Journal of Topology最新文献

{"title":"Knotted families from graspers","authors":"Danica Kosanović","doi":"10.1112/topo.12337","DOIUrl":"https://doi.org/10.1112/topo.12337","url":null,"abstract":"<p>For any smooth manifold <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> of dimension <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 4$</annotation>\u0000 </semantics></math>, we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math>, in every degree that is a multiple of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$d-3$</annotation>\u0000 </semantics></math>, and show that they are detected in the Taylor tower of Goodwillie and Weiss. The classes are obtained from families of string links constructed in the <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>-ball.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12337","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902759","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":"Picard sheaves, local Brauer groups, and topological modular forms","authors":"Benjamin Antieau,&nbsp;Lennart Meier,&nbsp;Vesna Stojanoska","doi":"10.1112/topo.12333","DOIUrl":"https://doi.org/10.1112/topo.12333","url":null,"abstract":"<p>We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of <span></span><math>\u0000 <semantics>\u0000 <mi>TMF</mi>\u0000 <annotation>$mathrm{TMF}$</annotation>\u0000 </semantics></math> is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of <span></span><math>\u0000 <semantics>\u0000 <mi>TMF</mi>\u0000 <annotation>$mathrm{TMF}$</annotation>\u0000 </semantics></math> and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2-torsion group.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12333","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140844924","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":"Convex co-compact representations of 3-manifold groups","authors":"Mitul Islam,&nbsp;Andrew Zimmer","doi":"10.1112/topo.12332","DOIUrl":"https://doi.org/10.1112/topo.12332","url":null,"abstract":"<p>A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean <span></span><math>\u0000 <semantics>\u0000 <mo>×</mo>\u0000 <annotation>$times$</annotation>\u0000 </semantics></math> Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12332","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140818858","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":"Koszul self-duality of manifolds","authors":"Connor Malin","doi":"10.1112/topo.12334","DOIUrl":"https://doi.org/10.1112/topo.12334","url":null,"abstract":"<p>We show that Koszul duality for operads in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Top</mi>\u0000 <mo>,</mo>\u0000 <mo>×</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(mathrm{Top},times)$</annotation>\u0000 </semantics></math> can be expressed via generalized Thom complexes. As an application, we prove the Koszul self-duality of the right module <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>E</mi>\u0000 <mi>M</mi>\u0000 </msub>\u0000 <annotation>$E_M$</annotation>\u0000 </semantics></math> associated to a framed manifold <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math>. We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140808033","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":"Brane structures in microlocal sheaf theory","authors":"Xin Jin,&nbsp;David Treumann","doi":"10.1112/topo.12325","DOIUrl":"10.1112/topo.12325","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> be an exact Lagrangian submanifold of a cotangent bundle <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <annotation>$T^* M$</annotation>\u0000 </semantics></math>, asymptotic to a Legendrian submanifold <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Λ</mi>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <annotation>$Lambda subset T^{infty } M$</annotation>\u0000 </semantics></math>. We study a locally constant sheaf of <math>\u0000 <semantics>\u0000 <mi>∞</mi>\u0000 <annotation>$infty$</annotation>\u0000 </semantics></math>-categories on <math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math>, called the sheaf of brane structures or <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Brane</mi>\u0000 <mi>L</mi>\u0000 </msub>\u0000 <annotation>$mathrm{Brane}_L$</annotation>\u0000 </semantics></math>. Its fiber is the <math>\u0000 <semantics>\u0000 <mi>∞</mi>\u0000 <annotation>$infty$</annotation>\u0000 </semantics></math>-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 <mo>(</mo>\u0000 <mi>L</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>Brane</mi>\u0000 <mi>L</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Gamma (L,mathrm{Brane}_L)$</annotation>\u0000 </semantics></math> to the <math>\u0000 <semantics>\u0000 <mi>∞</mi>\u0000 <annotation>$infty$</annotation>\u0000 </semantics></math>-category of sheaves of spectra on <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> with singular support in <math>\u0000 <semantics>\u0000 <mi>Λ</mi>\u0000 <annotation>$Lambda$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12325","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124243","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":"Equivariant Lagrangian Floer homology via cotangent bundles of \u0000 \u0000 \u0000 E\u0000 \u0000 G\u0000 N\u0000 \u0000 \u0000 $EG_N$","authors":"Guillem Cazassus","doi":"10.1112/topo.12328","DOIUrl":"https://doi.org/10.1112/topo.12328","url":null,"abstract":"&lt;p&gt;We provide a construction of equivariant Lagrangian Floer homology &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$HF_G(L_0, L_1)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, for a compact Lie group &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; acting on a symplectic manifold &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in a Hamiltonian fashion, and a pair of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-Lagrangian submanifolds &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L_0, L_1 subset M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$EG$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$H^*(BG)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-bimodules. In the case w","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12328","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140114186","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":"An \u0000 \u0000 h\u0000 $h$\u0000 -principle for embeddings transverse to a contact structure","authors":"Robert Cardona,&nbsp;Francisco Presas","doi":"10.1112/topo.12326","DOIUrl":"https://doi.org/10.1112/topo.12326","url":null,"abstract":"<p>Given a class of embeddings into a contact or a symplectic manifold, we give a sufficient condition, that we call isocontact or isosymplectic realization, for this class to satisfy a general <math>\u0000 <semantics>\u0000 <mi>h</mi>\u0000 <annotation>$h$</annotation>\u0000 </semantics></math>-principle. The flexibility follows from the <math>\u0000 <semantics>\u0000 <mi>h</mi>\u0000 <annotation>$h$</annotation>\u0000 </semantics></math>-principles for isocontact and isosymplectic embeddings, it provides a framework for classical results, and we give two new applications. Our main result is that embeddings transverse to a contact structure satisfy a full <math>\u0000 <semantics>\u0000 <mi>h</mi>\u0000 <annotation>$h$</annotation>\u0000 </semantics></math>-principle in two cases: if the complement of the embedding is overtwisted, or when the intersection of the image of the formal derivative with the contact structure is strictly contained in a proper symplectic subbundle. We illustrate the general framework on symplectic manifolds by studying the universality of Hamiltonian dynamics on regular level sets via a class of embeddings.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140104530","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":"On fillings of contact links of quotient singularities","authors":"Zhengyi Zhou","doi":"10.1112/topo.12329","DOIUrl":"https://doi.org/10.1112/topo.12329","url":null,"abstract":"<p>We study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co-fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non-existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>/</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Z</mi>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>${mathbb {C}}^n/({mathbb {Z}}/2)$</annotation>\u0000 </semantics></math> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$ngeqslant 3$</annotation>\u0000 </semantics></math>, which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mi>U</mi>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$SU(2)$</annotation>\u0000 </semantics></math>, and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the <i>orbifold</i> diffeomorphism type of <i>exact orbifold fillings</i> of contact links of some isolated terminal quotient singularities.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140069661","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":"A characterization of heaviness in terms of relative symplectic cohomology","authors":"Cheuk Yu Mak,&nbsp;Yuhan Sun,&nbsp;Umut Varolgunes","doi":"10.1112/topo.12327","DOIUrl":"https://doi.org/10.1112/topo.12327","url":null,"abstract":"<p>For a compact subset <math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> of a closed symplectic manifold <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>,</mo>\u0000 <mi>ω</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(M, omega)$</annotation>\u0000 </semantics></math>, we prove that <math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> is heavy if and only if its relative symplectic cohomology over the Novikov field is nonzero. As an application, we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results is also included.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12327","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140069758","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":"Some rational homology computations for diffeomorphisms of odd-dimensional manifolds","authors":"Johannes Ebert,&nbsp;Jens Reinhold","doi":"10.1112/topo.12324","DOIUrl":"https://doi.org/10.1112/topo.12324","url":null,"abstract":"&lt;p&gt;We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mo&gt;#&lt;/mo&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;∖&lt;/mo&gt;\u0000 &lt;mi&gt;int&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;D&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$U_{g,1}^n:= #^g(S^n times S^{n+1})setminus mathrm{int}(D^{2n+1})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, for large &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;annotation&gt;$g$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, up to degree &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$n-3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The answer is that it is a free graded commutative algebra on an appropriate set of Miller–Morita–Mumford classes. Our proof goes through the classical three-step procedure: (a) compute the cohomology of the homotopy automorphisms, (b) use surgery to compare this to block diffeomorphisms, and (c) use pseudoisotopy theory and algebraic &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 ","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12324","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139655252","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}