Pub Date : 2024-05-01DOI: 10.4310/hha.2024.v26.n1.a16
Martintxo Saralegi-Aranguren
In a previous work we proved the refinement invariance of several intersection (co)homologies existing in the literature. Specifically, we worked with a refinement $f : (X, mathcal{S}) to (X,mathcal{T})$ between two CS‑sets where the strata of $mathcal{S}$ were embedded in the strata of $mathcal{T}$. However, in this paper, we establish that this embedding condition is not a requirement for the refinement invariance property.
{"title":"Addendum to “Refinement invariance of intersection (co)homologies”","authors":"Martintxo Saralegi-Aranguren","doi":"10.4310/hha.2024.v26.n1.a16","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a16","url":null,"abstract":"In a previous work we proved the refinement invariance of several intersection (co)homologies existing in the literature. Specifically, we worked with a refinement $f : (X, mathcal{S}) to (X,mathcal{T})$ between two CS‑sets where the strata of $mathcal{S}$ were embedded in the strata of $mathcal{T}$. However, in this paper, we establish that this embedding condition is not a requirement for the refinement invariance property.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140837422","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 : 2024-05-01DOI: 10.4310/hha.2024.v26.n1.a18
Matthias Franz
We prove that Szczarba’s twisting cochain is comultiplicative. In particular, the induced map from the cobar construction $Omega C(X)$ of the chains on a $1$-reduced simplicial set $X$ to $C(GX)$, the chains on the Kan loop group of $X$, is a quasiisomorphism of $operatorname{dg}$ bialgebras. We also show that Szczarba’s twisted shuffle map is a $operatorname{dgc}$ map connecting a twisted Cartesian product with the associated twisted tensor product. This gives a natural $operatorname{dgc}$ model for fibre bundles.We apply our results to finite covering spaces and to the Serre spectral sequence.
{"title":"Szczarba’s twisting cochain is comultiplicative","authors":"Matthias Franz","doi":"10.4310/hha.2024.v26.n1.a18","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a18","url":null,"abstract":"We prove that Szczarba’s twisting cochain is comultiplicative. In particular, the induced map from the cobar construction $Omega C(X)$ of the chains on a $1$-reduced simplicial set $X$ to $C(GX)$, the chains on the Kan loop group of $X$, is a quasiisomorphism of $operatorname{dg}$ bialgebras. We also show that Szczarba’s twisted shuffle map is a $operatorname{dgc}$ map connecting a twisted Cartesian product with the associated twisted tensor product. This gives a natural $operatorname{dgc}$ model for fibre bundles.We apply our results to finite covering spaces and to the Serre spectral sequence.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140837262","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 : 2024-05-01DOI: 10.4310/hha.2024.v26.n1.a14
Andreas Kraft, Jonas Schnitzer
The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between $L_infty$-morphisms that are twisted with gauge equivalent Maurer–Cartan elements.
{"title":"The homotopy class of twisted $L_infty$-morphisms","authors":"Andreas Kraft, Jonas Schnitzer","doi":"10.4310/hha.2024.v26.n1.a14","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a14","url":null,"abstract":"The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative. We prove that the globalized formalities with respect to two different covariant derivatives are homotopic. More explicitly, we derive the statement by proving a more general homotopy equivalence between $L_infty$-morphisms that are twisted with gauge equivalent Maurer–Cartan elements.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140837430","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 : 2024-05-01DOI: 10.4310/hha.2024.v26.n1.a17
James Pascaleff
We show that the homotopy theories of differential graded categories and $A_infty$-categories over a field are equivalent at the $(infty, 1)$-categorical level. The results are corollaries of a theorem of Canonaco–Ornaghi–Stellari combined with general relationships between different versions of $(infty, 1)$-categories.
{"title":"Remarks on the equivalence between differential graded categories and A-infinity categories","authors":"James Pascaleff","doi":"10.4310/hha.2024.v26.n1.a17","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a17","url":null,"abstract":"We show that the homotopy theories of differential graded categories and $A_infty$-categories over a field are equivalent at the $(infty, 1)$-categorical level. The results are corollaries of a theorem of Canonaco–Ornaghi–Stellari combined with general relationships between different versions of $(infty, 1)$-categories.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140837433","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 : 2024-05-01DOI: 10.4310/hha.2024.v26.n1.a15
Connor Malin
$defEmbMN{operatorname{Emb}(M,N)}defEM{E_M}defEn{E_n}$ Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $EmbMN$ via the “embedding tower”, which is constructed using the homotopy theory of presheaves on $M$. The same theory allows us to study the stable homotopy type of $EmbMN$ via the “stable embedding tower”. By analyzing cubes of framed configuration spaces, we prove that the layers of the stable embedding tower are tangential homotopy invariants of $N$. If $M$ is framed, the moduli space of disks $EM$ is intimately connected to both the stable and unstable embedding towers through the $En$ operad. The action of $En$ on $EM$ induces an action of the Poisson operad poisn on the homology of configuration spaces $H_ast (F(M,-))$. In order to study this action, we introduce the notion of Poincaré–Koszul operads and modules and show that $En$ and $EM$ are examples. As an application, we compute the induced action of the Lie operad on $H_ast (F(M,-))$ and show it is a homotopy invariant of $M^+$.
{"title":"The stable embedding tower and operadic structures on configuration spaces","authors":"Connor Malin","doi":"10.4310/hha.2024.v26.n1.a15","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a15","url":null,"abstract":"$defEmbMN{operatorname{Emb}(M,N)}defEM{E_M}defEn{E_n}$ Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $EmbMN$ via the “embedding tower”, which is constructed using the homotopy theory of presheaves on $M$. The same theory allows us to study the stable homotopy type of $EmbMN$ via the “stable embedding tower”. By analyzing cubes of framed configuration spaces, we prove that the layers of the stable embedding tower are tangential homotopy invariants of $N$. If $M$ is framed, the moduli space of disks $EM$ is intimately connected to both the stable and unstable embedding towers through the $En$ operad. The action of $En$ on $EM$ induces an action of the Poisson operad poisn on the homology of configuration spaces $H_ast (F(M,-))$. In order to study this action, we introduce the notion of Poincaré–Koszul operads and modules and show that $En$ and $EM$ are examples. As an application, we compute the induced action of the Lie operad on $H_ast (F(M,-))$ and show it is a homotopy invariant of $M^+$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140837261","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 : 2024-03-20DOI: 10.4310/hha.2024.v26.n1.a11
Ngô A. Tuân
Let $p$ be an odd prime and let $M_n$ be the extra-special $p$-group of order$p^{2n+1} (n geqslant 1)$ and exponent $p^2$. We completely compute the $mod p$ Margolis homology of the image ImRes $(A, M_n)$ for every maximal elementary abelian $p$-subgroup $A$ of $M_n$.
{"title":"The Margolis homology of the cohomology restriction from an extra-special group to its maximal elementary abelian subgroups","authors":"Ngô A. Tuân","doi":"10.4310/hha.2024.v26.n1.a11","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a11","url":null,"abstract":"Let $p$ be an odd prime and let $M_n$ be the extra-special $p$-group of order$p^{2n+1} (n geqslant 1)$ and exponent $p^2$. We completely compute the $mod p$ Margolis homology of the image ImRes $(A, M_n)$ for every maximal elementary abelian $p$-subgroup $A$ of $M_n$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198772","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 : 2024-03-20DOI: 10.4310/hha.2024.v26.n1.a12
John Jones, Dmitriy Rumynin, Adam Thomas
We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of topological spaces are equivalent. This can also be interpreted as an equivalence of infinity categories.
{"title":"Compact Lie groups and complex reductive groups","authors":"John Jones, Dmitriy Rumynin, Adam Thomas","doi":"10.4310/hha.2024.v26.n1.a12","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a12","url":null,"abstract":"We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of topological spaces are equivalent. This can also be interpreted as an equivalence of infinity categories.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198452","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 : 2024-03-20DOI: 10.4310/hha.2024.v26.n1.a9
Samson Saneblidze, Ronald Umble
We prove that the formula for the diagonal approximation $Delta_K$ on J. Stasheff’s $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in $href{ https://dx.doi.org/10.4310/HHA.2004.v6.n1.a20}{[7]}$ agrees with the “magical formula” for the diagonal approximation $Delta^prime_K$ derived by Markl and Shnider in $href{ https://www.ams.org/journals/tran/2006-358-06/S0002-9947-05-04006-7/ }{[5]}$, by J.-L. Loday in $href{ https://doi.org/10.1007/978-0-8176-4735-3_13 }{[4]}$, and more recently by Masuda, Thomas, Tonks, and Vallette in $href{ https://doi.org/10.5802/jep.142}{[6]}$.
{"title":"Comparing diagonals on the associahedra","authors":"Samson Saneblidze, Ronald Umble","doi":"10.4310/hha.2024.v26.n1.a9","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a9","url":null,"abstract":"We prove that the formula for the diagonal approximation $Delta_K$ on J. Stasheff’s $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in $href{ https://dx.doi.org/10.4310/HHA.2004.v6.n1.a20}{[7]}$ agrees with the “magical formula” for the diagonal approximation $Delta^prime_K$ derived by Markl and Shnider in $href{ https://www.ams.org/journals/tran/2006-358-06/S0002-9947-05-04006-7/ }{[5]}$, by J.-L. Loday in $href{ https://doi.org/10.1007/978-0-8176-4735-3_13 }{[4]}$, and more recently by Masuda, Thomas, Tonks, and Vallette in $href{ https://doi.org/10.5802/jep.142}{[6]}$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198774","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 : 2024-03-20DOI: 10.4310/hha.2024.v26.n1.a10
Mahmoud Benkhalifa
Let $X$ be a $2$-connected and $6$-dimensional CW‑complex such that $H_3 (X) otimes mathbb{Z}_2 = 0$. This paper aims to describe the group $mathcal{E}(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $mathcal{E}_ast (X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES($X$), we define the group $Gamma S(X)$ of $Gamma$-automorphisms of WES($X$) and we prove that $mathcal{E}(X)/mathcal{E}_ast (X) cong Gamma mathcal{S}(X)$.
让 $X$ 是一个 2$ 连接且 $6$ 维的 CW 复数，使得 $H_3 (X) otimes mathbb{Z}_2 = 0$。本文旨在描述 $X$ 的自同调等价群 $mathcal{E}(X)$ modulo its normal subgroup $mathcal{E}_ast (X)$ of the elements that induce the identity on the homology groups.利用$X$的怀特海精确序列（用WES($X$)表示），我们定义了WES($X$)的$Gamma$自同调的群：$Gamma S(X)$，并证明了$mathcal{E}(X)/mathcal{E}_ast (X) cong Gamma mathcal{S}(X)$。
{"title":"On the group of self-homotopy equivalences of a 2-connected and 6-dimensional CW-complex","authors":"Mahmoud Benkhalifa","doi":"10.4310/hha.2024.v26.n1.a10","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a10","url":null,"abstract":"Let $X$ be a $2$-connected and $6$-dimensional CW‑complex such that $H_3 (X) otimes mathbb{Z}_2 = 0$. This paper aims to describe the group $mathcal{E}(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $mathcal{E}_ast (X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES($X$), we define the group $Gamma S(X)$ of $Gamma$-automorphisms of WES($X$) and we prove that $mathcal{E}(X)/mathcal{E}_ast (X) cong Gamma mathcal{S}(X)$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198701","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 : 2024-02-21DOI: 10.4310/hha.2024.v26.n1.a6
Marcin Chałupnik, Patryk Jaśniewski
$defPdn{mathcal{P}_{d,n}}$We introduce a new functor category: the category $Pdn$ of strict polynomial functors of degree $d$ with domain of dimension bounded by $n$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $mathrm{GL}_n$. We investigate in detail the homological algebra in $Pdn$ for $d = p$, where $p gt 0$ is the characteristic of a ground field. We also establish equivalences between certain subcategories of $Pdntextrm{’s}$ which resemble the Spanier–Whitehead duality in stable homotopy theory.
{"title":"On strict polynomial functors with bounded domain","authors":"Marcin Chałupnik, Patryk Jaśniewski","doi":"10.4310/hha.2024.v26.n1.a6","DOIUrl":"https://doi.org/10.4310/hha.2024.v26.n1.a6","url":null,"abstract":"$defPdn{mathcal{P}_{d,n}}$We introduce a new functor category: the category $Pdn$ of strict polynomial functors of degree $d$ with domain of dimension bounded by $n$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $mathrm{GL}_n$. We investigate in detail the homological algebra in $Pdn$ for $d = p$, where $p gt 0$ is the characteristic of a ground field. We also establish equivalences between certain subcategories of $Pdntextrm{’s}$ which resemble the Spanier–Whitehead duality in stable homotopy theory.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139924676","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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