Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3531
Keyao Peng
In this paper, we compute the (total) Milnor-Witt motivic cohomology of the complement of a hyperplane arrangement in an affine space.
{"title":"Milnor–Witt motivic cohomology of complements of hyperplane arrangements","authors":"Keyao Peng","doi":"10.2140/agt.2023.23.3531","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3531","url":null,"abstract":"In this paper, we compute the (total) Milnor-Witt motivic cohomology of the complement of a hyperplane arrangement in an affine space.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"30 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3553
Lennart Meier
The goal of this article is to construct and study connective versions of topological modular forms of higher level like $mathrm{tmf}_1(n)$. In particular, we use them to realize Hirzebruch's level-$n$ genus as a map of ring spectra.
{"title":"Connective models for topological modular forms of level n","authors":"Lennart Meier","doi":"10.2140/agt.2023.23.3553","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3553","url":null,"abstract":"The goal of this article is to construct and study connective versions of topological modular forms of higher level like $mathrm{tmf}_1(n)$. In particular, we use them to realize Hirzebruch's level-$n$ genus as a map of ring spectra.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"35 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135723944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3417
Andrew Putman
We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.
{"title":"Partial Torelli groups and homological stability","authors":"Andrew Putman","doi":"10.2140/agt.2023.23.3417","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3417","url":null,"abstract":"We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"35 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135723943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3655
Drew Heard
Following a suggestion of Hovey and Strickland, we study the category of $K(k) vee K(k+1) vee cdots vee K(n)$-local spectra. When $k = 0$, this is equivalent to the category of $E(n)$-local spectra, while for $k = n$, this is the category of $K(n)$-local spectra, both of which have been studied in detail by Hovey and Strickland. Based on their ideas, we classify the localizing and colocalizing subcategories, and give characterizations of compact and dualizable objects. We construct an Adams type spectral sequence and show that when $p gg n$ it collapses with a horizontal vanishing line above filtration degree $n^2+n-k$ at the $E_2$-page for the sphere spectrum. We then study the Picard group of $K(k) vee K(k+1) vee cdots vee K(n)$-local spectra, showing that this group is algebraic, in a suitable sense, when $p gg n$. We also consider a version of Gross--Hopkins duality in this category. A key concept throughout is the use of descent.
{"title":"The Spk,n–local stable homotopy category","authors":"Drew Heard","doi":"10.2140/agt.2023.23.3655","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3655","url":null,"abstract":"Following a suggestion of Hovey and Strickland, we study the category of $K(k) vee K(k+1) vee cdots vee K(n)$-local spectra. When $k = 0$, this is equivalent to the category of $E(n)$-local spectra, while for $k = n$, this is the category of $K(n)$-local spectra, both of which have been studied in detail by Hovey and Strickland. Based on their ideas, we classify the localizing and colocalizing subcategories, and give characterizations of compact and dualizable objects. We construct an Adams type spectral sequence and show that when $p gg n$ it collapses with a horizontal vanishing line above filtration degree $n^2+n-k$ at the $E_2$-page for the sphere spectrum. We then study the Picard group of $K(k) vee K(k+1) vee cdots vee K(n)$-local spectra, showing that this group is algebraic, in a suitable sense, when $p gg n$. We also consider a version of Gross--Hopkins duality in this category. A key concept throughout is the use of descent.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"2 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3805
Connor Sell
There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This paper provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type.
{"title":"Cusps and commensurability classes of hyperbolic 4–manifolds","authors":"Connor Sell","doi":"10.2140/agt.2023.23.3805","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3805","url":null,"abstract":"There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This paper provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135723947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3763
Paula Truöl
We provide explicit formulas for the integer-valued smooth concordance invariant $upsilon(K) = Upsilon_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsvath, Stipsicz and Szabo, by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. As an application, we show that for positive 3-braid knots $K$ several alternating distances all equal the sum $g(K) + upsilon(K)$, where $g(K)$ denotes the 3-genus of $K$. In particular, we compute the alternation number, the dealternating number and the Turaev genus for all positive 3-braid knots. We also provide upper and lower bounds on the alternation number and dealternating number of every 3-braid knot which differ by 1.
{"title":"The upsilon invariant at 1 of 3–braid knots","authors":"Paula Truöl","doi":"10.2140/agt.2023.23.3763","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3763","url":null,"abstract":"We provide explicit formulas for the integer-valued smooth concordance invariant $upsilon(K) = Upsilon_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsvath, Stipsicz and Szabo, by constructing cobordisms between 3-braid knots and (connected sums of) torus knots. As an application, we show that for positive 3-braid knots $K$ several alternating distances all equal the sum $g(K) + upsilon(K)$, where $g(K)$ denotes the 3-genus of $K$. In particular, we compute the alternation number, the dealternating number and the Turaev genus for all positive 3-braid knots. We also provide upper and lower bounds on the alternation number and dealternating number of every 3-braid knot which differ by 1.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"35 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135723945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3587
Panagiotis Tselekidis
We prove a new inequality for the asymptotic dimension of HNN-extensions. We deduce that the asymptotic dimension of every one relator group is at most two, confirming a conjecture of A.Dranishnikov. As another corollary we calculate the exact asymptotic dimension of Right-angled Artin groups. We prove a new upper bound for the asymptotic dimension of fundamental groups of graphs of groups. This leads to a partial result on the asymptotic Morita conjecture for finitely generated groups.
{"title":"Asymptotic dimension of graphs of groups and one-relator groups","authors":"Panagiotis Tselekidis","doi":"10.2140/agt.2023.23.3587","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3587","url":null,"abstract":"We prove a new inequality for the asymptotic dimension of HNN-extensions. We deduce that the asymptotic dimension of every one relator group is at most two, confirming a conjecture of A.Dranishnikov. As another corollary we calculate the exact asymptotic dimension of Right-angled Artin groups. We prove a new upper bound for the asymptotic dimension of fundamental groups of graphs of groups. This leads to a partial result on the asymptotic Morita conjecture for finitely generated groups.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"37 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135724069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3745
Naoki Kitazawa, Osamu Saeki
We show that a closed orientable 3--dimensional manifold admits a round fold map into the plane, i.e. a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, i.e. round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.
{"title":"Round fold maps on 3–manifolds","authors":"Naoki Kitazawa, Osamu Saeki","doi":"10.2140/agt.2023.23.3745","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3745","url":null,"abstract":"We show that a closed orientable 3--dimensional manifold admits a round fold map into the plane, i.e. a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, i.e. round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"1 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3615
Harrison Bray, Richard Canary, Lien-Yung Kao
In this paper, we produce a mapping class group invariant pressure metric on the space QF(S) of quasiconformal deformations of a co-finite area Fuchsian group uniformizing a surface S. Our pressure metric arises from an analytic pressure form on QF(S) which is degenerate only on pure bending vectors on the Fuchsian locus. Our techniques also show that the Hausdorff dimension of the limit set varies analytically over QF(S).
{"title":"Pressure metrics for deformation spaces of quasifuchsian groups with parabolics","authors":"Harrison Bray, Richard Canary, Lien-Yung Kao","doi":"10.2140/agt.2023.23.3615","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3615","url":null,"abstract":"In this paper, we produce a mapping class group invariant pressure metric on the space QF(S) of quasiconformal deformations of a co-finite area Fuchsian group uniformizing a surface S. Our pressure metric arises from an analytic pressure form on QF(S) which is degenerate only on pure bending vectors on the Fuchsian locus. Our techniques also show that the Hausdorff dimension of the limit set varies analytically over QF(S).","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"30 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-05DOI: 10.2140/agt.2023.23.3835
Shengkui Ye, Yanxin Zhao
We prove that the group $mathrm{QI}^{+}(mathbb{R})$ of orientation-preserving quasi-isometries of the real line is a left-orderable, non-simple group, which cannot act effectively on the real line $mathbb{R}.$
{"title":"The group of quasi-isometries of the real line cannot act effectively on the line","authors":"Shengkui Ye, Yanxin Zhao","doi":"10.2140/agt.2023.23.3835","DOIUrl":"https://doi.org/10.2140/agt.2023.23.3835","url":null,"abstract":"We prove that the group $mathrm{QI}^{+}(mathbb{R})$ of orientation-preserving quasi-isometries of the real line is a left-orderable, non-simple group, which cannot act effectively on the real line $mathbb{R}.$","PeriodicalId":50826,"journal":{"name":"Algebraic and Geometric Topology","volume":"2 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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