In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of arXiv:1910.14608, which are based on Quillen's rational homotopy theory. According to Sullivan, the rational homotopy type of a nilpotent space $X$ with finite Betti numbers is completely determined by a commutative differential graded algebra $A$ modelling the cup product on rational cohomology. In this article we extend this correspondence between topology and algebra to parametrised stable homotopy theory: for a space $X$ corresponding to the cdga $A$, we prove an equivalence between specific rational homotopy categories for parametrised spectra over $X$ and for differential graded $A$-modules. While not full, the rational homotopy categories we consider contain a large class of parametrised spectra. The simplicity of the approach that we develop enables direct calculations in parametrised stable homotopy theory using differential graded modules. To illustrate the usefulness of our approach, we build a comprehensive dictionary of algebraic translations of topological constructions; providing algebraic models for base change functors, fibrewise stabilisations, parametrised Postnikov sections, fibrewise smash products, and complexes of fibrewise stable maps.
在本文中，我们推广了Sullivan的PL de Rham理论，得到了参数化谱的有理同伦理论的简单代数模型。这简化并补充了arXiv:1910.14608基于Quillen的有理同伦理论的结果。根据Sullivan的研究，有限Betti数的幂零空间X的有理同伦类型完全由在有理上同调上对杯积建模的交换微分梯度代数a决定。在本文中，我们将拓扑与代数的这种对应关系推广到参数化稳定同伦理论:对于对应于cdga的空间X$，我们证明了X$上的参数化谱和微分梯度a $-模的特定有理同伦范畴之间的等价性。虽然不是满的，但我们考虑的有理同伦范畴包含了一大类参数化谱。我们开发的方法的简单性使我们能够使用微分梯度模块直接计算参数化稳定同伦理论。为了说明我们的方法的有用性，我们建立了一个拓扑结构的代数翻译的综合字典;提供了碱基变化函子，纤维稳定，参数化波士尼科夫截面，纤维粉碎产物和纤维稳定映射的复合体的代数模型。
{"title":"Strict algebraic models for rational parametrised spectra, I","authors":"V. Braunack-Mayer","doi":"10.2140/AGT.2021.21.917","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.917","url":null,"abstract":"In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of arXiv:1910.14608, which are based on Quillen's rational homotopy theory. \u0000According to Sullivan, the rational homotopy type of a nilpotent space $X$ with finite Betti numbers is completely determined by a commutative differential graded algebra $A$ modelling the cup product on rational cohomology. In this article we extend this correspondence between topology and algebra to parametrised stable homotopy theory: for a space $X$ corresponding to the cdga $A$, we prove an equivalence between specific rational homotopy categories for parametrised spectra over $X$ and for differential graded $A$-modules. While not full, the rational homotopy categories we consider contain a large class of parametrised spectra. The simplicity of the approach that we develop enables direct calculations in parametrised stable homotopy theory using differential graded modules. \u0000To illustrate the usefulness of our approach, we build a comprehensive dictionary of algebraic translations of topological constructions; providing algebraic models for base change functors, fibrewise stabilisations, parametrised Postnikov sections, fibrewise smash products, and complexes of fibrewise stable maps.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"18 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91498032","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}
Gromov and Ivanov established an analogue of Leray's theorem on cohomology of contractible covers for bounded cohomology of amenable covers. We present an alternative proof of this fact, using classifying spaces of families of subgroups.
{"title":"Bounded cohomology of amenable covers via classifying spaces","authors":"C. Loeh, Roman Sauer","doi":"10.4171/lem/66-1/2-8","DOIUrl":"https://doi.org/10.4171/lem/66-1/2-8","url":null,"abstract":"Gromov and Ivanov established an analogue of Leray's theorem on cohomology of contractible covers for bounded cohomology of amenable covers. We present an alternative proof of this fact, using classifying spaces of families of subgroups.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88024441","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 : 2019-10-14DOI: 10.2140/agt.2020.20.3147
F. Garc'ia, Tobias Dyckerhoff, Walker H. Stern
In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.
{"title":"A relative 2–nerve","authors":"F. Garc'ia, Tobias Dyckerhoff, Walker H. Stern","doi":"10.2140/agt.2020.20.3147","DOIUrl":"https://doi.org/10.2140/agt.2020.20.3147","url":null,"abstract":"In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76057606","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 give a fully faithful integral model for spaces in terms of $mathbb{E}_{infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete $mathbb{E}_{infty}$-rings for each prime $p$. Using this, we show that the data of a space $X$ is the data of its Spanier-Whitehead dual as an $mathbb{E}_{infty}$-ring together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen's $Q$-construction acts on the $infty$-category of $mathbb{E}_{infty}$-rings with "genuine equivariant multiplication," which we call global algebras. The second is a "pre-group-completed" variant of algebraic $K$-theory which we call partial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of $mathbb{F}_p$ up to $p$-completion.
{"title":"Integral models for spaces via the higher Frobenius","authors":"Allen Yuan","doi":"10.1090/jams/998","DOIUrl":"https://doi.org/10.1090/jams/998","url":null,"abstract":"We give a fully faithful integral model for spaces in terms of $mathbb{E}_{infty}$-ring spectra and the Nikolaus-Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete $mathbb{E}_{infty}$-rings for each prime $p$. Using this, we show that the data of a space $X$ is the data of its Spanier-Whitehead dual as an $mathbb{E}_{infty}$-ring together with a trivialization of the Frobenius action after completion at each prime. \u0000In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen's $Q$-construction acts on the $infty$-category of $mathbb{E}_{infty}$-rings with \"genuine equivariant multiplication,\" which we call global algebras. The second is a \"pre-group-completed\" variant of algebraic $K$-theory which we call partial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of $mathbb{F}_p$ up to $p$-completion.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82111521","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 term "motivic Moore spectrum" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the motivic sphere spectrum descends to a ring structure on a motivic Moore spectrum. This discussion requires an understanding of some Toda brackets in the motivic stable homotopy groups of spheres.
{"title":"Remarks on motivic Moore spectra.","authors":"O. Röndigs","doi":"10.1090/CONM/745/15026","DOIUrl":"https://doi.org/10.1090/CONM/745/15026","url":null,"abstract":"The term \"motivic Moore spectrum\" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the motivic sphere spectrum descends to a ring structure on a motivic Moore spectrum. This discussion requires an understanding of some Toda brackets in the motivic stable homotopy groups of spheres.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83454132","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 consider a functor from the category of groups to itself $Gmapsto mathbb Z_infty G$ that we call right exact $mathbb Z$-completion of a group. It is connected with the pronilpotent completion $hat G$ by the short exact sequence $1to {varprojlim}^1: M_n G to mathbb Z_infty G to hat G to 1,$ where $M_n G$ is $n$-th Baer invariant of $G.$ We prove that $mathbb Z_infty pi_1(X)$ is an invariant of homological equivalence of a space $X$. Moreover, we prove an analogue of Stallings' theorem: if $Gto G'$ is a 2-connected group homomorphism, then $mathbb Z_infty Gcong mathbb Z_infty G'.$ We give examples of $3$-manifolds $X,Y$ such that $ hat{pi_1(X)}cong hat{pi_1( Y)}$ but $mathbb Z_infty pi_1(X)not cong mathbb Z_infty pi_1(Y).$ We prove that for a finitely generated group $G$ we have $(mathbb Z_infty G)/ gamma_omega= hat G.$ So the difference between $hat G$ and $mathbb Z_infty G$ lies in $gamma_omega.$ This allows us to treat $mathbb Z_infty pi_1(X)$ as a transfinite invariant of $X.$ The advantage of our approach is that it can be used not only for $3$-manifolds but for arbitrary spaces.
{"title":"Right exact group completion as a transfinite invariant of homology equivalence","authors":"S. Ivanov, R. Mikhailov","doi":"10.2140/AGT.2021.21.447","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.447","url":null,"abstract":"We consider a functor from the category of groups to itself $Gmapsto mathbb Z_infty G$ that we call right exact $mathbb Z$-completion of a group. It is connected with the pronilpotent completion $hat G$ by the short exact sequence $1to {varprojlim}^1: M_n G to mathbb Z_infty G to hat G to 1,$ where $M_n G$ is $n$-th Baer invariant of $G.$ We prove that $mathbb Z_infty pi_1(X)$ is an invariant of homological equivalence of a space $X$. Moreover, we prove an analogue of Stallings' theorem: if $Gto G'$ is a 2-connected group homomorphism, then $mathbb Z_infty Gcong mathbb Z_infty G'.$ We give examples of $3$-manifolds $X,Y$ such that $ hat{pi_1(X)}cong hat{pi_1( Y)}$ but $mathbb Z_infty pi_1(X)not cong mathbb Z_infty pi_1(Y).$ We prove that for a finitely generated group $G$ we have $(mathbb Z_infty G)/ gamma_omega= hat G.$ So the difference between $hat G$ and $mathbb Z_infty G$ lies in $gamma_omega.$ This allows us to treat $mathbb Z_infty pi_1(X)$ as a transfinite invariant of $X.$ The advantage of our approach is that it can be used not only for $3$-manifolds but for arbitrary spaces.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83711155","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 some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.
{"title":"Relative phantom maps and rational homotopy","authors":"D. Kishimoto, Takahiro Matsushita","doi":"10.1090/proc/15431","DOIUrl":"https://doi.org/10.1090/proc/15431","url":null,"abstract":"We generalize some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86525569","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 obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted cohomology groups. As an application of our results we prove a tangent prolongation of Simpson's theorem, characterising tangent $infty$-categories of presentable $infty$-categories as accessible localisations of $infty$-categories of presheaves of parametrised spectra. Applying these results to the homotopy theory of smooth $infty$-stacks produces well-behaved (symmetric monoidal) model categories of smooth parametrised spectra. These models provide a concrete foundation for studying twisted differential cohomology, incorporating previous work of Bunke and Nikolaus.
{"title":"Combinatorial parametrised spectra","authors":"V. Braunack-Mayer","doi":"10.2140/AGT.2021.21.801","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.801","url":null,"abstract":"We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted cohomology groups. \u0000As an application of our results we prove a tangent prolongation of Simpson's theorem, characterising tangent $infty$-categories of presentable $infty$-categories as accessible localisations of $infty$-categories of presheaves of parametrised spectra. Applying these results to the homotopy theory of smooth $infty$-stacks produces well-behaved (symmetric monoidal) model categories of smooth parametrised spectra. These models provide a concrete foundation for studying twisted differential cohomology, incorporating previous work of Bunke and Nikolaus.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89932926","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}
Let $K$ be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller--Morita--Mumford classes for smooth bundles with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism $Diff(K)to pi_0 Diff(K)$ does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.
{"title":"Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem","authors":"Jeffrey Giansiracusa, A. Kupers, Bena Tshishiku","doi":"10.2140/TUNIS.2021.3.75","DOIUrl":"https://doi.org/10.2140/TUNIS.2021.3.75","url":null,"abstract":"Let $K$ be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller--Morita--Mumford classes for smooth bundles with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism $Diff(K)to pi_0 Diff(K)$ does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"94 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73618438","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 purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the first author is fully homotopical. This implies that every presentably symmetric monoidal $infty$-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.
{"title":"Homotopy Invariance of Convolution Products","authors":"S. Sagave, S. Schwede","doi":"10.1093/IMRN/RNZ334","DOIUrl":"https://doi.org/10.1093/IMRN/RNZ334","url":null,"abstract":"The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the first author is fully homotopical. This implies that every presentably symmetric monoidal $infty$-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76274415","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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