We determine the first homology group with coefficients in H₁(N; Z) for the mapping class group of a non-orientable surface N of genus three with two boundary components.

In this paper we generalize the Zero Divisor Conjecture and Rigidity Theorem for k-regular sequence. For this purpose for any k-regular M-sequence x₁, ..., x_n we prove that if , then , for all i ≥ 1. Also we show that if , then , for all integers i ≥ 0 (i ≠ n).

In this paper, we construct a sequence of homotopy invariants with respect to the model structure of Ozornova-Rovelli on the category of simplicial sets with marking and show that they are compatible with a construction of loop spaces.

Let be a mirror pair of a complex torus X and its mirror partner . This mirror pair is described as the trivial special Lagrangian torus fibrations X → B and → B on the same base space B by SYZ construction. Then, we can associate a holomorphic line bundle → X to a pair of a Lagrangian section s of → B and a unitary local system along it. In this paper, we first construct the deformation of X by a certain flat gerbe and its mirror partner from the mirror pair , and discuss deformations of objects and over the deformed mirror pair .

We prove that if a non-Sasakian contact metric 3-τ-a-manifold or contact metric 3-H-manifold is weakly Einstein, then it is locally isometric to a Lie group equipped with a left invariant contact metric structure.

In this paper, we provide a correspondence between certain 5-dimensional complex spacetimes and 4-dimensional twistor spaces. The spacetimes are almost contact manifolds whose curvature tensor satisfies certain conditions.

By using the correspondence, we show that a 5-dimensional K-contact manifold can be obtained from the Ren-Wang twistor space [10], which is obtained from two copies of identifying open subsets by a holomorphic map. From this result, the Ren-Wang twistor space can be interpreted in the framework of Itoh [5].