Journal of Topology最新文献

{"title":"Equivariant knots and knot Floer homology","authors":"Irving Dai, Abhishek Mallick, Matthew Stoffregen","doi":"10.1112/topo.12312","DOIUrl":"10.1112/topo.12312","url":null,"abstract":"We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and extend a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route toward establishing the noncommutativity of the equivariant concordance group.","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42949514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lagrangian cobordism functor in microlocal sheaf theory I","authors":"Wenyuan Li","doi":"10.1112/topo.12310","DOIUrl":"10.1112/topo.12310","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Λ</mi>\u0000 <mo>±</mo>\u0000 </msub>\u0000 <annotation>$Lambda _pm$</annotation>\u0000 </semantics></math> be Legendrian submanifolds in the cosphere bundle <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mrow>\u0000 <mo>∗</mo>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <annotation>$T^{*,infty }M$</annotation>\u0000 </semantics></math>. Given a Lagrangian cobordism <math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> of Legendrians from <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Λ</mi>\u0000 <mo>−</mo>\u0000 </msub>\u0000 <annotation>$Lambda _-$</annotation>\u0000 </semantics></math> to <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Λ</mi>\u0000 <mo>+</mo>\u0000 </msub>\u0000 <annotation>$Lambda _+$</annotation>\u0000 </semantics></math>, we construct a functor <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Φ</mi>\u0000 <mi>L</mi>\u0000 <mo>*</mo>\u0000 </msubsup>\u0000 <mo>:</mo>\u0000 <msubsup>\u0000 <mi>Sh</mi>\u0000 <msub>\u0000 <mi>Λ</mi>\u0000 <mo>+</mo>\u0000 </msub>\u0000 <mi>c</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>→</mo>\u0000 <msubsup>\u0000 <mi>Sh</mi>\u0000 <msub>\u0000 <mi>Λ</mi>\u0000 <mo>−</mo>\u0000 </msub>\u0000 <mi>c</mi>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <msub>\u0000 <mo>⊗</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mo>*</mo>\u0000 </mrow>\u0000 </m","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12310","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46583482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Smoothing finite-order bilipschitz homeomorphisms of 3-manifolds","authors":"Lucien Grillet","doi":"10.1112/topo.12309","DOIUrl":"10.1112/topo.12309","url":null,"abstract":"<p>We show that, for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 <mo>=</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>4000</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$varepsilon =frac{1}{4000}$</annotation>\u0000 </semantics></math>, any action of a finite cyclic group by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mi>ε</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(1+varepsilon )$</annotation>\u0000 </semantics></math>-bilipschitz homeomorphisms on a closed 3-manifold is conjugated to a smooth action.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47990972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The top homology group of the genus 3 Torelli group","authors":"Igor A. Spiridonov","doi":"10.1112/topo.12308","DOIUrl":"10.1112/topo.12308","url":null,"abstract":"<p>The Torelli group of a genus <math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math> oriented surface <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <annotation>$Sigma _g$</annotation>\u0000 </semantics></math> is the subgroup <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <annotation>$mathcal {I}_g$</annotation>\u0000 </semantics></math> of the mapping class group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Mod</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm Mod}(Sigma _g)$</annotation>\u0000 </semantics></math> consisting of all mapping classes that act trivially on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mi>Z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>${rm H}_1(Sigma _g, mathbb {Z})$</annotation>\u0000 </semantics></math>. The quotient group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Mod</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>/</mo>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>${rm Mod}(Sigma _g) / mathcal {I}_g$</annotation>\u0000 </semantics></math> is isomorphic to the symplectic group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Sp</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mi>g</mi>\u0000 <mo>,</mo>\u0000 <mi>Z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm Sp}(2g, mathbb {Z})$</annotation>\u0000 </semantics></math>. The cohomological dimension of the group <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mi>g</mi>\u0000 </msub>\u0000 <annotation>$mathcal {I}_g$</annotation>\u0000 </semantics></math> equ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43984189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical properties of convex cocompact actions in projective space","authors":"Theodore Weisman","doi":"10.1112/topo.12307","DOIUrl":"10.1112/topo.12307","url":null,"abstract":"<p>We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger–Guéritaud–Kassel: we show that convex cocompactness in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {R}mathrm{P}^d$</annotation>\u0000 </semantics></math> is equivalent to an expansion property of the group about its limit set, occurring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups that are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43304379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Automorphisms of procongruence curve and pants complexes","authors":"Marco Boggi,&nbsp;Louis Funar","doi":"10.1112/topo.12306","DOIUrl":"10.1112/topo.12306","url":null,"abstract":"<p>In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47340177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low-dimensional linear representations of mapping class groups","authors":"Mustafa Korkmaz","doi":"10.1112/topo.12305","DOIUrl":"https://doi.org/10.1112/topo.12305","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> be a compact orientable surface of genus <math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math> with marked points in the interior. Franks–Handel (<i>Proc. Amer. Math. Soc</i>. <b>141</b> (2013) 2951–2962)  proved that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>&lt;</mo>\u0000 <mn>2</mn>\u0000 <mi>g</mi>\u0000 </mrow>\u0000 <annotation>$n&lt;2g$</annotation>\u0000 </semantics></math> then the image of a homomorphism from the mapping class group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Mod</mi>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm Mod}(S)$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>GL</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>C</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm GL}(n,{mathbb {C}})$</annotation>\u0000 </semantics></math> is trivial if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$ggeqslant 3$</annotation>\u0000 </semantics></math> and is finite cyclic if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$g=2$</annotation>\u0000 </semantics></math>. The first result is our own proof of this fact. Our second main result shows that for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$ggeqslant 3$</annotation>\u0000 </semantics></math> up to conjugation there are only two homomorphisms from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Mod</mi>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm Mod}(S)$</annotation>\u0000 </semantics></math> to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>GL</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mi>g</mi>\u0000 <mo>,</mo>\u0000 <mi>C</mi>\u0000 <mo>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50132748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symplectic mapping class groups of blowups of tori","authors":"Gleb Smirnov","doi":"10.1112/topo.12304","DOIUrl":"10.1112/topo.12304","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math> be a Kähler form on the real 4-torus <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <annotation>$T^4$</annotation>\u0000 </semantics></math>. Suppose that <math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math> satisfies an irrationality condition that can be achieved by an arbitrarily small perturbation of <math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math>. This note shows that the smoothly trivial symplectic mapping class group of the one-point symplectic blowup of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <mo>,</mo>\u0000 <mi>ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(T^4,omega )$</annotation>\u0000 </semantics></math> is infinitely generated.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12304","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46244923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonnegative scalar curvature on manifolds with at least two ends","authors":"Simone Cecchini,&nbsp;Daniel Räde,&nbsp;Rudolf Zeidler","doi":"10.1112/topo.12303","DOIUrl":"10.1112/topo.12303","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> be an orientable connected <math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>-dimensional manifold with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>∈</mo>\u0000 <mo>{</mo>\u0000 <mn>6</mn>\u0000 <mo>,</mo>\u0000 <mn>7</mn>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$nin lbrace 6,7rbrace$</annotation>\u0000 </semantics></math> and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Y</mi>\u0000 <mo>⊂</mo>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <annotation>$Ysubset M$</annotation>\u0000 </semantics></math> be a two-sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> are either both spin or both nonspin. Using Gromov's <math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math>-bubbles, we show that <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> does not admit a complete metric of psc. We provide an example showing that the spin/nonspin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension 7, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension 2. We deduce as special cases that, if <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$Y$</annotation>\u0000 </semantics></math> does not admit a metric of psc and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dim</mo>\u0000 <mo>(</mo>\u0000 <mi>Y</mi>\u0000 <mo>)</mo>\u0000 <mo>≠</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$dim (Y) ne 4$</annotation>\u0000 </semantics></math>, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <mo>:</mo>\u0000 <mo>=</mo>\u0000 <mi>Y</mi>\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12303","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45500999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Group and Lie algebra filtrations and homotopy groups of spheres","authors":"Laurent Bartholdi,&nbsp;Roman Mikhailov","doi":"10.1112/topo.12301","DOIUrl":"10.1112/topo.12301","url":null,"abstract":"<p>We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>,</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$s,d$</annotation>\u0000 </semantics></math> the torsion of the homotopy group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>π</mi>\u0000 <mi>s</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$pi _s(S^d)$</annotation>\u0000 </semantics></math> into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, as for every prime <math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>, there is some <math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-torsion in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>π</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$pi _{2p}(S^2)$</annotation>\u0000 </semantics></math> by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>π</mi>\u0000 <mn>4</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>Z</mi>\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12301","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44038174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}