Pub Date : 2019-07-10DOI: 10.4310/HHA.2021.v23.n2.a17
M. Franz
A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces.
{"title":"Homotopy Gerstenhaber formality of Davis–Januszkiewicz spaces","authors":"M. Franz","doi":"10.4310/HHA.2021.v23.n2.a17","DOIUrl":"https://doi.org/10.4310/HHA.2021.v23.n2.a17","url":null,"abstract":"A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces of Davis-Januszkiewicz spaces.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74281334","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 : 2019-06-28DOI: 10.1007/978-3-030-61163-7
Araminta Amabel, A. Kalmykov, L. Muller, Hiro Tanaka
{"title":"Lectures on Factorization Homology, ∞-Categories, and Topological Field Theories","authors":"Araminta Amabel, A. Kalmykov, L. Muller, Hiro Tanaka","doi":"10.1007/978-3-030-61163-7","DOIUrl":"https://doi.org/10.1007/978-3-030-61163-7","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90302756","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 prove the $K$- and $L$-theoretic Farrell--Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $Artimes mathbb{Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell--Jones Conjecture for free-by-cyclic groups.
{"title":"The Farrell–Jones conjecture for normally poly-free groups","authors":"B. Bruck, Dawid Kielak, Xiaolei Wu","doi":"10.1090/proc/15357","DOIUrl":"https://doi.org/10.1090/proc/15357","url":null,"abstract":"We prove the $K$- and $L$-theoretic Farrell--Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $Artimes mathbb{Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell--Jones Conjecture for free-by-cyclic groups.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"395 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89056048","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 study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1leq lleq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $ngeq 2$ we have $mathrm{cd}([P_n,P_n])=n-2$.
{"title":"On the homology of the commutator subgroup of the pure braid group","authors":"Andrea Bianchi","doi":"10.1090/proc/15404","DOIUrl":"https://doi.org/10.1090/proc/15404","url":null,"abstract":"We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1leq lleq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $ngeq 2$ we have $mathrm{cd}([P_n,P_n])=n-2$.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87387505","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 : 2019-05-01DOI: 10.4310/HHA.2020.V22.N1.A21
S. Kaji, S. Kuroki, Eunjeong Lee, D. Suh
In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type.
{"title":"Flag Bott manifolds of general Lie type and their equivariant cohomology rings","authors":"S. Kaji, S. Kuroki, Eunjeong Lee, D. Suh","doi":"10.4310/HHA.2020.V22.N1.A21","DOIUrl":"https://doi.org/10.4310/HHA.2020.V22.N1.A21","url":null,"abstract":"In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82559831","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}
The thesis deals with holomorphic germs $ Phi: (mathbb{C}^2, 0) to (mathbb{C}^3,0) $ singular only at the origin, with a special emphasis on the distinguished class of finitely determined germs. The results are published in two articles (arXiv:1404.2853 and arXiv:1902.01229), joint with Andras Nemethi. In Chapter 3 of the thesis we study the associated immersion $ S^3 looparrowright S^5 $, while Chapter 5 contains an algorithm providing the Milnor fibre boundary of the non-isolated hypersurface singularity determined by the image of $ Phi $. These results create bridges between different areas of complex singularity theory and immersion theory. The background of these topics is summerized in Chapter 1, 2 and 4.
{"title":"On certain complex surface singularities","authors":"Gergo Pintér","doi":"10.15476/ELTE.2018.041","DOIUrl":"https://doi.org/10.15476/ELTE.2018.041","url":null,"abstract":"The thesis deals with holomorphic germs $ Phi: (mathbb{C}^2, 0) to (mathbb{C}^3,0) $ singular only at the origin, with a special emphasis on the distinguished class of finitely determined germs. The results are published in two articles (arXiv:1404.2853 and arXiv:1902.01229), joint with Andras Nemethi. In Chapter 3 of the thesis we study the associated immersion $ S^3 looparrowright S^5 $, while Chapter 5 contains an algorithm providing the Milnor fibre boundary of the non-isolated hypersurface singularity determined by the image of $ Phi $. These results create bridges between different areas of complex singularity theory and immersion theory. The background of these topics is summerized in Chapter 1, 2 and 4.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85960331","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 : 2019-04-22DOI: 10.1016/J.LAA.2019.07.034
E. Mac'ias-Virg'os, M. J. Pereira-Sáez, Daniel Tanr'e
{"title":"Relative singular value decomposition and applications to LS-category","authors":"E. Mac'ias-Virg'os, M. J. Pereira-Sáez, Daniel Tanr'e","doi":"10.1016/J.LAA.2019.07.034","DOIUrl":"https://doi.org/10.1016/J.LAA.2019.07.034","url":null,"abstract":"","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83731623","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 prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears in the statement of the topological Riemann-Roch theorem.
{"title":"Coassembly is a homotopy limit map","authors":"Cary Malkiewich, M. Merling","doi":"10.2140/AKT.2020.5.373","DOIUrl":"https://doi.org/10.2140/AKT.2020.5.373","url":null,"abstract":"We prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears in the statement of the topological Riemann-Roch theorem.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81300356","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 : 2019-03-25DOI: 10.2140/AGT.2020.20.2129
Andrew H. Baker, Tilman Bauer
The Joker is an important finite cyclic module over the mod-$2$ Steenrod algebra $mathcal A$. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory ($2$-compact groups, topological modular forms) and may be of independent interest.
{"title":"The realizability of some finite-length modules over the Steenrod algebra by spaces","authors":"Andrew H. Baker, Tilman Bauer","doi":"10.2140/AGT.2020.20.2129","DOIUrl":"https://doi.org/10.2140/AGT.2020.20.2129","url":null,"abstract":"The Joker is an important finite cyclic module over the mod-$2$ Steenrod algebra $mathcal A$. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory ($2$-compact groups, topological modular forms) and may be of independent interest.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73608470","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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