We determine the first homology group with coefficients in $H_1(N;{\mathbf{Z}})$ 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 ${x_1},...,{x_n}$ we prove that if $\dim{\rm Tor}_2^R({\frac{R}{{({{x_1},...,{x_n}} )}},M}) \le k$, then $\dim{\rm Tor}_i^R({\frac{R}{{({{x_1},...,{x_n}})}},M}) \le k$, for all $i \ge 1$. Also we show that if $\dim{\rm Ext}_R^{n + 2}({\frac{R}{{({{x_1},...,{x_n}})}},M}) \le k$, then $\dim{\rm Ext}_R^{i}({\frac{R}{{({{x_1},...,{x_n}})}},M}) \le k$, for all integers $i \ge 0$ $({i \ne n})$.

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 $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\rightarrow X$ to a pair$(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\rightarrow B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ and its mirror partner $\check{X}_{\mathcal{G}}$ from the mirror pair $(X,\check{X})$, and discuss deformations of objects $E(s,\mathcal{L})$ and $(s,\mathcal{L})$ over the deformed mirror pair $(X_{\mathcal{G}},\check{X}_{\mathcal{G}})$.

We prove that if a non-Sasakian contact metric 3-$\tau$-$a$-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 $\mathbb{C}^4$ identifying open subsets by a holomorphic map. From this result, the Ren-Wang twistor space can be interpreted in the framework of Itoh [5].