As a consequence of the algebraicity of chromatic homotopy at large primes, we show that the Hopkins' Picard group of the $K(n)$-local category coincides with the algebraic one when $2p-2 > n^{2}+n$.
{"title":"Chromatic Picard groups at large primes","authors":"Piotr Pstrkagowski","doi":"10.1090/proc/16004","DOIUrl":"https://doi.org/10.1090/proc/16004","url":null,"abstract":"As a consequence of the algebraicity of chromatic homotopy at large primes, we show that the Hopkins' Picard group of the $K(n)$-local category coincides with the algebraic one when $2p-2 > n^{2}+n$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82341987","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 : 2018-11-01DOI: 10.2140/agt.2020.20.2609
Rachael Boyd
We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.
{"title":"The low-dimensional homology of finite-rank Coxeter groups","authors":"Rachael Boyd","doi":"10.2140/agt.2020.20.2609","DOIUrl":"https://doi.org/10.2140/agt.2020.20.2609","url":null,"abstract":"We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78708861","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}
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent homology (PH) provides a delicate balance between data simplification and intrinsic structure characterization, and has been applied to various areas successfully. However, the combination of PH and machine learning has been hindered greatly by three challenges, namely topological representation of data, PH-based distance measurements or metrics, and PH-based feature representation. With the development of topological data analysis, progresses have been made on all these three problems, but widely scattered in different literatures. In this paper, we provide a systematical review of PH and PH-based supervised and unsupervised models from a computational perspective. Our emphasis is the recent development of mathematical models and tools, including PH softwares and PH-based functions, feature representations, kernels, and similarity models. Essentially, this paper can work as a roadmap for the practical application of PH-based machine learning tools. Further, we consider different topological feature representations in different machine learning models, and investigate their impacts on the protein secondary structure classification.
{"title":"Persistent-Homology-Based Machine Learning and Its Applications -- A Survey","authors":"Chi Seng Pun, Kelin Xia, S. Lee","doi":"10.2139/SSRN.3275996","DOIUrl":"https://doi.org/10.2139/SSRN.3275996","url":null,"abstract":"A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent homology (PH) provides a delicate balance between data simplification and intrinsic structure characterization, and has been applied to various areas successfully. However, the combination of PH and machine learning has been hindered greatly by three challenges, namely topological representation of data, PH-based distance measurements or metrics, and PH-based feature representation. With the development of topological data analysis, progresses have been made on all these three problems, but widely scattered in different literatures. In this paper, we provide a systematical review of PH and PH-based supervised and unsupervised models from a computational perspective. Our emphasis is the recent development of mathematical models and tools, including PH softwares and PH-based functions, feature representations, kernels, and similarity models. Essentially, this paper can work as a roadmap for the practical application of PH-based machine learning tools. Further, we consider different topological feature representations in different machine learning models, and investigate their impacts on the protein secondary structure classification.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"22 7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82715591","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}
We describe an $E_k$-cell structure on the free $E_{k+1}$-algebra on a point, and more generally describe how the May-Milgram filtration of $Omega^m Sigma^m S^{k}$ lifts to a filtration of the free $E_{k+m}$-algebra on a point by iterated pushouts of free $E_k$-algebras.
{"title":"The May–Milgram filtration and\u0000ℰk–cells","authors":"Inbar Klang, A. Kupers, Jeremy Miller","doi":"10.2140/AGT.2021.21.105","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.105","url":null,"abstract":"We describe an $E_k$-cell structure on the free $E_{k+1}$-algebra on a point, and more generally describe how the May-Milgram filtration of $Omega^m Sigma^m S^{k}$ lifts to a filtration of the free $E_{k+m}$-algebra on a point by iterated pushouts of free $E_k$-algebras.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76224265","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}
We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the E-homology of THH(A;B), where E is a ring spectrum, A is a commutative S-algebra and B is a connective commutative Aalgebra. The input of the spectral sequence are the topological Hochschild homology groups of B with coefficients in the E-homology groups of B ∧A B. The mod p and v1 topological Hochschild homology of connective complex K-theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We present an alternative, short computation using the generalized Brun spectral sequence.
{"title":"On the Brun spectral sequence for topological Hochschild homology","authors":"Eva Höning","doi":"10.2140/AGT.2020.20.817","DOIUrl":"https://doi.org/10.2140/AGT.2020.20.817","url":null,"abstract":"We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the E-homology of THH(A;B), where E is a ring spectrum, A is a commutative S-algebra and B is a connective commutative Aalgebra. The input of the spectral sequence are the topological Hochschild homology groups of B with coefficients in the E-homology groups of B ∧A B. The mod p and v1 topological Hochschild homology of connective complex K-theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We present an alternative, short computation using the generalized Brun spectral sequence.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75778931","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 : 2018-07-31DOI: 10.4310/hha.2021.v23.n2.a15
Ai Guan
We introduce a notion of left homotopy for Maurer--Cartan elements in $L_{infty}$-algebras and $A_{infty}$-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger--Stasheff's theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincar'e lemma for differential forms taking values in an $L_{infty}$-algebra.
{"title":"Gauge equivalence for complete $L_infty$-algebras","authors":"Ai Guan","doi":"10.4310/hha.2021.v23.n2.a15","DOIUrl":"https://doi.org/10.4310/hha.2021.v23.n2.a15","url":null,"abstract":"We introduce a notion of left homotopy for Maurer--Cartan elements in $L_{infty}$-algebras and $A_{infty}$-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger--Stasheff's theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincar'e lemma for differential forms taking values in an $L_{infty}$-algebra.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"176 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85545173","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}
We show that the natural algebraic structure of the singular chains on a path connected topological space determines the fundamental group functorially. Moreover, we describe a notion of weak equivalence for the relevant algebraic structure under which the data of the fundamental group is preserved.
{"title":"Singular chains and the fundamental group","authors":"M. Rivera, M. Zeinalian","doi":"10.4064/fm734-6-2020","DOIUrl":"https://doi.org/10.4064/fm734-6-2020","url":null,"abstract":"We show that the natural algebraic structure of the singular chains on a path connected topological space determines the fundamental group functorially. Moreover, we describe a notion of weak equivalence for the relevant algebraic structure under which the data of the fundamental group is preserved.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77792692","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}
We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) rightarrow THH(E(2))rightarrow overline{THH}(E(2))$ and describe $overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 leq i leq n$. In particular, we compute $K(i)_*THH(E(n))$.
我们提供了一个完整的描述$THH(E(2))$的假设下，约翰逊-威尔逊光谱$E(2)$在一个选定的奇素数携带$E_infty$ -结构。我们还将$THH(E(2))$放在共纤维序列$E(2) rightarrow THH(E(2))rightarrow overline{THH}(E(2))$中，并假设$E(2)$是一个$E_3$环谱来描述$overline{THH}(E(2))$。我们陈述了关于所有$n$和$0 leq i leq n$的$THH(E(n))$的$K(i)$ -局部行为的一般结果。特别地，我们计算$K(i)_*THH(E(n))$。
{"title":"Towards topological Hochschild homology of\u0000Johnson–Wilson spectra","authors":"Christian Ausoni, Birgit Richter","doi":"10.2140/agt.2020.20.375","DOIUrl":"https://doi.org/10.2140/agt.2020.20.375","url":null,"abstract":"We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) rightarrow THH(E(2))rightarrow overline{THH}(E(2))$ and describe $overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 leq i leq n$. In particular, we compute $K(i)_*THH(E(n))$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72995445","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}
This paper is a continuation of the version I of the same title, which intends to clarify and expand the results in the last chapter of `the green book' by the second author. In particular, we give the stable homotopy groups of $p$-local spectra $T(m)_{(1)}$ for $m>0$. This is a part of a program to compute the $p$-components of $pi_{*}(S^{0})$ through dimension $2p^{4}(p-1)$ for $p>2$. We will refer to the results from the version I freely as if they were in the first four sections of this paper, which begins with section 5.
{"title":"The Method of Infinite Descent in Stable Homotopy Theory II","authors":"Hirofumi Nakai, D. Ravenel","doi":"10.1090/conm/293/04951","DOIUrl":"https://doi.org/10.1090/conm/293/04951","url":null,"abstract":"This paper is a continuation of the version I of the same title, which intends to clarify and expand the results in the last chapter of `the green book' by the second author. In particular, we give the stable homotopy groups of $p$-local spectra $T(m)_{(1)}$ for $m>0$. This is a part of a program to compute the $p$-components of $pi_{*}(S^{0})$ through dimension $2p^{4}(p-1)$ for $p>2$. We will refer to the results from the version I freely as if they were in the first four sections of this paper, which begins with section 5.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"114 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75740853","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}
Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We will show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tmf_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.
{"title":"Topological modular forms with level structure: Decompositions and duality","authors":"Lennart Meier","doi":"10.1090/tran/8514","DOIUrl":"https://doi.org/10.1090/tran/8514","url":null,"abstract":"Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We will show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tmf_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83653973","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}
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