Real analytic lift of foliations of Thurston and Tsuboi

Thurston constructed codimension one foliations on $S^3$ thereby proved that the homomorphism $gv: \pi_3(B\overline{\Gamma}^\infty_1)\rightarrow \mathbb{R}$ induced by the Godbillon-Vey invariant is surjective. By another real analytic construction, he proved that the homomorphism $gv: H_3(B\overline{\Gamma}^\omega_1)\rightarrow \mathbb{R}$ is also surjective where $B\overline{\Gamma}^\omega_1$ is a $K(\pi,1)$ space by Haefliger. Tsuboi proved that the former surjection splits so that $\pi_3(B\overline{\Gamma}^\infty_1)= \mathbb{R}\oplus \mathrm{Ker}\,gv$. He further showed that the subgroup of $H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})$ generated by all the Thurston's constructions coincides with his direct summand $\mathbb{R}$. In this paper, we prove that Thurston's second surjection splits and also that the subgroup of $H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})$ generated by all the Thurston's cycles is equal to our direct summand $\mathbb{R}$ which is a lift of Tsuboi's one. To show this, we modify the arguments of Thurston and Tsuboi by replacing Reeb components with a real analytic construction. We prove certain {\it uniqueness} of them by showing acyclicity of the affine group in the Haefliger group $\pi_1(B\overline{\Gamma}^\omega_1)$. We also prove the existence of a new kind of characteristic class of foliations in $H^4(B\overline{\Gamma}^\omega_1;\mathbb{Z})$.