1 Introduction↩︎

More than \(50\) years ago, in a celebrated paper [1] Thurston constructed plenty of codimension one foliations on \(S^3\) by making use of classical hyperbolic geometry governed by the group \(\mathit{\mathit{PSL}}(2,\mathbb{R})\) as well as the Reeb foliation on the solid torus. By this he showed that the Godbillon-Vey class is a continuously varying characteristic class and proved a ground breaking result that the homomorphism \[gv: \pi_3(B\overline{\Gamma}^\infty_1)\rightarrow \mathbb{R} \label{eq:GVs}\tag{1}\] induced by the Godbillon-Vey class is surjective. Here \(B\overline{\Gamma}^\infty_1\) denotes Haefliger’s classifying space for his \(\overline{\Gamma}_1^\infty\) structures.

In the real analytic category, Thurston also proved that the homomorphism \[gv: H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})\rightarrow \mathbb{R} \label{eq:GV}\tag{2}\] is surjective. The method for this was to glue his first examples as above with similar ones, associated to finite covering groups \(\mathit{\mathit{PSL}}(2,\mathbb{R})^{(n)}\;(n=2,3,\ldots)\) of \(\mathit{\mathit{PSL}}(2,\mathbb{R})\), along the common boundaries which are various linear foliations on the torus, thus avoiding the use of the Reeb component which is not real analytic. In his paper [1], Thurston did not describe details of his proof. However, fortunately Brooks wrote a proof in [2].

In [3], Tsuboi generalized Thurston’s former construction as follows. He identified building blocks of Thurston’s construction as elements of the second homology group of \(\mathit{\mathit{PSL}}(2,\mathbb{R})\), considered as a discrete group, relative to its rotation subgroup \(\mathit{PSO}(2)\). By a very nice argument, he proved that \[H_2(\mathit{\mathit{PSL}}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\cong \mathbb{R}. \label{eq:rel}\tag{3}\] By utilizing this he was able to prove that the above surjective homomorphism splits so that \(\pi_3(B\overline{\Gamma}^\infty_1)\cong \mathbb{R}\oplus \mathrm{Ker}\,gv\). Tsuboi further constructed many more \(3\)-cycles of \(B\overline{\Gamma}^\infty_1\) by considering all the finite covering groups \(\mathit{\mathit{PSL}}(2,\mathbb{R})^{(n)}\) of \(\mathit{\mathit{PSL}}(2,\mathbb{R})\) and proving similar isomorphisms as in 3 for all \(n\). He then proved the remarkable result that the subspace of \(H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})\) generated by all of his cycles obtained in this way remains the same as his direct summand so that the Godbillon-Vey invariant is the complete invariant for such foliations.

In this paper, we enhance the above constructions of Thurston and Tsuboi to show that Thurston’s surjective homomorphism 2 also splits. In our construction, instead of Reeb component to bound linear foliations on the torus, we slightly enlarge the group of parallel translations of the real line \(\mathbb{R}\), which is canonically isomorphic to \(\mathbb{R}\) itself, by adding just two homothety mappings of it to obtain a group isomorphic to \(\mathbb{R}\rtimes \mathbb{Z}^2\) as a subgroup of \(\pi_1(B\overline{\Gamma}^\omega_1)\). Then we show that any linear foliation on the torus, which can be considered as an element of \(H_2(\mathbb{R};\mathbb{Z})\), is the boundary of a \(3\)-chain of the above group \(\mathbb{R}\rtimes \mathbb{Z}^2\) which is uniquely defined up to homology. Our construction of these \(3\)-chains is an explicit and systematic one and the Godbillon-Vey class vanishes identically on them at the chain level. By combining these properties, we obtain the desired direct summand \(\mathbb{R}\subset H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})\). Furthermore we prove that all of Thurston’s analytic cycles of \(H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})\) mentioned above are contained in this direct summand.

Convention: In this paper, for any group \(G\), including \(\mathit{\mathit{PSL}}(2,\mathbb{R})\) and \(\mathbb{R}\), \(H_*(G;\mathbb{Z})\) will denote the integral homology group of \(G\) considered as a discrete group.

2 Statement of the main results↩︎

We give two proofs of this theorem. One proof described in \(\S 5\) is given by constructing a homomorphism \[H_2(\mathit{\mathit{PSL}}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\overset{\mathit{Tsuboi}}{\cong}\mathbb{R}\rightarrow H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\] such that its composition with the homomorphism \(gv\) above is an isomorphism. The other proof described in \(\S 6\) is closer to Thurston’s original analytic construction. More precisely, we show that the image of the natural homomorphism \[H_2(\mathit{\mathit{PSL}}(2,\mathbb{R})*_{\mathit{SO}(2)} \mathit{SL}(2,\mathbb{R});\mathbb{Z}) \rightarrow H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\] considered by him is already isomorphic to \(\mathbb{R}\) such that the restriction of \(gv\) on it gives an isomorphism.

Now in order to treat Thurston’s construction systematically, Tsuboi defined in [3] a group \(G_T\) to be the free product of all the \(n\)-fold covering groups \(\mathit{\mathit{PSL}}(2,\mathbb{R})^{(n)}\;(n=1,2,\cdots)\) of \(\mathit{\mathit{PSL}}(2,\mathbb{R})\) amalgamated along their maximal compact subgroups all of which are canonically identified with the rotation group \(\mathit{SO}(2)\). There is a natural homomorphism \[\rho_T: G_T\rightarrow \mathrm{Diff}_+^\infty S^1.\] By using this, he constructed a homomorphism \[H_2(G_T,\mathit{SO}(2);\mathbb{Z})\rightarrow H_3(B\overline{\Gamma}_1^\infty;\mathbb{Z}) \label{eq:th}\tag{4}\] and proved that its image is equal to his direct summand (Theorem 1.2 of the above cited paper). The following is a real analytic enhancement of this result.

As a corollary to Theorem 1, we show the existence of a huge amount of non-trivial cohomology classes in \(H^4(B\overline{\Gamma}^\omega_1;\mathbb{Z})\). These classes serve as a new kind of characteristic class of foliations in the following sense. They vanish on any space with \(\overline{\Gamma}^\omega_1\) structure if its third homology group is finitely generated. On the other hand, they detect uniquely divisible (i.e. \(\mathbb{Q}\) vector space) subgroup in the third integral homology group of the space on which the Godbillon-Vey class evaluates non-trivially. The same is true for the smooth case as well.

3 Strategy↩︎

Our strategy is simple. As already mentioned in \(\S 1\), in the construction of Thurston he used the Reeb component to bound linear foliations on the torus. Tsuboi generalized this construction to yield many more examples. However these constructions are not real analytic. In order to obtain real analytic cycles, we add just two homothety mappings of \(\mathbb{R}\) to the group of parallel translations on it and then prove that any linear foliation on the torus bounds a \(3\)-chain in this slightly extended group.

To be more precise, first let us fix our notations. Let \(\mathrm{Diff}_+^\infty S^1\) denote the group of orientation preserving \(C^\infty\) diffeomorphisms of \(S^1\) and let \(\widetilde{\mathrm{Diff}}_+^\infty S^1\) be its universal cover. Let \(\mathrm{Diff}_+^\omega S^1, \widetilde{\mathrm{Diff}}_+^\omega S^1\) be the subgroups of \(\mathrm{Diff}_+^\infty S^1, \widetilde{\mathrm{Diff}}_+^\infty S^1\) consisting of real analytic elements. Now consider the central extension \[0\rightarrow \mathbb{Z}\rightarrow \widetilde{\mathrm{Diff}}_+^\omega S^1 \rightarrow \mathrm{Diff}_+^\omega S^1\rightarrow 1.\] It has the following sub central extension \[0\rightarrow \mathbb{Z}\rightarrow \mathbb{R} \rightarrow \mathit{SO}(2)\rightarrow 1\] where \(\mathit{SO}(2)\) is the subgroup of \(\mathrm{Diff}_+^\omega S^1\) consisting of rotations. On the other hand, there is a natural identification \[\widetilde{\mathrm{Diff}}_+^\omega S^1= \{f\in \mathrm{Diff}_+^\omega \mathbb{R}; Tf=fT\}\quad \text{where T(x)=x+1\;(x\in \mathbb{R})}\] and the subgroup \(\mathbb{R}\subset \widetilde{\mathrm{Diff}}_+^\omega S^1\) can be written as \[\mathbb{R}=\{T_s; s\in \mathbb{R}\}\quad \text{where T_s(x)=x+s\;(x\in \mathbb{R})}.\] Henceforth we identify \(\widetilde{\mathrm{Diff}}_+^\omega S^1\) as a subgroup of \(\mathrm{Diff}_+^\omega \mathbb{R}\).

Now consider the subgroup \[\mathrm{Diff}_+^\omega \mathbb{R}\supset \mathbb{R}^+=\{S_r;r\in \mathbb{R}^+\}\quad \text{where S_r(x)=rx\;(x\in \mathbb{R})}.\] Thus we have two subgroups \[\mathbb{R}=\{T_s; s\in \mathbb{R}\},\mathbb{R}^+=\{S_r; r\in \mathbb{R}^+\}\subset \mathrm{Diff}_+^\omega \mathbb{R}.\]

It follows that the semi-direct product \(\mathbb{R}\rtimes \mathbb{R}^+\) can be considered as a subgroup of \(\mathrm{Diff}_+^\omega \mathbb{R}\) (the orientation preserving affine transformation group of \(\mathbb{R}\)). In [8], the outer endomorphisms of the group \(\widetilde{\mathrm{Diff}}_+^\infty S^1\) induced by the conjugation action of homothety transformations \(S_{1/n}\) for \(n=2,3,\ldots\), which we denoted by \(\varphi_n\), were introduced. We studied them further in our paper [9] to investigate the non-triviality question of powers of the Euler class. The present work began when we noticed that these elements may have a role here again.

For later use, we recall the explicit form of \(\varphi_n\) (as well as its inverse): \[\varphi_n(f)=S_{\frac{1}{n}}\circ f\circ S_n,\;\varphi^{-1}_n(f)=S_n\circ f\circ S_{\frac{1}{n}} \quad (f\in \mathrm{Diff}_+^\omega \mathbb{R}). \label{eq:varphi}\tag{5}\]

Let \(Z_{(2,3)}\subset\mathbb{R}^+\) be the subgroup generated by two elements \(S_2,S_3\). Clearly \(Z_{(2,3)}\cong\mathbb{Z}^2\). We consider the small subgroup \[\mathbb{R}\rtimes Z_{(2,3)}\subset \mathbb{R}\rtimes\mathbb{R}^+.\]

Now recall that Haefliger [10] proved that his classifying space \(B\overline{\Gamma}^\omega_1\) for \(\overline{\Gamma}^\omega_1\)-structures is a \(K(\pi,1)\) space. We would like to call the group \(\Gamma_H=\pi_1(B\overline{\Gamma}^\omega_1)\) the Haefliger group. \(\mathrm{Diff}_+^\omega \mathbb{R}\) is a subgroup of \(\Gamma_H\) so that we can write \[\mathbb{R}\rtimes Z_{(2,3)}\subset \mathbb{R}\rtimes\mathbb{R}^+\subset \mathrm{Diff}_+^\omega \mathbb{R}\subset \Gamma_H.\]

Now we are ready to state our strategy. Any linear foliation on the torus can be described as a \(2\)-cycle \[\sigma=1 \wedge s=[(1,s)-(s,1)]\in H_2(\mathbb{R};\mathbb{Z})\cong\wedge^2_\mathbb{Z}\mathbb{R}\] of the subgroup \(\mathbb{R}\subset \mathbb{R}\rtimes\mathbb{R}^+\subset \Gamma_H\). The crucial point of our work is to show that any such \(2\)-cycle bounds a \(3\)-chain in the above small subgroup \(\mathbb{R}\rtimes Z_{(2,3)}\subset \Gamma_H\), whose relative homology class in \(H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z})\) is uniquely determined, and furthermore we obtain explicit formula for such \(3\)-chains systematically. This gives rise to the desired direct summand \(\mathbb{R}\subset H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})=H_3(\Gamma_H;\mathbb{Z})\). One more important fact here is that the Godbillon-Vey cocycle given in [11] vanishes identically on \(\mathbb{R}\rtimes\mathbb{R}^+\). This is because of the form of the above cocycle combined with the obvious fact that the second derivative of any affine transformation \(ax+b\) vanishes identically. This implies that the Godbillon-Vey invariant does not change by our closing of the linear foliations on the torus in a similar way as in the case of Reeb components which were used in the construction of Thurston and also by Tsuboi.

4 Preliminary on inner automorphisms of groups↩︎

Let \(G\) be a discrete group and let \(BG\) be its classifying space. It is a \(K(G,1)\) space. For any element \(\gamma\in G\), let \[\varphi: G\rightarrow G\] be the inner automorphism of \(G\) defined by \(\varphi(\alpha)=\gamma\alpha\gamma^{-1}\, (\alpha\in G)\). Let \(C_*(G)\) be the chain complex of \(G\) with integral coefficients so that \(H_*(C_*(G))=H_*(G;\mathbb{Z})\). Let \[B\varphi:BG\rightarrow BG\] be the continuous map induced by \(\varphi\). As is well known, this map is homotopic to the identity map. Therefore, the chain map \[\varphi_*:C_*(G)\rightarrow C_*(G)\] induced by \(\varphi\) is chain homotopic to the identity. Namely there exists a homomorphism \[H_\gamma: C_*(G)\rightarrow C_{*+1}(G)\] such that \[\varphi_*-\mathrm{id}=\partial H_\gamma+H_\gamma\partial \label{eq:H}\tag{6}\] where \(\partial\) denotes the boundary map of \(C_*(G)\). In fact, the following formula which realizes the homotopy \(B\varphi\sim \mathrm{id.}\) simplicially, gives an explicit expression for \(H_\gamma\).

5 Proof of Theorem 1↩︎

As already mentioned in \(\S\) 1, Tsuboi [3] proved that \(H_2(\mathit{\mathit{PSL}}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\cong\mathbb{R}\). The parameter \(s\in\mathbb{R}\) in this isomorphism has the following geometric meaning. Any homology class \(\sigma\) in this group can be represented as a foliated \(S^1\)-bundle over a compact oriented surface with a boundary whose monodromy is contained in \(\mathit{\mathit{PSL}}(2,\mathbb{R})\) and restricts to a rotation on the boundary so that the restriction of this foliated \(S^1\)-bundle to the boundary is a linear foliation on the torus. Since the Euler class of this foliated \(S^1\)-bundle is trivial (because the base surface has a boundary), the monodromy group can be lifted to \(\widetilde{\mathit{\mathit{PSL}}}(2,\mathbb{R})\) whose restriction to the boundary is contained in the subgroup \(\mathbb{R}\subset \widetilde{\mathit{\mathit{PSL}}}(2,\mathbb{R})\) consisting of parallel translations of \(\mathbb{R}\). The parameter \(s\) of this cycle is the translation length of the boundary linear foliation as a foliated \(S^1\)-product over the circle. In the case when monodromy of the boundary is trivial so that the foliated \(S^1\)-bundle extends to the closed surface, the parameter \(s\) is the same as the Euler class evaluated on the fundamental class which is an integer. More concretely, for a given homology class \(\sigma\in H_2(\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\), its parameter \(s(\sigma)\in\mathbb{R}\) can be determined as follows. In the setting of group homology, \(\sigma\) is represented by \(2k\) elements \(\alpha_i,\beta_i\in \mathit{PSL}(2,\mathbb{R})\;(i=1,\ldots,k)\) such that \[[\alpha_1,\beta_1]\cdots [\alpha_k,\beta_k]= \partial(\sigma)\in H_1(\mathit{PSO}(2);\mathbb{Z})=\mathit{PSO}(2)\;(=\mathbb{R}/\mathbb{Z}).\] Now choose lifts \(\tilde{\alpha}_i,\tilde{\beta}_i\in \widetilde{\mathit{PSL}}(2,\mathbb{R})\) of the elements \(\alpha_i,\beta_i\). Then the value \[[\tilde{\alpha}_1,\tilde{\beta}_1]\cdots [\tilde{\alpha}_k,\tilde{\beta}_k]\in \mathbb{R}\subset \widetilde{\mathit{PSL}}(2,\mathbb{R})\] is well defined independent of the choices made and this is the parameter \(s(\sigma)\).

To make a real analytic relative \(3\)-cycle which bounds linear foliations on the torus, we consider the homology group of the pair \((\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R})\). Consider the long exact sequence \[\cdots \rightarrow H_3(\mathbb{R};\mathbb{Z})\rightarrow H_3(\mathbb{R}\rtimes Z_{(2,3)};\mathbb{Z})\rightarrow H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z})\overset{\partial}{\rightarrow} H_2(\mathbb{R};\mathbb{Z})\rightarrow \cdots \label{eq:les}\tag{7}\]

Now we would like to give an explicit formula for the homomorphism \[\Psi:=\partial^{-1}: H_2(\mathbb{R};\mathbb{Z})\rightarrow H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z})\] at the chain level. This is the heart of our construction of analytic cycles because it plays the role of the Reeb component in the works of Thurston and Tsuboi. We would like to call elements of the image of the above homomorphism at the chain level “real analytic caps" (see Definition 1 below).

To do this, we prepare a technical lemma. In the proof of Proposition 12 the following fact was used. For each \(s_1,s_2\in\mathbb{R}\), the cycles \(\langle 2s_1,2s_2\rangle, \langle 3s_1,3s_2\rangle\) of the abelian group \(\mathbb{R}\) are homologous to \(4\langle s_1,s_2\rangle ,\;9 \langle s_1,s_2\rangle\), respectively. In fact we have the following formula which we use in our explicit description of \(\Psi\).

By combining Proposition 12 and Proposition 14, we obtain the following.

Let \(\widetilde{\mathit{PSL}}(2,\mathbb{R})\) denote the universal covering group of \(\mathit{PSL}(2,\mathbb{R})\). We consider \(\mathit{PSL}(2,\mathbb{R})\) as a subgroup of \(\mathrm{Diff}_+^\omega S^1\) in a way that its subgroup \(\mathit{PSO}(2)\) is identified with \(\mathit{SO}(2)\subset \mathrm{Diff}_+^\omega S^1\). Also we consider \(\widetilde{\mathit{PSL}}(2,\mathbb{R})\) as a subgroup of \(\mathrm{Diff}_+^\omega \mathbb{R}\) through the embedding \(\widetilde{\mathrm{Diff}}_+^\omega S^1\subset \mathrm{Diff}_+^\omega \mathbb{R}\) mentioned in \(\S\) 3. Recall that \(\mathbb{R}\rtimes Z_{(2,3)}\subset \mathbb{R}\rtimes\mathbb{R}^+\subset \mathrm{Diff}_+^\omega \mathbb{R}\) is the subgroup generated by the translation group \(\mathbb{R}\) and two elements \(S_2,S_3\in \mathbb{R}^+\). We set \[\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}=\widetilde{\mathit{PSL}}(2,\mathbb{R}) *_{\mathbb{R}} (\mathbb{R}\rtimes Z_{(2,3)}).\] Then there is a canonical (presumably injective) homomorphism \[\rho: \widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}\rightarrow \mathrm{Diff}_+^\omega \mathbb{R}\subset \Gamma_H. \label{eq:rho}\tag{9}\]

We consider the following commutative diagram. \[\begin{CD} H_2(\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z}) @>{\mu}>{\text{Gysin map}}> H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R}),\mathbb{R};\mathbb{Z}) @. \\ @V{\partial}VV @V{\partial}VV @.\\ H_1(\mathit{PSO}(2);\mathbb{Z}) @>{\mu}>{\text{Gysin map}}> H_2(\mathbb{R};\mathbb{Z}) @>{\Psi=\partial^{-1}}>\cong> H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z}). \end{CD} \label{eq:cdg}\tag{10}\] Here the upper homomorphism \(\mu\) is a part of the relative Gysin exact sequence of the pair of the central extension \((\widetilde{\mathit{PSL}}(2,\mathbb{R}),\mathbb{R})\rightarrow (\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2))\) and the lower \(\mu\) is its image under the boundary map. In both cases, a homology class \(\sigma\) on the left hand side is represented by a foliated \(S^1\)-bundle with prescribed monodromy and its image under the homomorphism \(\mu\), which we denote by \(\tilde{\sigma}\), is represented by the associated total space which is a foliated \(S^1\)-product with canonical trivialization as a smooth \(S^1\)-bundle. Based on this, we make the following definition.

Here the expression \(\tilde{\sigma}-\Psi(\partial\tilde{\sigma})\) is a somewhat simplified notation and its precise meaning will be given in the proof of the following Proposition.

To proceed to the proof of Theorem 1, we replace \(\mathit{PSL}(2,\mathbb{R})\) with its double cover \(\mathit{SL}(2,\mathbb{R})\) in Definition 2. Here we recall that Tsuboi [3] proved that \[H_2(\mathit{PSL}(2,\mathbb{R})^{(n)},\mathit{SO}(2);\mathbb{Z})\cong\mathbb{R}\quad (n=1,2,\cdots)\] where the meaning of the parameter \(s\in\mathbb{R}\) for \(n>1\) is similar to the case \(n=1\). Namely, in the description of the meaning of \(s\) for the case \(n=1\) given in the beginning of this section, we just replace \(\mathit{PSL}(2,\mathbb{R})\) by \(\mathit{PSL}(2,\mathbb{R})^{(n)}\). Observe that the center of \(\widetilde{\mathit{PSL}}(2,\mathbb{R})^{(n)}\) is the same as the center \(\mathbb{Z}\) of \(\widetilde{\mathit{PSL}}(2,\mathbb{R})\) for all \(n\). Also we have the following commutative diagram \[\begin{CD} \widetilde{\mathit{PSL}}(2,\mathbb{R}) @>{\subset}>> \widetilde{\mathrm{Diff}}_+^\omega S^1 @>{\subset}>>\Gamma_H \\ @V{\varphi_2}V{\cong}V @V{\varphi_2}V{}V @V{\varphi_2}V{}V\\ \widetilde{\mathit{SL}}(2,\mathbb{R}) @>{\subset}>> \widetilde{\mathrm{Diff}}_+^\omega S^1 @>{\subset}>> \Gamma_H. \end{CD}\] Namely the universal cover \(\widetilde{\mathit{SL}}(2,\mathbb{R})\) of \(\mathit{SL}(2,\mathbb{R})\), considered as a subgroup of \(\widetilde{\mathrm{Diff}}_+^\omega S^1\), is equal to the image under \(\varphi_2\) of \(\widetilde{\mathit{PSL}}(2,\mathbb{R})\) (cf. [8]). Since the action of \(\varphi_2\) is equal to the conjugation by the element \(S_2^{-1}\), we can conclude that \(\widetilde{\mathit{SL}}(2,\mathbb{R})\) is contained in the group \(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}\). Thus we obtain the following definition.

Proof of Theorem 1
In [3], Tsuboi defined two homomorphisms \[\begin{align} &t_1: H_2(\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\cong \mathbb{R}\rightarrow H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})\\ &t_2: H_2(\mathit{SL}(2,\mathbb{R}),\mathit{SO}(2);\mathbb{Z})\cong \mathbb{R}\rightarrow H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z}) \end{align}\] and proved the following facts. \[\begin{align} (i)&\, \text{\mathrm{Im}\, t_1 is a direct summand of the target and gv is an isomorphism on it}\\ (ii)&\, \text{t_2(s)=4t_1(s) for any s\in\mathbb{R} so that \mathrm{Im}\, t_2=\mathrm{Im}\, t_1}. \label{eq:tsu} \end{align}\tag{12}\] Our task is to compare our homomorphism \(\Phi\) (see Definition 2) with Tsuboi’s homomorphism \(t_1\) mentioned above. If we can show directly that the homomorphism \(\Phi\) followed by the projection \(H_3(\Gamma_H;\mathbb{Z})\rightarrow H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})\) coincides with \(t_1\), then we are done. In other words, we have to show that the \(\Gamma_1\)-structure of our real analytic chain \(C(\sigma)\) which bounds linear foliations on the torus is homotopic to that of Thurston’s and Tsuboi’s constructions using Reeb foliations. This seems to be a rather delicate problem. Instead of showing this directly, we adopt an alternate method where we compare the two constructions via Thurston’s real analytic cycles as follows.

For each parameter \(s\in\mathbb{R}\), we consider the difference \(\Phi_2(s)-\Phi(s)\). At the chain level, the added \(3\)-chains \(C\langle 1, s\rangle\) of the group \(\mathbb{R}\rtimes Z_{(2,3)}\) cancel so that only chains of the two groups \(\widetilde{\mathit{PSL}}(2,\mathbb{R}),\widetilde{\mathit{SL}}(2,\mathbb{R})\) remain which make a cycle of the group \(\widetilde{\mathit{PSL}}(2,\mathbb{R})*_{\mathbb{R}} \widetilde{\mathit{SL}}(2,\mathbb{R})\). This is exactly the cycle that Thurston adopted to show that the homomorphism 2 is surjective. On the other hand, in the corresponding difference \(t_2(s)-t_1(s)\) in Tsuboi’s construction, the added part containing Reeb foliations also cancel and the remaining cycle is Thurston’s cycle projected to \(H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})\).

Now by Proposition 18, \(\Phi_2(s)-\Phi(s)=3\Phi(s)\). Similarly by 12 , \(t_2(s)-t_1(s)=3t_1(s)\). It follows that \(3\Phi(s)\in H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})\) projects to \(3t_1(s)\in H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})\). Since both of these two classes are members of uniquely divisible groups \(\mathrm{Im}\, \Phi, \mathrm{Im}\, t_1\), we can now conclude that \(\Phi(s)\) projects to \(t_1(s)\) for all \(s\). This is because in any uniquely divisible group, the equality \(3x=3y\) implies \(x=y\). This completes the proof. 0◻

6 Alternative proof of Theorem 1↩︎

In this section, we give an alternative proof of our main result Theorem 1. In the previous section \(\S 5\), we proved this theorem by constructing a homomorphism \[H_2(\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\overset{\mathit{Tsuboi}}{\cong}\mathbb{R}\rightarrow H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\] such that its composition with the homomorphism \(gv: H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\rightarrow \mathbb{R}\) is an isomorphism. Our alternative proof of Theorem 1 is given along the line of Thurston’s original construction. He considered the group \[\Gamma_T:=\mathit{PSL}(2,\mathbb{R})*_{\mathit{SO}(2)} \mathit{SL}(2,\mathbb{R}) \label{eq:TG}\tag{13}\] which acts naturally on the circle so that there is a homomorphism \[\rho_T: \Gamma_T\rightarrow \mathrm{Diff}_+^\omega S^1.\] This induces the following associated homomorphism \[\tilde{\rho}_T: H_2(\Gamma_T;\mathbb{Z})\rightarrow H_2(\mathrm{Diff}_+^\omega S^1;\mathbb{Z})\rightarrow H_3(\widetilde{\mathrm{Diff}}_+^\omega S^1;\mathbb{Z})\rightarrow H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z}).\] We prove the following stronger result which immediately implies Theorem 1.

This shows that Thurston’s surjection \(H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\rightarrow\mathbb{R}\) is already achieved by considering only the group \(\Gamma_T=\mathit{PSL}(2,\mathbb{R})*_{\mathit{SO}(2)} \mathit{SL}(2,\mathbb{R})\) without considering higher covering groups \(\mathit{PSL}(2,\mathbb{R})^{(n)}\) for \(n\geq 3\).

As a byproduct of this theorem, we show another result (see Theorem 21 below) which may have independent meaning. Here is the background for this result. Recall that we have two characteristic classes \[\begin{align} & \chi\in H^2(B\mathrm{Diff}_+^{\infty,\delta} S^1;\mathbb{Z}) \\ & \alpha \in H^2(B\mathrm{Diff}_+^{\infty,\delta} S^1;\mathbb{R}) \end{align}\] for foliated \(S^1\)-bundles (see [8] e.g.). The former is the Euler class while the latter is the Godbillon-Vey class integrated along the fiber. These two classes are also defined as cohomology classes of the group \(\Gamma_T\) through the homomorphism \(\rho_T\). Although Thurston did not mention in his paper [1], by his construction of real analytic cycles mentioned in \(\S 1\), he essentially proved that the homomorphism \[(\chi,\alpha): H_2(B\mathrm{Diff}_+^{\omega,\delta} S^1;\mathbb{Z})\rightarrow \mathbb{Z}\oplus\mathbb{R}\] is surjective. In the proof of the following theorem, we also describe a detailed proof of this result of Thurston.

The proofs of Theorem 20 and Theorem 21 are based on the following Proposition 23 and the fact that the inclusion \(\mathbb{R}\subset \widetilde{\mathit{PSL}}(2,\mathbb{R})*_{\mathbb{R}}(\mathbb{R}\rtimes Z_{(2,3)})\) is homologically trivial up to degree \(3\) which we showed in \(\S 5\).

Recall that Sah and Wagoner [13] proved that there is an isomorphism \[H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\cong K^0_2(\mathbb{R})\oplus \mathbb{Z}\] where \(K^0_2(\mathbb{R})\) denotes the second algebraic \(K_2\) group of the real field \(\mathbb{R}\) omitting the factor \(\mathbb{Z}/2\) so that it is a \(\mathbb{Q}\) vector space, and the second factor is detected by the Euler class. Tsuboi [3] extended this theorem to show that there exist similar isomorphisms \[H_2(\mathit{PSL}(2,\mathbb{R})^{(n)};\mathbb{Z})\cong K^0_2(\mathbb{R})\oplus \mathbb{Z}\] for all the finite covering group \(\mathit{PSL}(2,\mathbb{R})^{(n)}\) of \(\mathit{PSL}(2,\mathbb{R})\). Here the second factors \(\mathbb{Z}\) are all detected by the Euler class and the first factor is the image of the natural homomorphism \[H_2(\mathit{SO}(2);\mathbb{Z}) \rightarrow H_2(\mathit{PSL}(2,\mathbb{R})^{(n)};\mathbb{Z})\] induced by the inclusion \(\mathit{SO}(2)\subset \mathit{PSL}(2,\mathbb{R})^{(n)}\) of the maximal compact subgroup each of which is canonically identified with the rotation group \(\mathit{SO}(2)\).

As preparations for the proof of Theorem 20, we consider the subgroup \(K^0_2(\mathbb{R})\subset H_2(\Gamma_T;\mathbb{Z})\) and the direct summand \(\mathbb{Z}_U\subset U=H_2(\Gamma_T;\mathbb{Z})/K^0_2(\mathbb{R})\). First we consider the former one. As mentioned above, Sah-Wagoner [13] proved that \[H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\cong K^0_2(\mathbb{R})\oplus\mathbb{Z}.\] Here the summand \(K^0_2(\mathbb{R})\) denotes the algebraic \(K_2\) group of the real field \(\mathbb{R}\) omitting the \(\mathbb{Z}/2\)-factor and described as follows. Define a subgroup \(A\subset \mathit{SL}(2,\mathbb{R})\) by \[A=\left\{\begin{pmatrix} r&0\\ 0&r^{-1} \end{pmatrix}; r\in \mathbb{R}^+ \right\}\;\subset \mathit{SL}(2,\mathbb{R}),\] which appears in the Iwasawa decomposition \(\mathit{SL}(2,\mathbb{R})=KAN\). Then \[K^0_2(\mathbb{R})=\mathrm{Im}\left(H_2(A;\mathbb{Z})\rightarrow H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\right).\] Later Tsuboi [3] showed that this subgroup coincides with the image of the second homology group of the maximal compact subgroup \(K=\mathit{SO}(2)\subset \mathit{SL}(2,\mathbb{R})\). Namely \[K^0_2(\mathbb{R})=\mathrm{Im}\left(H_2(\mathit{SO}(2);\mathbb{Z})\rightarrow H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\right).\] Now, Parry-Sah [14] (p. 191, (2.40)) asserted the following statement. In the Gysin exact sequence \[\cdots\rightarrow H_4(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\overset{\cap \chi}{\rightarrow}H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z}) \overset{\mu}{\rightarrow} H_3(\widetilde{\mathit{SL}}(2,\mathbb{R});\mathbb{Z}) \rightarrow H_3(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\rightarrow 0 \label{eq:gy}\tag{14}\] corresponding to the central extension \(0\rightarrow \mathbb{Z}\rightarrow \widetilde{\mathit{SL}}(2,\mathbb{R})\rightarrow \mathit{SL}(2,\mathbb{R})\rightarrow 1\), the homomorphism \(\cap\chi\) is the \(0\)-map. It implies that the homomorphism \[\mu: H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\cong K^0_2(\mathbb{R})\oplus \mathbb{Z}\subset H_3(\widetilde{\mathit{SL}}(2,\mathbb{R});\mathbb{Z}) \label{eq:mui}\tag{15}\] is injective. The authors mentioned that this result “would follow from the vanishing of the square of \(e\) (the Euler class)". This is true for the summand \(\mathbb{Z}\subset H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\). However we think that the injectivity under \(\mu\) in 15 of the other submodule \(K^0_2(\mathbb{R})\subset H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\) is not a consequence of it. This does not affect our argument below, but see Appendix in this section for this point.

Now we consider the latter direct summand \(\mathbb{Z}_U\subset U=H_2(\Gamma_T;\mathbb{Z})/K^0_2(\mathbb{R})\). A homology class \(\sigma\in H_2(\Gamma_T;\mathbb{Z})\) which projects to the class \((4,1)\in \mathbb{Z}_U\) with Euler number \(3\) was described in our former note [9] as follows. Let \[\rho:\pi_1(\Sigma_2)\rightarrow \mathit{PSL}(2,\mathbb{R})\subset \Gamma_T\] be a Fuchsian representation, corresponding to a hyperbolic structure on a closed surface \(\Sigma_2\) of genus \(2\). Since \(\Sigma_2\) has a spin structure, this representation lifts to \(\mathit{SL}(2,\mathbb{R})\) so that there is a representation \[\rho^s:\pi_1(\Sigma_2)\rightarrow \mathit{SL}(2,\mathbb{R})\subset\Gamma_T\] which projects to \(\rho\) under the natural projection \(\mathit{SL}(2,\mathbb{R})\rightarrow \mathit{PSL}(2,\mathbb{R})\). Then the Euler numbers are given by \(\chi(\rho_*([\Sigma_2]))=-2\) while \(\chi(\rho^s_*([\Sigma_2]))=-1\). Now set \(\sigma=\rho^s_*([\Sigma_2])-2\rho_*([\Sigma_2])\in H_2(\Gamma_T;\mathbb{Z})\) so that \(\chi(\sigma)=3\). In fact, by Remark 24 it projects to the element \((4,1)\in \mathbb{Z}_U\) as required and also \(\alpha(\sigma)=0\).

We wonder whether an answer to the following question is known or not.

It seems natural to expect that the above map is non-trivial and we tried to prove this. However, it required complicated computations involving Steinberg symbols so that we could not obtain an answer.

Proof of Theorem 20

Consider the following sequence of homomorphisms \[H_2(\Gamma_T;\mathbb{Z})\overset{\mu}{\rightarrow} H_3(\widetilde{\Gamma}_T;\mathbb{Z})\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\rightarrow H_3(\Gamma_H;\mathbb{Z}).\] Here recall the very important fact that the central extension \(\widetilde{\Gamma}_T\) of the Thurston’s group \(\Gamma_T\) associated with the Euler class is contained in \(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}\). Also \(\mu\) denotes a part of the Gysin exact sequence.

First step

Here we show \(\tilde{\mu}(K^0_2(\mathbb{R}))=0\in H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\) where \[\tilde{\mu}:H_2(\Gamma_T;\mathbb{Z})\overset{\mu}{\rightarrow} H_3(\widetilde{\Gamma}_T;\mathbb{Z})\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\] denotes the composition of the first two homomorphisms above. This is enough to prove Theorem 1.

The central extension \[0\rightarrow \mathbb{Z}\rightarrow \widetilde{\Gamma}_T\rightarrow \Gamma_T\rightarrow 1\] restricts to the central extension \[0\rightarrow \mathbb{Z}\rightarrow\mathbb{R}\rightarrow \mathbb{R}/\mathbb{Z}\rightarrow 1\] where \(\mathbb{R}/\mathbb{Z}\) is the same as the rotation group \(\mathit{SO}(2)\subset \Gamma_T\) and \(\mathbb{R}\subset \widetilde{\Gamma}_T\) denotes the set of translations. By the naturality of the Gysin map, the following diagram is commutative. \[\begin{CD} H_2(\mathbb{R}/\mathbb{Z};\mathbb{Z}) @>{\mu}>> H_3(\mathbb{R};\mathbb{Z}) \\ @V{i_*}VV @VV{j_*}V\\ H_2(\Gamma_T;\mathbb{Z}) @>{\mu}>>H_3(\widetilde{\Gamma}_T;\mathbb{Z}). \end{CD}\] where \(j:\mathbb{R}\rightarrow \widetilde{\Gamma}_T\) denotes the inclusion. Now by Tsuboi \[H_2(\Gamma_T;\mathbb{Z})\supset K^0_2(\mathbb{R})=\mathrm{Im}\, i_*.\] Hence, by the commutativity of the above diagram, we have \[\mu(K^0_2(\mathbb{R}))\subset \mathrm{Im}\, j_*.\] Now consider the following sequence \[H_3(\widetilde{\Gamma}_T;\mathbb{Z})\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\rightarrow H_3(\Gamma_H;\mathbb{Z})\] and the commutative diagram \[\begin{CD} H_3(\mathbb{R}) @>{=}>> H_3(\mathbb{R}) \\ @V{j_*}VV @VV{k_*}V\\ H_3(\widetilde{\Gamma}_T;\mathbb{Z}) @>{\ell_*}>>H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z}). \end{CD}\] By the commutativity, we conclude that \[\ell_*(\mu(K^0_2(\mathbb{R})))\subset k_*(H_3(\mathbb{R};\mathbb{Z}))\] where \(\ell: \widetilde{\Gamma}_T\rightarrow \widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}\) denotes the inclusion. Now the other very important fact here which we proved in \(\S 5\) is that \[\text{H_3(\mathbb{R};\mathbb{Z}) vanishes in H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})}.\] The point of the proof is the property that \(2^3-1=7\) and \(3^3-1=26\) are coprime to each other. Therefore the subgroup \(K^0_2(\mathbb{R})\subset H_2(\Gamma_T;\mathbb{Z})\) vanishes in \(H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\) and hence also in \(H_3(\Gamma_H;\mathbb{Z})\).

On the other hand, the quotient \(H_2(\Gamma_T;\mathbb{Z})/K^0_2(\mathbb{R})\) is isomorphic to \(\mathbb{Z}\oplus \mathbb{R}\) by Proposition 23. We can finally conclude that \[\mathrm{Im}(H_2(\Gamma_T;\mathbb{Z})\rightarrow H_3(\Gamma_H;\mathbb{Z}))\] contains a direct summand \(\mathbb{R}\) which maps isomorphically onto \(\mathbb{R}\) by the homomorphism gv. This completes the First step of the proof of Theorem 20.

Second step

By the first step above, the homomorphism \(\tilde{\mu}:H_2(\Gamma_T;\mathbb{Z})\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\) factors through a homomorphism \[U:=H_2(\Gamma_T;\mathbb{Z})/K^0_2(\mathbb{R})=\mathbb{Z}_U\oplus \mathbb{R}_U\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z}).\] Here we prove that the direct summand \(\mathbb{Z}_U\) of \(U\) vanishes in the above homomorphism. This will finish the proof of Theorem 20. We first prove that the element \(\sigma=\rho^s_*([\Sigma_2])-2\rho_*([\Sigma_2])\in H_2(\Gamma_T;\mathbb{Z})\), which projects to \(3\in \mathbb{Z}_U\), vanishes in \(H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\) under the above homomorphism. This proof can be given using only the property of the endomorphism \(\varphi_2:\widetilde{\mathrm{Diff}}_+^\omega S^1\rightarrow\widetilde{\mathrm{Diff}}_+^\omega S^1\) without making use of our technology developed in the preceding sections. Recall that \(\varphi_2\) sends the subgroup \(\widetilde{\mathit{PSL}}(2,\mathbb{R})\subset\widetilde{\mathrm{Diff}}_+^\omega S^1\) isomorphically onto the subgroup \(\widetilde{\mathit{SL}}(2,\mathbb{R})\subset\widetilde{\mathrm{Diff}}_+^\omega S^1\) (see \(\S 5\)). Also recall that \(\Gamma_T=\mathit{PSL}(2,\mathbb{R})*_{\mathit{SO}(2)} \mathit{SL}(2,\mathbb{R})\) so that \[\widetilde{\Gamma}_T=\widetilde{\mathit{PSL}}(2,\mathbb{R})*_\mathbb{R}\widetilde{\mathit{SL}}(2,\mathbb{R}).\] Now, if we denote by \(S^1(\rho)\) (resp. \(S^1(\rho^s)\)) the total space of the flat \(S^1\)-bundle induced by \(\rho\) (resp. \(\rho^s\)), then there exists a fiberwise \(2\)-fold covering map \(S^1(\rho^s)\rightarrow S^1(\rho)\). Therefore the endomorphism \(\varphi_2\) appears in the following commutative diagram. \[\begin{CD} \pi_1(S^1(\rho^s))@>{\tilde{\rho^s}}>> \widetilde{\mathit{SL}}(2,\mathbb{R}) @>{\subset}>> \widetilde{\mathrm{Diff}}_+^\omega S^1 \\ @V{\cap}V{\text{index 2}}V @A{\varphi_2}A{\cong}A @AA{\varphi_2}A\\ \pi_1(S^1(\rho)) @>{\tilde{\rho}}>>\widetilde{\mathit{PSL}}(2,\mathbb{R})@>{\subset}>>\widetilde{\mathrm{Diff}}_+^\omega S^1 \end{CD} \label{eq:cd}\tag{16}\] where \(\tilde{\rho}:\pi_1(S^1(\rho))\rightarrow \widetilde{\mathit{PSL}}(2,\mathbb{R})\) (resp. \(\tilde{\rho^s}:\pi_1(S^1(\rho^s))\rightarrow \widetilde{\mathit{SL}}(2,\mathbb{R})\)) denotes the representation induced by \(\rho\) (resp. \(\rho^s\)). Since \(S^1(\rho)\) is a closed oriented \(3\)-manifold, its fundamental class represents a homology class \([S^1(\rho)]\in H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R});\mathbb{Z})\). In fact, it is the Gysin image of the class \([\Sigma_2]\in H_2(\mathit{PSL}(2,\mathbb{R});\mathbb{Z})\) so that \([S^1(\rho)]=\mu_{\mathit{PSL}}(\rho_*([\Sigma_2]))\). Similarly \([S^1(\rho^s)]\in H_3(\widetilde{\mathit{SL}}(2,\mathbb{R});\mathbb{Z})\) and \([S^1(\rho^s)]=\mu_{\mathit{SL}}(\rho^s_*([\Sigma_2]))\). Here \(\mu_{\mathit{PSL}}\) denotes the Gysin homomorphism \[\mu_{\mathit{PSL}}: H_2(\mathit{PSL}(2,\mathbb{R});\mathbb{Z})\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R});\mathbb{Z})\] and \(\mu_{\mathit{SL}}\) denotes similar one for the group \(\mathit{SL}(2,\mathbb{R})\). Also let \(\mu_\Gamma\) denotes the Gysin homomorphism for the group \(\Gamma_T\). Now by the naturality of the Gysin homomorphisms for the group \(\Gamma_T\) and its subgroups \(\mathit{PSL}(2,\mathbb{R})\) and \(\mathit{SL}(2,\mathbb{R})\), we have \[\begin{align} \mu_\Gamma(\sigma)&=\mu_\Gamma(\rho^s_*([\Sigma_2])-2\rho_*([\Sigma_2]))\\ &=\tilde{\rho^s}_*(\mu_{\mathit{SL}}([\Sigma_2]))-2\tilde{\rho}_*(\mu_{\mathit{PSL}}([\Sigma_2]))\\ &=(i_{\mathit{SL}})_*[S^1(\rho^s)]-2(i_{\mathit{PSL}})_*[S^1(\rho)]\in H_3(\widetilde{\Gamma}_T;\mathbb{Z})\; \end{align}\] where \(i_{\mathit{PSL}}: \widetilde{\mathit{PSL}}(2,\mathbb{R})\rightarrow \widetilde{\Gamma}_T\) (resp. \(i_{\mathit{SL}}: \widetilde{\mathit{SL}}(2,\mathbb{R})\rightarrow \widetilde{\Gamma}_T\)) denotes the inclusion.

Now by the above commutative diagram 16 , we have \[(\varphi_2)_*(2 [S^1(\rho)])=[S^1(\rho^s)]. \label{eq:ss}\tag{17}\] If we consider everything in the larger group \(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}\) which contains \(\widetilde{\Gamma}_T\), then we obtain the following commutative diagram \[\begin{CD} \widetilde{\mathit{SL}}(2,\mathbb{R})@>{i_{\mathit{SL}}}>> \widetilde{\Gamma}_T @>{j^s}>{\subset}> \widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)} \\ @A{\varphi_2}A{\cong}A @. @A{\cong}A{\varphi_2}A\\ \widetilde{\mathit{PSL}}(2,\mathbb{R}) @>{i_{\mathit{PSL}}}>>\widetilde{\Gamma}_T@>j>{\subset}>\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}, \end{CD} \label{eq:cd2}\tag{18}\] where \(j, j^s\) denote inclusions. We can now conclude that \[\begin{align} \tilde{\mu}(\sigma)&=(j^s\circ i_{\mathit{SL}})_*[S^1(\rho^s)]-2(j\circ i_{\mathit{PSL}})_*[S^1(\rho)]\\ &=(j^s\circ i_{\mathit{SL}})_*(\varphi_2)_*(2 [S^1(\rho)])-2(j\circ i_{\mathit{PSL}})_*[S^1(\rho)]\quad \text{(by \eqref{eq:ss})}\\ &=2(\varphi_2)_*(j\circ i_{\mathit{PSL}})_*[S^1(\rho)]-2(j\circ i_{\mathit{PSL}})_*[S^1(\rho)]\quad (\text{commutativity of \eqref{eq:cd2}})\\ &=0. \end{align}\] The last equality follows from the fact that \(\varphi_2\) acts on the group \(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)}\) by conjugation. This finishes the proof that \(3\mathbb{Z}_U\subset U\) vanishes under the homomorphism \(\tilde{\mu}\).

To proceed to prove the vanishing of the whole \(\mathbb{Z}_U\subset U\), we have to enhance the above argument by employing our technique developed in the preceding sections. It is a refinement of the previous one comparing certain \(2\)-cycles of \(\mathit{PSL}(2,\mathbb{R})\) and \(\mathit{SL}(2,\mathbb{R})\) adapted to the relative \(2\)-cycles of \(\mathit{PSL}(2,\mathbb{R})\) and \(\mathit{SL}(2,\mathbb{R})\) with respect to their maximal compact subgroups. To do this, we consider the homomorphisms \(\Phi\) and \(\Phi_2\) \[\begin{align} \Phi&: H_2(\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\cong \mathbb{R}\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\rightarrow H_3(\Gamma_H;\mathbb{Z})\\ \Phi&_2: H_2(\mathit{SL}(2,\mathbb{R}),\mathit{SO}(2);\mathbb{Z})\cong \mathbb{R}\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3)};\mathbb{Z})\rightarrow H_3(\Gamma_H;\mathbb{Z}) \end{align}\] introduced in \(\S 5\). We set \[\tau=\left(\frac{4}{3},\frac{1}{3}\right)\in U\] so that \(\chi(\tau)=1, \alpha(\tau)=0\) and \(\tau=1\in \mathbb{Z}_U\subset U\). We can write \(\tau=(\tau_1,\tau_2)\) where \[\begin{align} \tau_1&= \frac{4}{3}\in H_2(\mathit{PSL}(2,\mathbb{R}),\mathit{PSO}(2);\mathbb{Z})\cong \mathbb{R}\\ \tau_2&= \frac{1}{3}\in H_2(\mathit{SL}(2,\mathbb{R}),\mathit{SO}(2);\mathbb{Z})\cong \mathbb{R}. \end{align}\] Then there are compact oriented surfaces with one boundary component of genus \(g_i\;(i=1,2)\), denoted by \(\Sigma_{g_i}^0=\Sigma_{g_i}\setminus \mathrm{Int}\, D^2\;(i=1,2)\), and homomorphisms \[\begin{align} \rho_1&: \pi_1(\Sigma_{g_1}^0,p_1)\rightarrow \mathit{PSL}(2,\mathbb{R})\\ \rho_2&: \pi_1(\Sigma_{g_2}^0,p_2)\rightarrow \mathit{SL}(2,\mathbb{R}) \end{align}\] such that they represent the classes \(\tau_1, \tau_2\) respectively, where \(p_i\in \partial \Sigma_{g_i}^0\;(i=1,2)\) denote base points. Therefore the restrictions of the homomorphisms \(\rho_i\) to the boundaries \[\begin{align} \rho_1|_{\partial \Sigma_{g_1}}&: \pi_1(\partial \Sigma_{g_1}^0,p_1)\cong\mathbb{Z}\rightarrow \mathit{PSO}(2)\\ \rho_2|_{\partial \Sigma_{g_2}}&: \pi_1(\partial \Sigma_{g_2}^0,p_2)\cong \mathbb{Z}\rightarrow \mathit{SO}(2) \end{align}\] are given by \[\begin{align} \mathbb{Z}\ni 1\mapsto \frac{1}{3}\;(=\text{120^\circ\;rotation})\;\in \mathit{PSO}(2)\cong \mathbb{R}/\mathbb{Z}\\ \mathbb{Z}\ni 1\mapsto \frac{1}{3}\;(=\text{120^\circ\;rotation})\;\in \mathit{SO}(2)\cong \mathbb{R}/\mathbb{Z} \end{align}\] where both \(\mathit{PSO}(2)\) and \(\mathit{SO}(2)\) are canonically identified with \(\mathbb{R}/\mathbb{Z}\). Furthermore the homomorphisms \(\rho_i\) can be lifted to \[\begin{align} \tilde{\rho}_1&: \pi_1(\Sigma_{g_1}^0,p_1)\rightarrow \widetilde{\mathit{PSL}}(2,\mathbb{R})\\ \tilde{\rho}_2&: \pi_1(\Sigma_{g_2}^0,p_2)\rightarrow \widetilde{\mathit{SL}}(2,\mathbb{R}) \end{align}\] whose restrictions to the boundaries \[\begin{align} \tilde{\rho}_1|_{\partial \Sigma_{g_1}}&: \pi_1(\partial \Sigma_{g_1}^0,p_1)\cong\mathbb{Z}\rightarrow \mathbb{Z}\subset \widetilde{\mathit{PSO}}(2)\\ \tilde{\rho}_2|_{\partial \Sigma_{g_2}}&: \pi_1(\partial \Sigma_{g_2}^0,p_2)\cong \mathbb{Z}\rightarrow \mathbb{Z}\subset \widetilde{SO}(2) \end{align}\] satisfy the conditions \[\begin{align} \mathbb{Z}\ni 1\mapsto \frac{4}{3}\in \mathbb{Z}\subset \widetilde{\mathit{PSO}}(2)\\ \mathbb{Z}\ni 1\mapsto \frac{1}{3}\in \mathbb{Z}\subset \widetilde{SO}(2) \end{align}\] (the translation lengths). Now we glue the two compact surfaces \(\Sigma_{g_i}^0\;(i=1,2)\) along their boundaries, matching the base points and orientations, to make a closed oriented surface \(\Sigma_g\;(g=g_1+g_2)\). Here the orientation on it is given as \(\Sigma_{g_1}^0-\Sigma_{g_2}^0\). Then we can construct a homomorphism \[\rho: \pi_1(\Sigma_g,p)\rightarrow \Gamma_T=\mathit{PSL}(2,\mathbb{R})*_{\mathit{PSO}(2)=\mathit{SO}(2)} \mathit{SL}(2,\mathbb{R})\] where \(p\in \Sigma_g\) denotes the base point obtained by glueing \(p_1\) and \(p_2\). By the theorem of van Kampen, we have \[\pi_1(\Sigma_g,p)=\pi_1(\Sigma_{g_1}^0,p_1)*_{\pi_1(\partial \Sigma_{g_1}^0,p_1)=\pi_1(\partial \Sigma_{g_2}^0,p_2)}\pi_1(\Sigma_{g_2}^0,p_2).\] Then we can define \(\rho\) by setting \[\rho|_{\pi_1(\Sigma_{g_i}^0,p_i)}=\rho_i\;(i=1,2)\] because \[\rho_1|_{\partial \Sigma_{g_1}^0}=\rho_2|_{\partial \Sigma_{g_2}^0}\] once we identify \(\mathit{PSO}(2)=\mathit{SO}(2)\). Now we set \[\tilde{\tau}=\rho_*([\Sigma_g])\in H_2(\Gamma_T;\mathbb{Z}).\] By the above construction, we see that \[H_2(\Gamma_T;\mathbb{Z})\ni \tilde{\tau}\mapsto \tau\in H_2(\Gamma_T;\mathbb{Z})/K^0_2(\mathbb{R})\] under the natural projection. Now, by the construction of the homomorphism \(\rho\), we see that \[\tilde{\mu}(\tilde{\tau})=\tilde{\tau}_1-\tilde{\tau}_2.\] Namely, the homology class \(\tilde{\mu}(\tilde{\tau})\) is represented by the \(3\)-cycle \(\tilde{\tau}_1-\tilde{\tau}_2\). On the other hand, we can also see that the following equality holds \[\tilde{\mu}(\tilde{\tau})=\Phi\left(\frac{4}{3}\right)-\Phi_2\left(\frac{1}{3}\right). \label{eq:mm}\tag{19}\] This is the key fact in our argument. The point here is that the “real analytic cap" appearing in the first term is exactly equal to that of the second term so that they cancel each other in the above expression. (cf. the commutative diagram 10 ). More explicitly, this can be seen as follows. By the definitions of \(\Phi\) and \(\Phi_2\) (see Definition 2 and Definition 3), we have \[\begin{align} \Phi\left(\frac{4}{3}\right)&=\tilde{\tau}_1-\Psi(\partial \tilde{\tau}_1)\\ \Phi_2\left(\frac{1}{3}\right)&=\tilde{\tau}_2-\Psi(\partial \tilde{\tau}_2). \end{align}\] Since \[\frac{4}{3}\equiv \frac{1}{3}\;\mathrm{mod}\;1,\] we have \[1\wedge \frac{4}{3} =1\wedge \frac{1}{3}\in H_2(\mathbb{R};\mathbb{Z})\;\Rightarrow \; \partial \tilde{\tau}_1=\partial \tilde{\tau}_2.\] Hence \[\Psi(\partial \tilde{\tau}_1)=\Psi(\partial \tilde{\tau}_2).\] This verifies the above cancellation and thereby confirms the above equality 19 .

Now by Proposition 18, we know that \(\Phi_2(s)=4\Phi(s)\) for any \(s\). We can now conclude that \[\tilde{\mu}(\tilde{\tau})=\Phi\left(\frac{4}{3}\right)-\Phi_2\left(\frac{1}{3}\right) =4\Phi\left(\frac{1}{3}\right)-\Phi_2\left(\frac{1}{3}\right)=0.\] Since \(\tau\) represents the generator of the summand \(\mathbb{Z}_U\subset U\), this completes the proof of Second step and hence that of Theorem 20. 0◻

Proof of Theorem 21
By Proposition 23, there exists a short exact sequence \[0\rightarrow K^0_2(\mathbb{R})\rightarrow H_2(\Gamma_T;\mathbb{Z}) \overset{(\chi,\alpha)}{\rightarrow}\mathbb{Z}\oplus\mathbb{R}\rightarrow 0. \label{eq:kh}\tag{20}\] It follows that there is a short exact sequence \[0\rightarrow K^0_2(\mathbb{R})\rightarrow \mathrm{Ker}\;\chi\;(\subset H_2(\Gamma_T;\mathbb{Z})) \overset{\alpha}{\rightarrow}\mathbb{R}\rightarrow 0.\] Recall that \(K^0_2\) is a \(\mathbb{Q}\) vector space. It is easy to see that group \(\mathrm{Ker}\;\chi\) is a torsion free divisible group and so that it is a \(\mathbb{Q}\) vector space. Now it is easy to see that the extension 20 is a split extension if we assume the Axiom of Choice. The required result follows from this. 0◻

As for the first question \((i)\) above, since \(K^0_2(\mathbb{R})\) comes from the inclusion \(i: \mathit{SO}(2)\rightarrow \mathrm{Diff}_+^\omega S^1\) by Tsuboi’s result, we may more generally ask the following.

Thurston proved the injectivity of the homomorphism 21 in degree \(2\) by considering his group \(\Gamma_T\) described above (see 13 ), and in the smooth case, he also proved the non-triviality of all the powers of \(\alpha\) based on results of Mather. Then it was proved in [8] that the above homomorphism with the target replaced by \(H^*(\mathrm{Diff}_+^\infty S^1;\mathbb{R})\) is injective.

However, in the real analytic case, almost nothing is known beyond Thurston’s results. Even the problem of determining non-triviality of the square of the rational Euler class \(\chi\) is already very difficult and that of higher powers of \(\chi\) gets more and more difficult (as for the integral Euler class, Nariman [15] proved non-triviality of all the powers of it). Moreover non-triviality of the powers \(\alpha^k\;(k=2,3,\ldots)\) of \(\alpha\) should be considered as a big mystery. Certainly completely new ideas must be necessary to challenge this problem.

The content of this appendix is not used in other part of the present paper.

We consider the assertion of Parry-Sah [14] (p. 191, (2.40)) mentioned above which implies that the homomorphism \[\mu: H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\cong K^0_2(\mathbb{R})\oplus \mathbb{Z}\subset H_3(\widetilde{\mathit{SL}}(2,\mathbb{R});\mathbb{Z}) \label{eq:mui2}\tag{22}\] is injective. The authors mentioned that this result “would follow from the vanishing of the square of \(e\) (the Euler class)".

This is true for the summand \(\mathbb{Z}\subset H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\) because by the well known property of the Gysin exact sequence, the injectivity under \(\mu\) of this summand \(\mathbb{Z}\), which is detected by the Euler class, is equivalent to the vanishing of the square of the rational Euler class \(\chi_\mathbb{Q}^2\) in \(H^4(\mathit{SL}(2,\mathbb{R});\mathbb{Q})\). But this vanishing follows from the fact that the Lie algebra cohomology of \(\mathit{SL}(2,\mathbb{R})\) is given as \(H^*(\mathfrak{sl}(2,\mathbb{R}),\mathit{SO}(2))\cong \mathbb{R}[\chi]/(\chi^2)\) (cf. Milnor [16]). We can extend this argument to the cases of larger groups such as \(\Gamma_T\) and \(G_T\) so that the vanishing \(\chi^2_\mathbb{Q}=0\) holds in \(H^4(\Gamma_T;\mathbb{Q})\) and \(H^4(G_T;\mathbb{Q})\) as well. It is an important problem to determine whether this continues to hold in \(H^4(\mathrm{Diff}_+^\omega S^1;\mathbb{Q})\) or not (see Problem 28 above).

On the other hand however, we think that the injectivity under \(\mu\) in 22 of the other submodule \(K^0_2(\mathbb{R})\subset H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\) is not a consequence of the vanishing of the square of the Euler class. A simple counter-example: in the \(S^1\)-bundle \(\text{(Hopf map,id)}: S^3\times S^2\rightarrow S^2\times S^2\), \(\chi^2=0\) but \(\mu: H_2(S^2\times S^2;\mathbb{Z})\rightarrow H_3(S^3\times S^2;\mathbb{Z})\) is not an injection. It needs another argument. The authors gave, what they called, a direct argument to prove this result. However, they only gave the homomorphism \(\mu\) at the cycle level by constructing \(3\)-dimensional cycles in the target \(H_3(\widetilde{\mathit{SL}}(2,\mathbb{R});\mathbb{Z})\) associated to any \(2\)-dimensional cycle in \(H_2(A;\mathbb{Z})\) which represents elements of \(K^0_2(\mathbb{R})\) as well as a \(2\)-dimensional cycle representing the generator of the summand \(\mathbb{Z}\subset H_2(\mathit{SL}(2,\mathbb{R});\mathbb{Z})\). The authors called the resulting \(3\)-cycles “lifts" of the original \(2\)-cycles. As far as we understand, this does not prove the required results. It is necessary to show that these cycles are homologically non-trivial. We could not find a proof for this injectivity. However, this does not affect our argument for the proof of Theorem 20.

7 Proof of Theorem 2↩︎

Tsuboi’s homomorphism 4 mentioned in \(\S\) 2, was defined as follows. First he proved that \[\begin{align} H_2(G_T,\mathit{SO}(2);\mathbb{Z})&\cong \bigoplus_{n=1}^\infty H_2(\mathit{PSL}(2,\mathbb{R})^{(n)},\mathit{SO}(2);\mathbb{Z})\\ H_2(\mathit{PSL}(2,\mathbb{R})^{(n)}&,\mathit{SO}(2);\mathbb{Z})\cong \mathbb{R}\quad (n=1,2,\cdots) \end{align}\] and then constructed homomorphisms \[t_n: H_2(\mathit{PSL}(2,\mathbb{R})^{(n)},\mathit{SO}(2);\mathbb{Z})\rightarrow H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})\] for all \(n\). Finally he proved that the equality \(t_n(s)=n^2 t_1(s)\, (s\in\mathbb{R})\) holds for any \(n\) which implies his main theorem of [3].

To make a real analytic lift of the homomorphism \(t_n\), we observe that the following commutative diagram holds: \[\begin{CD} \widetilde{\mathit{PSL}}(2,\mathbb{R}) @>{\subset}>> \widetilde{\mathrm{Diff}}_+^\omega S^1 @>{\subset}>>\Gamma_H \\ @V{\varphi_n}VV @V{\varphi_n}V{}V @V{\varphi_n}V{}V\\ \widetilde{\mathit{PSL}}(2,\mathbb{R})^{(n)} @>{\subset}>> \widetilde{\mathrm{Diff}}_+^\omega S^1 @>{\subset}>> \Gamma_H. \end{CD}\] Namely the universal cover \(\widetilde{\mathit{PSL}}(2,\mathbb{R})^{(n)}\) of \(\mathit{PSL}(2,\mathbb{R})^{(n)}\), considered as a subgroup of \(\widetilde{\mathrm{Diff}}_+^\omega S^1\), is equal to the image under \(\varphi_n\) of \(\widetilde{\mathit{PSL}}(2,\mathbb{R})\) (cf. [8]). Keeping this in mind, we define \[\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3,n)}=\widetilde{\mathit{PSL}}(2,\mathbb{R})*_{\mathbb{R}} (\mathbb{R}\rtimes Z_{(2,3,n)})\] where \(Z_{(2,3,n)}\) be the subgroup of \(\mathbb{R}^+\) generated by three elements \(S_2,S_3,S_n\) (if \(n\) does not have prime factor other than \(2,3\), then \(Z_{(2,3,n)}=Z_{(2,3)}\). But this does not affect the following argument). Since the action of \(\varphi_n\) is equal to the conjugation by the element \(S_n^{-1}\), we can conclude that \(\widetilde{\mathit{PSL}}(2,\mathbb{R})^{(n)}\) is contained in the group \(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3,n)}\).

Based on these facts, we make the following definition of our real analytic lift of Tsuboi’s homomorphism \(t_n\).

Proof of Theorem 2
Since we have \[H_2(G_T,\mathit{SO}(2);\mathbb{Z})\cong \bigoplus_{n=1}^\infty H_2(\mathit{PSL}(2,\mathbb{R})^{(n)},\mathit{SO}(2);\mathbb{Z}),\] in order to prove Theorem 2, it is now enough to prove the following statement which is a generalization of Proposition 18 to the case of \(n\)-fold covering.

This completes the proof of Theorem 2. 0◻

8 Uniqueness of our construction within the affine group↩︎

In section \(\S 5\), we construct what we call real analytic caps (see Definition 1) which are \(3\)-chains bounding \(2\)-cycles representing linear foliations on the torus whose monodromy groups are contained in the group \(\mathbb{R}\) acting on itself by translations. They play the role of Reeb components in the real analytic context. In this section, we show that our construction of real analytic caps are uniquely defined within the affine group \(\mathrm{Aff}^+ (1)=\mathbb{R}\rtimes \mathbb{R}^+\) (see Proposition 39 below).

We mentioned in Remark 9 that K. Rozhe (C. Roger) [12] proved that the natural projection \(\mathrm{Aff}^+ (n)\rightarrow \mathrm{GL}^+ (n,\mathbb{R})\) induces an isomorphism \[H_*(\mathrm{Aff}^+ (n);\mathbb{Z})\cong H_*(\mathrm{GL}^+ (n,\mathbb{R});\mathbb{Z})\] for any \(n\). In particular, the homomorphism \(H_*(\mathbb{R}^n ;\mathbb{Z})\rightarrow H_*(\mathrm{Aff}^+ (n);\mathbb{Z})\) is trivial except for the degree \(0\) part. In the case of \(n=1\), the affine group \(\mathrm{Aff}^+ (1)=\mathbb{R}\rtimes \mathbb{R}^+\) is a subgroup of \(\mathrm{Diff}_+^\omega \mathbb{R}\) so that it can also be considered as a subgroup of the Haefliger group \(\Gamma_H\). More precisely, \(\mathbb{R}=\{T_s; s\in \mathbb{R}\},\mathbb{R}^+=\{S_r; r\in \mathbb{R}^+\}\subset \mathrm{Diff}_+^\omega \mathbb{R}\) where \(T_s(x)=x+s, S_r(x)=rx \;(x\in\mathbb{R})\) (see \(\S 3\)). The following Proposition exhibits a certain “flexibility" of the Haefliger group.

With above preparations in mind, we proceed to state the main result of this section. Our real analytic caps was a chain-level realization of the homomorphism \[\Psi:H_2(\mathbb{R};\mathbb{Z})\overset{\cong}{\rightarrow} H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z})\] which is the inverse of the boundary homomorphism \(\partial:H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z})\overset{\cong}{\rightarrow} H_2(\mathbb{R};\mathbb{Z})\). We consider what will happen if we replace \(Z_{(2,3)}\) with the group \(\mathbb{R}^+\) of all the homotheties.

Since both of the groups \(H_3(\mathbb{R}^+;\mathbb{Z})\) and \(H_2(\mathbb{R};\mathbb{Z})\) are \(\mathbb{Q}\) vector spaces, \(H_3(\mathbb{R}\rtimes \mathbb{R}^+,\mathbb{R};\mathbb{Z})\) is also a \(\mathbb{Q}\) vector space. Therefore the boundary operator \(\partial:H_3(\mathbb{R}\rtimes \mathbb{R}^+,\mathbb{R};\mathbb{Z}) \rightarrow H_2(\mathbb{R};\mathbb{Z})\) has a huge amount of right inverses among which our homomorphism \(\Psi:H_2(\mathbb{R};\mathbb{Z})\rightarrow H_3(\mathbb{R}\rtimes Z_{(2,3)},\mathbb{R};\mathbb{Z})\subset H_3(\mathbb{R}\rtimes \mathbb{R}^+,\mathbb{R};\mathbb{Z})\) is the simplest one. Associated to any choice \(\Psi':H_2(\mathbb{R};\mathbb{Z})\rightarrow H_3(\mathbb{R}\rtimes \mathbb{R}^+,\mathbb{R};\mathbb{Z})\) of right inverse to this boundary operator \(\partial\), we can define an analytic cap and then the homomorphisms \[\Phi'_n: H_2(\mathit{PSL}(2,\mathbb{R})^{(n)},\mathit{SO}(2);\mathbb{Z})\cong \mathbb{R}\rightarrow H_3(\widetilde{\mathit{PSL}}(2,\mathbb{R})_{(2,3,n)};\mathbb{Z})\rightarrow H_3(\Gamma_H;\mathbb{Z})\quad\] for any \(n=1,2,\ldots\). However, we have the following uniqueness result.

9 A new kind of characteristic class of foliations↩︎

In this section, we prove Theorem 5.

We begin by recalling an old theorem of Wiegold in [20] which was published in the late \(1960\)’s.

To be a bit more precise, let us consider the central extension \[0\rightarrow \mathbb{Z}\rightarrow\mathbb{Q}\rightarrow \mathbb{Q}/\mathbb{Z}\rightarrow 1\] which yields the Gysin exact sequence \[H^1(\mathbb{Q};\mathbb{Z})=0\rightarrow H^0(\mathbb{Q}/\mathbb{Z};\mathbb{Z})\rightarrow H^2(\mathbb{Q}/\mathbb{Z};\mathbb{Z})\rightarrow H^2(\mathbb{Q};\mathbb{Z})\rightarrow H^1(\mathbb{Q}/\mathbb{Z};\mathbb{Z})=0.\] By the universal coefficient theorem, we see that \[H^2(\mathbb{Q}/\mathbb{Z};\mathbb{Z})\cong \mathrm{Ext}(\mathbb{Q}/\mathbb{Z},\mathbb{Z}),\quad H^2(\mathbb{Q};\mathbb{Z})\cong \mathrm{Ext}(\mathbb{Q},\mathbb{Z}).\] Wiegold showed in the above cited paper that \[\mathrm{Ext}(\mathbb{Q}/\mathbb{Z},\mathbb{Z})\cong\hat{\mathbb{Z}},\quad \mathrm{Ext}(\mathbb{Q},\mathbb{Z})\cong \mathbb{R}\] which gives rise to a short exact sequence \[0\rightarrow \mathbb{Z}\rightarrow\hat{\mathbb{Z}}\rightarrow \mathbb{R}\rightarrow 0.\] We would like to quote the final sentence of the above cited paper: “I would not have imagined that killing a cyclic subgroup could have so profound an effect".

Proof of Theorem 5
Let \(gv: B\overline{\Gamma}_1^\omega\rightarrow K(\mathbb{R},3)\) be the classifying map of the Godbillon-Vey class. Our task is to show that the induced homomorphism \[gv^*: H^4(K(\mathbb{R},3),\mathbb{Z})\cong \prod_{\lambda\in\mathcal{H}} \mathbb{R}_\lambda\rightarrow H^4(B\overline{\Gamma}_1^\omega,\mathbb{Z})\] is injective.

By the universal coefficient theorem for cohomology, we have the following split short exact sequence \[0\rightarrow \mathrm{Ext}(H_3(B\overline{\Gamma}_1^\omega,\mathbb{Z}),\mathbb{Z})\rightarrow H^{4}(B\overline{\Gamma}_1^\omega;\mathbb{Z})\rightarrow \mathrm{Hom}(H_{4}(B\overline{\Gamma}_1^\omega,\mathbb{Z});\mathbb{Z})\rightarrow 0.\]

By comparing this with the corresponding short exact sequence for the space \(K(\mathbb{R},3)\), keeping the above Proposition 41 in mind, we obtain the following commutative diagram \[\begin{CD} \mathrm{Ext}(H_3(K(\mathbb{R},3),\mathbb{Z}),\mathbb{Z}) @>{\cong}>> H^{4}(K(\mathbb{R},3);\mathbb{Z}) @>>> \mathrm{Hom}(H_{4}(K(\mathbb{R},3),\mathbb{Z});\mathbb{Z})=0\\ @V{gv^*}VV @V{gv^*}VV @VVV\\ \mathrm{Ext}(H_3(B\overline{\Gamma}_1^\omega,\mathbb{Z}),\mathbb{Z}) @>{\subset}>> H^{4}(B\overline{\Gamma}_1^\omega;\mathbb{Z}) @>>> \mathrm{Hom}(H_{4}(B\overline{\Gamma}_1^\omega,\mathbb{Z});\mathbb{Z}). \end{CD} \label{eq:cd3}\tag{25}\]

Now as was already mentioned in the Introduction, Thurston proved that the homomorphism \[gv: H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\rightarrow \mathbb{R}\] is surjective. On the other hand, by our main Theorem 1, this homomorphism splits so that there exists a direct summand \(\mathbb{R}\subset H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})\) such that \[H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z})= \mathbb{R}\oplus \mathrm{Ker}\, gv=H_3(K(\mathbb{R},3);\mathbb{Z})\oplus \mathrm{Ker}\, gv.\] By the property of the functor \(\mathrm{Ext}\), we have \[\mathrm{Ext}(H_3(B\overline{\Gamma}_1^\omega;\mathbb{Z}),\mathbb{Z})\cong \mathrm{Ext}(H_3(K(\mathbb{R},3);\mathbb{Z}),\mathbb{Z})\oplus \mathrm{Ext}(\mathrm{Ker}\, gv,\mathbb{Z}).\] If we put this into the above commutative diagram 25 , we obtain the required result. This completes the proof of the former part of our theorem.

Next we show the following facts to prove the latter claims of our Theorem. First, let \(X\) be any CW complex such that the homology group \(H_3(X;\mathbb{Z})\) is finitely generated. Then for any continuous map \(f: X\rightarrow B\overline{\Gamma}_1^\omega\), the induced homomorphism \[f^*: H^4(B\overline{\Gamma}_1^\omega;\mathbb{Z})\rightarrow H^4(X,\mathbb{Z})\] is trivial on \(\mathrm{Im}\, (gv^*: H^4(K(\mathbb{R},3);\mathbb{Z})\rightarrow H^4(B\overline{\Gamma}_1^\omega;\mathbb{Z}))\). To prove this, it is enough to show that for any continuous map \(h: X\rightarrow K(\mathbb{R},3)\), the induced homomorphism \[h^*:H^4(K(\mathbb{R},3);\mathbb{Z})\rightarrow H^4(X;\mathbb{Z})\] is trivial. By the proof of Proposition 41, this is equivalent to show that the homomorphism \[h^*:\mathrm{Ext(H_3(K(\mathbb{R},3);\mathbb{Z}),\mathbb{Z})}\rightarrow \mathrm{Ext(H_3(X;\mathbb{Z}),\mathbb{Z})}\] is trivial. By the assumption, \(H_3(X;\mathbb{Z})\) is a finitely generated abelian group. As is well-known, \(\mathrm{Ext}(\mathbb{Z},\mathbb{Z})=0\) and \(\mathrm{Ext}(\mathbb{Z}/n,\mathbb{Z})\cong \mathbb{Z}/n\) so that \(\mathrm{Ext(H_3(X;\mathbb{Z}),\mathbb{Z})}\) is a finite abelian group. On the other hand, the source group of \(h^*\) is \(\mathrm{Ext(H_3(K(\mathbb{R},3);\mathbb{Z}),\mathbb{Z})}\) which is isomorphic to \(\prod_{\lambda\in\mathcal{H}} \mathbb{R}_\lambda\) essentially by the Theorem of Wiegold [20]. We can now conclude that the homomorphism \(h^*\) is trivial. This completes the proof.

Next assume that there is given a \(\overline{\Gamma}_1^\omega\)-structure on \(X\) and there exists a uniquely divisible subgroup \(V\subset H_3(X;\mathbb{Z})\) such that the homomorphism \(gv:H_3(X;\mathbb{Z})\rightarrow \mathbb{R}\) induced by the Godbillon-Vey class is non-trivial on \(V\). By the assumption, \(V\) is uniquely divisible so that it is a \(\mathbb{Q}\) vector space. Then, since the target of \(gv\) is also a \(\mathbb{Q}\) vector space, it is easy to see that \(gv|_{V}\) is a \(\mathbb{Q}\) linear homomorphism. It follows that \(V\cong \left(\mathrm{Ker}\, gv\cap V\right )\oplus W\) for some subgroup \(W\subset H_3(X;\mathbb{Z})\) such that \(gv: W\cong \mathrm{Im}\, gv\subset \mathbb{R}\) (here we use the Axiom of Choice if \(\dim_\mathbb{Q}V=\infty\)). This then implies that \[H_3(X;\mathbb{Z})= \mathrm{Ker}\, gv\oplus W.\] If we use a similar argument as in the former part of this proof by replacing \(\mathbb{R}\) with its subspace \(\mathrm{Im}\, gv(\cong W) \subset \mathbb{R}\), then we can deduce that the homomorphism \[gv^*: H^4(\mathrm{K(Im}\, gv,3);\mathbb{Z}) \cong \prod_{\lambda\in\mathcal{H}'} \mathbb{R}_\lambda\rightarrow H^4(X,\mathbb{Z})\] is injective. Here \(\mathcal{H}'\) denotes a Hamel basis for the \(\mathbb{Q}\) subspace \(\mathrm{Im}\, gv\, (\cong W) \subset \mathbb{R}\). Since the homomorphism \(H^4(K(\mathbb{R},3);\mathbb{Z})\rightarrow H^4(\mathrm{K(Im}\, gv,3);\mathbb{Z})\) induced by the inclusion \(\mathrm{Im}\, gv\subset \mathbb{R}\) is surjective, we can conclude that our characteristic classes detect the subgroup \(W\subset H_3(X;\mathbb{Z})\) on which the Godbillon-Vey class gives an injection into its target \(\mathbb{R}\) as required.

This completes the proof. 0◻

References↩︎