Twisted Alexander polynomials and incompressible surfaces given by ideal points

We study incompressible surfaces constructed by Culler-Shalen theory in the context of twisted Alexander polynomials. For a $1$st cohomology class of a $3$-manifold the coefficients of twisted Alexander polynomials induce regular functions on the $SL_2(\mathbb{C})$-character variety. We prove that if an ideal point gives a Thurston norm minimizing non-separating surface dual to the cohomology class, then the regular function of the highest degree has a finite value at the ideal point.Comment: 10 pages, to appear in "The special issue for the 20th anniversary", the Journal of Mathematical Sciences, the University of Toky

The virtual fibering theorem for 3-manifolds

In 2007 Agol showed that if N is an aspherical compact 3-manifold with empty or toroidal boundary such that its fundamental group is virtually RFRS, then $N$ is virtually fibered. We give a largely self-contained proof of Agol's theorem using complexities of sutured manifolds.Comment: 25 pages, 6 figures v3: slightly revised an argument, otherwise minor change

Cosmological Implications of Number Counts of Clusters of Galaxies: logN-logS in X-Ray and Submm Bands

We compute the number counts of clusters of galaxies, the logN-logS relation, in several X-ray and submm bands on the basis of the Press-Schechter theory. We pay particular attention to a set of theoretical models which well reproduce the ROSAT 0.5-2 keV band logN-logS, and explore possibilities to further constrain the models from future observations with ASCA and/or at submm bands. The latter is closely related to the European PLANCK mission and the Japanese Large Millimeter and Submillimeter Array (LMSA) project. We exhibit that one can break the degeneracy in an acceptable parameter region on the $\Omega_0 - \sigma_8$ plane by combining the ROSAT logN-logS and the submm number counts. Models which reproduce the ROSAT band logN-logS will have $N(>S) \sim (150-300) (S/10^{-12} erg cm^{-2} s^{-1})^{-1.3}$ str$^{-1}$ at $S > 10^{-12} erg cm^{-2} s^{-1}$ in the ASCA 2-10 keV band, and $N(>S_\nu) \sim (10^2-10^4) (S_\nu/100 mJy)^{-1.5} str^{-1}$ at $S_\nu > 100 mJy$ in the submm (0.85mm) band. The amplitude of the logN-logS is very sensitive to the model parameters in the submm band. We also compute the redshift evolution of the cluster number counts and compare with that of the X-ray brightest Abell-type clusters. The results, although still preliminary, point to low density ($\Omega_0\sim 0.3$) universes. The contribution of clusters to the X-ray and submm background radiations is shown to be insignificant in any model compatible with the ROSAT logN-logS.Comment: 18 pages, 6 figures and 1 table. Figures revised. Accepted for publication in PAS