Pub Date : 2022-01-01DOI: 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}
Pub Date : 2022-01-01DOI: 10.1142/9789811248368_0009
P. Nguyen
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)
{"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}
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.
{"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}
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.
{"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}
Pub Date : 2021-01-01DOI: 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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：