Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a12
Gregory Lupton, Samuel Bruce Smith
We show a Gottlieb element in the rational homotopy of a simply connected space $X$ implies a structural result for the Sullivan minimal model, with different results depending on parity. In the even-degree case, we prove a rational Gottlieb element is a terminal homotopy element. This fact allows us to complete an argument of Dupont to prove an even-degree Gottlieb element gives a free factor in the rational cohomology of a formal space of finite type. We apply the odd-degree result to affirm a special case of the $2N$-conjecture on Gottlieb elements of a finite complex. We combine our results to make a contribution to the realization problem for the classifying space $Bmathrm{aut}_1(X)$. We prove a simply connected space $X$ satisfying $Bmathrm{aut}_1(X_{mathbb{Q}}) simeq S_{mathbb{Q}}^{2n}$ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree $2n-1$ for $n= 1, 2, 3.$
{"title":"The structuring effect of a Gottlieb element on the Sullivan model of a space","authors":"Gregory Lupton, Samuel Bruce Smith","doi":"10.4310/hha.2023.v25.n2.a12","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a12","url":null,"abstract":"We show a Gottlieb element in the rational homotopy of a simply connected space $X$ implies a structural result for the Sullivan minimal model, with different results depending on parity. In the even-degree case, we prove a rational Gottlieb element is a terminal homotopy element. This fact allows us to complete an argument of Dupont to prove an even-degree Gottlieb element gives a free factor in the rational cohomology of a formal space of finite type. We apply the odd-degree result to affirm a special case of the $2N$-conjecture on Gottlieb elements of a finite complex. We combine our results to make a contribution to the realization problem for the classifying space $Bmathrm{aut}_1(X)$. We prove a simply connected space $X$ satisfying $Bmathrm{aut}_1(X_{mathbb{Q}}) simeq S_{mathbb{Q}}^{2n}$ must have infinite-dimensional rational homotopy and vanishing rational Gottlieb elements above degree $2n-1$ for $n= 1, 2, 3.$","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n1.a15
E. Herscovich
{"title":"Cyclic $A_infty$-algebras and cyclic homology","authors":"E. Herscovich","doi":"10.4310/hha.2023.v25.n1.a15","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a15","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n1.a16
Gabriel Angelini-Knoll
{"title":"Complex orientations and $mathrm{TP}$ of complete DVRs","authors":"Gabriel Angelini-Knoll","doi":"10.4310/hha.2023.v25.n1.a16","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a16","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a5
Marek Golasiński
{"title":"The homotopy solvability of compact Lie groups and homogenous topological spaces","authors":"Marek Golasiński","doi":"10.4310/hha.2023.v25.n2.a5","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a5","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136004371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n1.a5
P. Hu, I. Kríz, Yunzeng Lu
{"title":"Stable equivariant complex cobordism of the symmetric group on three elements","authors":"P. Hu, I. Kríz, Yunzeng Lu","doi":"10.4310/hha.2023.v25.n1.a5","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a5","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a8
Lukas Katthän, Sean Tilson
Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;mathbb{F}_p)$ for any prime $p$ and $n leq 4$ up to possible multiplicative extensions. In order to accomplish this we show that the Kunneth spectral sequence based on an $E_3$-algebra $R$ is multiplicative when the $R$-modules in question are commutative $S$-algebras. We then apply this result by working over $BP$ which is known to be an $E_4$-algebra.
{"title":"The homology of connective Morava $E$-theory with coefficients in $mathbb{F}_p$","authors":"Lukas Katthän, Sean Tilson","doi":"10.4310/hha.2023.v25.n2.a8","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a8","url":null,"abstract":"Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_*(e_n;mathbb{F}_p)$ for any prime $p$ and $n leq 4$ up to possible multiplicative extensions. In order to accomplish this we show that the Kunneth spectral sequence based on an $E_3$-algebra $R$ is multiplicative when the $R$-modules in question are commutative $S$-algebras. We then apply this result by working over $BP$ which is known to be an $E_4$-algebra.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a9
Victor Carmona
We adopt semimodel categories to extend fundamental results related to Bousfield localizations of model categories. More specifically, we generalize Bousfield-Friedlander Theorem and Hirschhorn Localization Theorem of cellular model categories to settings where their classical formulation does not apply. We use such results to answer, in the world of semimodel categories, an open problem posed by May-Ponto about the existence of Bousfield localizations for Hurewicz and mixed type model structures.
{"title":"When Bousfield localizations and homotopy idempotent functors meet again","authors":"Victor Carmona","doi":"10.4310/hha.2023.v25.n2.a9","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a9","url":null,"abstract":"We adopt semimodel categories to extend fundamental results related to Bousfield localizations of model categories. More specifically, we generalize Bousfield-Friedlander Theorem and Hirschhorn Localization Theorem of cellular model categories to settings where their classical formulation does not apply. We use such results to answer, in the world of semimodel categories, an open problem posed by May-Ponto about the existence of Bousfield localizations for Hurewicz and mixed type model structures.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a3
Carlos Segovia
The 1+1 G-cobordism category, with G a finite group, is important in the construction of G-topological field theories which are completely determined by a G-Frobenius algebra. We give a description of the classifying space of this category generalizing the work of Ulrike Tillmann. Moreover, we compute the connected components and the fundamental group of this classifying space and we give a complete description of the classifying spaces of some important subcategories. Finally, we present some relations between the rank of the fundamental group of the G-cobordism category and the number of subgroups of the group G.
{"title":"The classifying space of the 1+1 dimensional $G$-cobordism category","authors":"Carlos Segovia","doi":"10.4310/hha.2023.v25.n2.a3","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a3","url":null,"abstract":"The 1+1 G-cobordism category, with G a finite group, is important in the construction of G-topological field theories which are completely determined by a G-Frobenius algebra. We give a description of the classifying space of this category generalizing the work of Ulrike Tillmann. Moreover, we compute the connected components and the fundamental group of this classifying space and we give a complete description of the classifying spaces of some important subcategories. Finally, we present some relations between the rank of the fundamental group of the G-cobordism category and the number of subgroups of the group G.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135701253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n2.a6
Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier
The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $mathcal{E}$ is a presentable $infty$-category, then we give a concrete formula for the constant hypersheaf functor $mathcal{E}to mathrm{Sh}^{mathrm{hyp}}(X;mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical K"unneth formula for locally constant hypersheaves.
{"title":"The homotopy-invariance of constructible sheaves","authors":"Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier","doi":"10.4310/hha.2023.v25.n2.a6","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a6","url":null,"abstract":"The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $mathcal{E}$ is a presentable $infty$-category, then we give a concrete formula for the constant hypersheaf functor $mathcal{E}to mathrm{Sh}^{mathrm{hyp}}(X;mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical K\"unneth formula for locally constant hypersheaves.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/hha.2023.v25.n1.a6
Lóránt Szegedy
We classify invertible 2-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. By a recent result of Kreck, Stolz and Teichner the first homotopy groups are given by the so called SKK groups. We compute them explicitly using the combinatorial model of framed and $r$-spin surfaces of Novak, Runkel and the author.
{"title":"On invertible $2$-dimensional framed and $r$-spin topological field theories","authors":"Lóránt Szegedy","doi":"10.4310/hha.2023.v25.n1.a6","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a6","url":null,"abstract":"We classify invertible 2-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. By a recent result of Kreck, Stolz and Teichner the first homotopy groups are given by the so called SKK groups. We compute them explicitly using the combinatorial model of framed and $r$-spin surfaces of Novak, Runkel and the author.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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