We determine the first homology group with coefficients in $H_1(N;mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.
{"title":"The first homology group with twisted coefficients for the mapping class group of a non-orientable surface of genus three with two boundary components","authors":"Piotr Pawlak, Michał Stukow","doi":"10.2996/kmj46301","DOIUrl":"https://doi.org/10.2996/kmj46301","url":null,"abstract":"We determine the first homology group with coefficients in $H_1(N;mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Some weakly Einstein contact metric 3-manifolds","authors":"Yaning Wang, Pei Wang","doi":"10.2996/kmj46305","DOIUrl":"https://doi.org/10.2996/kmj46305","url":null,"abstract":"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.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 $i ge 0$ $({i ne n})$.
本文推广了$k$ -正则序列的零因子猜想和刚性定理。为了这个目的，对于任何$k$ -正则$M$ -序列${x_1},...,{x_n}$，我们证明如果$dim{rm Tor}_2^R({frac{R}{{({{x_1},...,{x_n}} )}},M}) le k$，那么$dim{rm Tor}_i^R({frac{R}{{({{x_1},...,{x_n}})}},M}) le k$，对于所有$i ge 1$。我们也证明了如果$dim{rm Ext}_R^{n + 2}({frac{R}{{({{x_1},...,{x_n}})}},M}) le k$，那么$dim{rm Ext}_R^{i}({frac{R}{{({{x_1},...,{x_n}})}},M}) le k$，对于所有整数$i ge 0$$({i ne n})$。
{"title":"$k$-regular sequences to extension functors and local cohomology modules","authors":"Sajjad Arda, Seadat Ollah Faramarzi, Khadijeh Ahmadi Amoli","doi":"10.2996/kmj46302","DOIUrl":"https://doi.org/10.2996/kmj46302","url":null,"abstract":"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 $i ge 0$ $({i ne n})$.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"On loop spaces of simplicial sets with marking","authors":"Ryo Horiuchi","doi":"10.2996/kmj46303","DOIUrl":"https://doi.org/10.2996/kmj46303","url":null,"abstract":"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.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 $Xrightarrow B$ 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 $(X_{mathcal{G}},check{X}_{mathcal{G}})$.
{"title":"A gerby deformation of complex tori and the homological mirror symmetry","authors":"Kazushi Kobayashi","doi":"10.2996/kmj46304","DOIUrl":"https://doi.org/10.2996/kmj46304","url":null,"abstract":"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 $Xrightarrow B$ 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 $(X_{mathcal{G}},check{X}_{mathcal{G}})$.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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].
{"title":"Almost contact structures on the set of rational curves in a 4-dimensional twistor space","authors":"Michifumi Teruya","doi":"10.2996/kmj46306","DOIUrl":"https://doi.org/10.2996/kmj46306","url":null,"abstract":"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].","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on the Yotsutani-Zhou condition for relative K-instability","authors":"Yasufumi Nitta, Shunsuke Saito","doi":"10.2996/kmj46205","DOIUrl":"https://doi.org/10.2996/kmj46205","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48072011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Infinite series around multinomial coefficients and harmonic numbers","authors":"W. Chu","doi":"10.2996/kmj46201","DOIUrl":"https://doi.org/10.2996/kmj46201","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47598995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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