ∘ p ∘ f_{1}(x) = h ∘ R_{H} ∘ q(x) = v_{1} ∘ q(x).

Such homotopy is equivalent to a map H̃: M_{0} → J^{I}, where J^{I} denotes the continuous (free) maps from I to J, given by H̃(x)(t) = H(x, t). Hence we have the following commutative diagram

\xymatrix{∂M_{0} \ar[r]^{ι} \ar[d] & M_{0} \ar[r]^{H̃} \ar[d]^{q} & J^{I} \ar[d]^{e} \\ ∗ \ar[r] & M_{+} \ar[r]^{v_{0}×v_{1}} \ar@{.>}[ur] & J × J,}

where e is evaluation on {0, 1} ⊂ I. To find a homotopy between v_{0} and v_{1} is equivalent to give a lifting of v into J^{I}, this is possible if the pair of maps (H̃ ∘ ι, ∗) is nullhomotopic [11, Lemma 11.1’], where ∗ denotes the constant map to the basepoint. The map e is a fibration with fibre ΩJ, the loop space of J, and since e∘H̃ ∘ι is the constant map, the map H̃ ∘ ι has its image in ΩJ. The condition that the pair of maps (H̃ ∘ ι, ∗) is nullhomotopic is equivalent to H̃ ∘ ι being nullhomotopic in ΩJ.

Putting F̃: BG → J^{I} as F̃(z)(t) = F (z, t) we have the following commutative diagram

\xymatrix{∂M_{0} \ar[r]^{ι} \ar[d]_{g} & M_{0} \ar[r]^{H̃} \ar[d]_{f_{1}} & J^{I} \ar[d]^{e} \\ F \ar[r]_{j} & BG \ar[ur]^{F̃} \ar[r] & J × J.}

Since e ∘ F̃ ∘ j is also the constant map, the map F̃ ∘ j also has its image in ΩJ, therefore we have the following commutative diagram

\xymatrix{∂M_{0} \ar[r]^{H̃∘ι} \ar[d]_{g} & ΩJ \\ F \ar[ur]_{F̃∘j}}

By lemma 8.4 π_{1}(ΩJ) = π_{2}(J) = 0. Hence the map induced on fundamental groups π_{1}(F̃ ∘ j) is zero and since F is acyclic by [10, Cor. (3.2)] F̃ ∘ j is nullhomotopic in ΩJ. Therefore H̃ ∘ ι is nullhomotopic in ΩJ which implies that v_{0} and v_{1} are homotopic.

Now it is a corollary the fact that the invariant α(M) (and so the Bloch invariant) lies in the Bloch group, compare with [13, §1, §5].

Corollary 8.7. α(M) ∈ B(ℂ).

Proof. The diagram of Proposition 8.6 induces the following commutative diagram in homology

\xymatrix{H_{3}(M_{+}; ℤ) \ar[r]^{^{R_{H}^{∗}}} \ar[dr]_{_{ψ_{1}_{∗}}} & H_{3}(BSL_{2}(ℂ)^{+}; ℤ) \ar[d]^{h_{∗}=τ} \\  & H_{3}(J(S^{2})_{SL_{2}(ℂ)}; ℤ) = P(ℂ)}  (28)

By Remark 8.5 we have that h = k^{+}, hence h and k induce the same homomorphism in homology. By Proposition 7.2 we have that k_{∗} = τ where τ is the map (10). Hence h_{∗} = τ and

α(M) = ψ_{1}_{∗}([M_{+}]) = τ ∘ R_{H}_{∗}([M̂]) ∈ B(ℂ),

since by the Bloch-Wigner exact sequence (11), the image of τ is the Bloch group B(ℂ).

Corollary 8.8. Let M be a non-compact orientable complete hyperbolic 3-manifold of finite volume which has an ideal triangulation by ideal tetrahedra △_{1}, …, △_{n}. Let z_{i} ∈ ℂ be the parameter of △_{i} for each i. Then

2 ∑_{i=1}^{n} z_{i} ∧ (1 − z_{i}) = 0 ∈ ∧_{ℤ}^{2}(ℂ^{×}).  (29)

Proof. By Corollary 8.7 and the Bloch-Wigner exact sequence (11) β(M) is in the kernel of the homomorphism λ given in (6) and the left-hand-side of (29) is precisely λ(β(M)).

Remark 8.9. By a result of Sah [15, Prop. 2.9 (a), Thm. 4.1 (b)] we have that K_{3}(ℂ) ≅ H_{3}(SL(ℂ); ℤ) ≅ H_{3}(SL_{2}(ℂ); ℤ)⊕K_{3}^{M}(ℂ) and K_{3}^{ind}(ℂ) ≅ H_{3}(SL_{2}(ℂ); ℤ). Here K_{3}^{M}(ℂ) is Milnor’s K-theory and K_{3}^{ind}(ℂ) is the idecomposable part of K_{3}(ℂ). We can construct an element in K_{3}^{ind}(ℂ) mapping the fundamental class [M_{+}] to H_{3}(SL_{2}(ℂ); ℤ) with the homomorphism R_{H}_{∗} in (28) and we have that such element is mapped to the Bloch invariant under τ. Composing R_{H} with the map s: BSL_{2}(ℂ)^{+} → BSL(ℂ)^{+} induced by the inclusion, we get an element in K_{3}(ℂ) and it can be proved that such an element is also mapped to the Bloch invariant under a homomorphism H_{3}(SL(ℂ); ℤ) → B(ℂ) given by τ.

The present paper is the result of an attempt to lift the Bloch invariant to K_{3}(ℂ) in this way. Unfortunately the map R_{H} depends on a specific choice of nullhomotopy of p∘f ∘ι in (14), and there is a ℚ/ℤ indeterminacy due to such choice. To get rid of the indeterminacy we should quotient H_{3}(SL_{2}(ℂ); ℤ) by ℚ/ℤ but we get back the Bloch group B(ℂ) by the Bloch-Wigner exact sequence (11). Thus, to lift the Bloch invariant to K_{3}(ℂ) one should look for a canonical way to construct an element in K_{3}(ℂ) from M.

Acknowledgments: The first author would like to thank Michel Matthey for many fruitful discussions.

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2003/n1a14/v5n1a14.(dvi,ps,pdf)

José Luis Cisneros-Molina jlcm@matcuer.unam.mx

Instituto de Matemáticas UNAM, Unidad Cuernavaca Av. Universidad s/n, Col. Lomas de Chamilpa Cuernavaca, Morelos, 62210 MEXICO

John D. S. Jones jdsj@maths.warwick.ac.uk

Mathematics Institute University of Warwick Coventry CV4 7AL England