arXiv: Algebraic Topology最新文献

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Complements on categories and topology 补充了类别和拓扑
arXiv: Algebraic Topology Pub Date : 2022-01-01 DOI: 10.1142/9789811248368_0008
{"title":"Complements on categories and topology","authors":"","doi":"10.1142/9789811248368_0008","DOIUrl":"https://doi.org/10.1142/9789811248368_0008","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79633900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An introduction to homotopy groups 同伦群的介绍
arXiv: Algebraic Topology Pub Date : 2022-01-01 DOI: 10.1142/9789811248368_0007
{"title":"An introduction to homotopy groups","authors":"","doi":"10.1142/9789811248368_0007","DOIUrl":"https://doi.org/10.1142/9789811248368_0007","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"221 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89129938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Solution of the exercises 习题解答
arXiv: Algebraic Topology Pub Date : 2022-01-01 DOI: 10.1142/9789811248368_0009
P. Nguyen
{"title":"Solution of the exercises","authors":"P. Nguyen","doi":"10.1142/9789811248368_0009","DOIUrl":"https://doi.org/10.1142/9789811248368_0009","url":null,"abstract":"1 Find the trigonometric functions Note that the Fourier transform can only be computed when the input is squaresummable (finite energy). This is not the case with the tangent. The tangent does not have a maximum or a minimum. Here are some ideas of features which may be useful: • Number of zeros (of times it crosses zero) • Average at different time scales • Some measure of periodicity (e.g. LP coefficients if you are familiar with them)","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89885587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Relative singular homology and homology theories 相对奇异同调及其同调理论
arXiv: Algebraic Topology Pub Date : 2022-01-01 DOI: 10.1142/9789811248368_0004
{"title":"Relative singular homology and homology theories","authors":"","doi":"10.1142/9789811248368_0004","DOIUrl":"https://doi.org/10.1142/9789811248368_0004","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85158232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Introducing Algebraic Topology 代数拓扑介绍
arXiv: Algebraic Topology Pub Date : 2022-01-01 DOI: 10.1142/9789811248368_0002
{"title":"Introducing Algebraic Topology","authors":"","doi":"10.1142/9789811248368_0002","DOIUrl":"https://doi.org/10.1142/9789811248368_0002","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77244274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Structure of the space of $GL_4(mathbb Z_2)$-coinvariants $mathbb Z_2otimes_{GL_4(mathbb Z_2)} PH_*(mathbb Z_2^4, mathbb Z_2)$ in some generic degrees and its application to Singer's cohomological transfer $GL_4(mathbb Z_2)$-协变量$mathbb Z_2otimes_{GL_4(mathbb Z_2)} PH_*(mathbb Z_2^4, mathbb Z_2)$在某些一般度上的空间结构及其在Singer上同调转移中的应用
arXiv: Algebraic Topology Pub Date : 2021-05-27 DOI: 10.31219/osf.io/4ckf8
Dang Vo Phuc
{"title":"Structure of the space of $GL_4(mathbb Z_2)$-coinvariants $mathbb Z_2otimes_{GL_4(mathbb Z_2)} PH_*(mathbb Z_2^4, mathbb Z_2)$ in some generic degrees and its application to Singer's cohomological transfer","authors":"Dang Vo Phuc","doi":"10.31219/osf.io/4ckf8","DOIUrl":"https://doi.org/10.31219/osf.io/4ckf8","url":null,"abstract":"Let $A$ denote the Steenrod algebra at the prime 2 and let $k = mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{otimes q} := kotimes_{A} P_q$ in each non-negative degree $n.$ This problem is the content of \"hit problem\" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariant $kotimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{otimes q}$ to the Adams $E_2$-term, ${rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $kotimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and the representation of the fourth transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. These new results confirmed Singer's conjecture for the monomorphism of the rank $4$ transfer. Our approach is different from that of Singer.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81787133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Endo-trivial modules for cyclic p-groups and generalized quaternion groups via Galois descent 基于伽罗瓦下降的循环p群和广义四元数群的内平凡模
arXiv: Algebraic Topology Pub Date : 2021-05-07 DOI: 10.26153/TSW/13645
J. V. D. Meer, R. Wong
{"title":"Endo-trivial modules for cyclic p-groups and generalized quaternion groups via Galois descent","authors":"J. V. D. Meer, R. Wong","doi":"10.26153/TSW/13645","DOIUrl":"https://doi.org/10.26153/TSW/13645","url":null,"abstract":"In this paper, we investigate the group of endotrivial modules for certain $p$-groups. Such groups were already been computed by Carlson-Thevenaz using the theory of support varieties; however, we provide novel homotopical proofs of their results for cyclic $p$-groups, the quaternion group of order 8, and for generalized quaternion groups using Galois descent and Picard spectral sequences, building on results of Mathew and Stojanoska. Our computations provide conceptual insights into the classical work of Carlson-Thevenaz.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83609785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Cosets, Normal Subgroups, and Quotient Groups 协集、正规子群和商群
arXiv: Algebraic Topology Pub Date : 2021-01-01 DOI: 10.1007/978-3-030-70608-1_7
Clark Bray, Adrian Butscher, Simon Rubinstein-Salzedo
{"title":"Cosets, Normal Subgroups, and Quotient Groups","authors":"Clark Bray, Adrian Butscher, Simon Rubinstein-Salzedo","doi":"10.1007/978-3-030-70608-1_7","DOIUrl":"https://doi.org/10.1007/978-3-030-70608-1_7","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81129322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Mayer–Vietoris Sequence Mayer-Vietoris序列
arXiv: Algebraic Topology Pub Date : 2021-01-01 DOI: 10.1007/978-3-030-70608-1_14
Clark Bray, Adrian Butscher, Simon Rubinstein-Salzedo
{"title":"The Mayer–Vietoris Sequence","authors":"Clark Bray, Adrian Butscher, Simon Rubinstein-Salzedo","doi":"10.1007/978-3-030-70608-1_14","DOIUrl":"https://doi.org/10.1007/978-3-030-70608-1_14","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82586542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Surface Preliminaries 表面预赛
arXiv: Algebraic Topology Pub Date : 2021-01-01 DOI: 10.1007/978-3-030-70608-1_1
Clark Bray, Adrian Butscher, Simon Rubinstein-Salzedo
{"title":"Surface Preliminaries","authors":"Clark Bray, Adrian Butscher, Simon Rubinstein-Salzedo","doi":"10.1007/978-3-030-70608-1_1","DOIUrl":"https://doi.org/10.1007/978-3-030-70608-1_1","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73341196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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