{"title":"闭可定向三流形上同调代数的Hochschild上同调","authors":"Qiufen Wang","doi":"10.26443/msurj.v16i1.61","DOIUrl":null,"url":null,"abstract":"Let F be a field of characteristic other than 2. We show that the zeroth Hochschild cohomology vector space HH0(A) of a degree 3 graded commutative Frobenius F-algebra A = iAi, where we will always assume A0 = F, is isomorphic to the direct sum of the degree 0, 2 and 3 graded components and the kernel of a certain natural evaluation map ιμ : A1 Λ2(A1). In particular, this holds forA = H∗(M; F) the cohomology algebra of a closed orientable 3-manifoldM. In Theorem A of [1], Charette proves the vanishing of a certain discriminantΔassociated to a closed orientable 3-manifold L with vanishing cup product 3-form. It turns out that if we could show that HH2,−2(A) = 0for A = H∗(L;C), we would have found a more elementary proof of this part of Charette’s theorem. We show that for any β 3, the degree 3 graded commutative Frobenius algebra A with μA = 0and dim(A1) = β satisfiesHH2,−2(A) = 0. Thus Charette’s theorem is not simplified.","PeriodicalId":91927,"journal":{"name":"McGill Science undergraduate research journal : MSURJ","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hochschild Cohomology of the Cohomology Algebra of Closed Orientable Three- Manifolds\",\"authors\":\"Qiufen Wang\",\"doi\":\"10.26443/msurj.v16i1.61\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let F be a field of characteristic other than 2. We show that the zeroth Hochschild cohomology vector space HH0(A) of a degree 3 graded commutative Frobenius F-algebra A = iAi, where we will always assume A0 = F, is isomorphic to the direct sum of the degree 0, 2 and 3 graded components and the kernel of a certain natural evaluation map ιμ : A1 Λ2(A1). In particular, this holds forA = H∗(M; F) the cohomology algebra of a closed orientable 3-manifoldM. In Theorem A of [1], Charette proves the vanishing of a certain discriminantΔassociated to a closed orientable 3-manifold L with vanishing cup product 3-form. It turns out that if we could show that HH2,−2(A) = 0for A = H∗(L;C), we would have found a more elementary proof of this part of Charette’s theorem. We show that for any β 3, the degree 3 graded commutative Frobenius algebra A with μA = 0and dim(A1) = β satisfiesHH2,−2(A) = 0. Thus Charette’s theorem is not simplified.\",\"PeriodicalId\":91927,\"journal\":{\"name\":\"McGill Science undergraduate research journal : MSURJ\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-04-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"McGill Science undergraduate research journal : MSURJ\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.26443/msurj.v16i1.61\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"McGill Science undergraduate research journal : MSURJ","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26443/msurj.v16i1.61","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Hochschild Cohomology of the Cohomology Algebra of Closed Orientable Three- Manifolds
Let F be a field of characteristic other than 2. We show that the zeroth Hochschild cohomology vector space HH0(A) of a degree 3 graded commutative Frobenius F-algebra A = iAi, where we will always assume A0 = F, is isomorphic to the direct sum of the degree 0, 2 and 3 graded components and the kernel of a certain natural evaluation map ιμ : A1 Λ2(A1). In particular, this holds forA = H∗(M; F) the cohomology algebra of a closed orientable 3-manifoldM. In Theorem A of [1], Charette proves the vanishing of a certain discriminantΔassociated to a closed orientable 3-manifold L with vanishing cup product 3-form. It turns out that if we could show that HH2,−2(A) = 0for A = H∗(L;C), we would have found a more elementary proof of this part of Charette’s theorem. We show that for any β 3, the degree 3 graded commutative Frobenius algebra A with μA = 0and dim(A1) = β satisfiesHH2,−2(A) = 0. Thus Charette’s theorem is not simplified.