In this work we study the homotopy theory of the category $mathsf{RMod}_{mathbf{P}}$ of right modules over a simplicial operad $mathbf{P}$ via the formalism of forest spaces $mathsf{fSpaces}$, as introduced by Heuts, Hinich and Moerdijk. In particular, we show that, for $mathbf{P}$ is closed and $Sigma$-free, there exists a Quillen equivalence between the projective model structure on $mathsf{RMod}_{mathbf{P}}$, and the contravariant model structure on the slice category $mathsf{fSpaces}_{/Nmathbf{P}}$ over the dendroidal nerve of $mathbf{P}$. As an application, we comment on how this result can be used to compute derived mapping spaces of between operadic right modules.
{"title":"Operadic right modules via the dendroidal formalism","authors":"Miguel Barata","doi":"arxiv-2409.01188","DOIUrl":"https://doi.org/arxiv-2409.01188","url":null,"abstract":"In this work we study the homotopy theory of the category\u0000$mathsf{RMod}_{mathbf{P}}$ of right modules over a simplicial operad\u0000$mathbf{P}$ via the formalism of forest spaces $mathsf{fSpaces}$, as\u0000introduced by Heuts, Hinich and Moerdijk. In particular, we show that, for\u0000$mathbf{P}$ is closed and $Sigma$-free, there exists a Quillen equivalence\u0000between the projective model structure on $mathsf{RMod}_{mathbf{P}}$, and the\u0000contravariant model structure on the slice category\u0000$mathsf{fSpaces}_{/Nmathbf{P}}$ over the dendroidal nerve of $mathbf{P}$. As\u0000an application, we comment on how this result can be used to compute derived\u0000mapping spaces of between operadic right modules.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195567","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 describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $mathcal{A}$ on its subalgebra $mathcal{A}(2)$. These arise as the cohomology of $text{v}_2$ self maps $Sigma^7 Z longrightarrow Z$, as in arXiv:1608.06250 [math.AT]. There are $256$ $mathbb{F}_2$ points in this variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$ actions of $Sq^{16}$. We describe in similar fashion the 1600 $mathcal{A}$ actions on $mathcal{A}(2)$ found by Roth(1977) and the inclusion of the variety of symmetric actions into the variety of all actions. We also describe two related varieties of $mathcal{A}$ actions, the maps between these and the behavior of Spanier-Whitehead duality on these varieties. Finally, we note that the actions which have been used in the literature correspond to the simplest choices, in which all the coordinates equal zero.
{"title":"Symmetric A actions on $mathcal{A}(2)$","authors":"Robert R. Bruner","doi":"arxiv-2408.16980","DOIUrl":"https://doi.org/arxiv-2408.16980","url":null,"abstract":"We describe the variety of `symmetric' left actions of the mod 2 Steenrod\u0000algebra $mathcal{A}$ on its subalgebra $mathcal{A}(2)$. These arise as the\u0000cohomology of $text{v}_2$ self maps $Sigma^7 Z longrightarrow Z$, as in\u0000arXiv:1608.06250 [math.AT]. There are $256$ $mathbb{F}_2$ points in this\u0000variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$\u0000actions of $Sq^{16}$. We describe in similar fashion the 1600 $mathcal{A}$\u0000actions on $mathcal{A}(2)$ found by Roth(1977) and the inclusion of the\u0000variety of symmetric actions into the variety of all actions. We also describe\u0000two related varieties of $mathcal{A}$ actions, the maps between these and the\u0000behavior of Spanier-Whitehead duality on these varieties. Finally, we note that\u0000the actions which have been used in the literature correspond to the simplest\u0000choices, in which all the coordinates equal zero.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195571","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 show that a certain class of categorical operads give rise to $E_n$-operads after geometric realization. The main arguments are purely combinatorial and avoid the technical topological assumptions otherwise found in the literature.
{"title":"Combinatorial and homotopical aspects of $E_n$-operads","authors":"Christian Schlichtkrull","doi":"arxiv-2408.17236","DOIUrl":"https://doi.org/arxiv-2408.17236","url":null,"abstract":"We show that a certain class of categorical operads give rise to\u0000$E_n$-operads after geometric realization. The main arguments are purely\u0000combinatorial and avoid the technical topological assumptions otherwise found\u0000in the literature.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195570","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}
In this paper we present an algorithm for computing the matrix representation $Delta_{q, mathrm{up}}^{K, L}$ of the up persistent Laplacian $triangle_{q, mathrm{up}}^{K, L}$ over a pair of non-branching and orientation-compatible simplicial complexes $Khookrightarrow L$, which has quadratic time complexity. Moreover, we show that the matrix representation $Delta_{q, mathrm{up}}^{K, L}$ can be identified as the Laplacian of a weighted oriented hypergraph, which can be regarded as a higher dimensional generalization of the Kron reduction. Finally, we introduce a Cheeger-type inequality with respect to the minimal eigenvalue $lambda_{mathbf{min}}^{K, L}$ of $Delta_{q, mathrm{up}}^{K, L}$.
{"title":"A faster algorithm of up persistent Laplacian over non-branching simplicial complexes","authors":"Rui Dong","doi":"arxiv-2408.16741","DOIUrl":"https://doi.org/arxiv-2408.16741","url":null,"abstract":"In this paper we present an algorithm for computing the matrix representation\u0000$Delta_{q, mathrm{up}}^{K, L}$ of the up persistent Laplacian $triangle_{q,\u0000mathrm{up}}^{K, L}$ over a pair of non-branching and orientation-compatible\u0000simplicial complexes $Khookrightarrow L$, which has quadratic time complexity.\u0000Moreover, we show that the matrix representation $Delta_{q, mathrm{up}}^{K,\u0000L}$ can be identified as the Laplacian of a weighted oriented hypergraph, which\u0000can be regarded as a higher dimensional generalization of the Kron reduction.\u0000Finally, we introduce a Cheeger-type inequality with respect to the minimal\u0000eigenvalue $lambda_{mathbf{min}}^{K, L}$ of $Delta_{q, mathrm{up}}^{K, L}$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195573","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 Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers. Sheehy's subdivision-Rips bifiltration $mathcal{SR}(-)$ is a density-sensitive refinement that is robust to outliers in a strong sense, but whose 0-skeleton has exponential size. For $X$ a finite metric space of constant doubling dimension and fixed $epsilon>0$, we construct a $(1+epsilon)$-homotopy interleaving approximation of $mathcal{SR}(X)$ whose $k$-skeleton has size $O(|X|^{k+2})$. For $kgeq 1$ constant, the $k$-skeleton can be computed in time $O(|X|^{k+3})$.
{"title":"Sparse Approximation of the Subdivision-Rips Bifiltration for Doubling Metrics","authors":"Michael Lesnick, Kenneth McCabe","doi":"arxiv-2408.16716","DOIUrl":"https://doi.org/arxiv-2408.16716","url":null,"abstract":"The Vietoris-Rips filtration, the standard filtration on metric data in\u0000topological data analysis, is notoriously sensitive to outliers. Sheehy's\u0000subdivision-Rips bifiltration $mathcal{SR}(-)$ is a density-sensitive\u0000refinement that is robust to outliers in a strong sense, but whose 0-skeleton\u0000has exponential size. For $X$ a finite metric space of constant doubling\u0000dimension and fixed $epsilon>0$, we construct a $(1+epsilon)$-homotopy\u0000interleaving approximation of $mathcal{SR}(X)$ whose $k$-skeleton has size\u0000$O(|X|^{k+2})$. For $kgeq 1$ constant, the $k$-skeleton can be computed in\u0000time $O(|X|^{k+3})$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225157","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 determine explicitly the stable homotopy groups of Moore spaces up to the range 7, using an equivalence of categories which allows to consider each Moore space as an exact couple of $mathbb Z$-modules.
{"title":"Stable Homotopy Groups of Moore Spaces","authors":"Inès Saihi","doi":"arxiv-2408.15709","DOIUrl":"https://doi.org/arxiv-2408.15709","url":null,"abstract":"We determine explicitly the stable homotopy groups of Moore spaces up to the\u0000range 7, using an equivalence of categories which allows to consider each Moore\u0000space as an exact couple of $mathbb Z$-modules.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195574","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}
In this thesis, we construct a new version of orthogonal calculus for functors $F$ from $C_2$-representations to $C_2$-spaces, where $C_2$ is the cyclic group of order 2. For example, the functor $BO(-)$, which sends a $C_2$-representation $V$ to the classifying space of its orthogonal group $BO(V)$. We obtain a bigraded sequence of approximations to $F$, called the strongly $(p,q)$-polynomial approximations $T_{p,q}F$. The bigrading arises from the bigrading on $C_2$-representations. The homotopy fibre $D_{p,q}F$ of the map from $T_{p+1,q}T_{p,q+1}F$ to $T_{p,q}F$ is such that the approximation $T_{p+1,q}T_{p,q+1}D_{p,q}F$ is equivalent to the functor $D_{p,q}F$ itself and the approximation $T_{p,q}D_{p,q}F$ is trivial. A functor with these properties is called $(p,q)$-homogeneous. Via a zig-zag of Quillen equivalences, we prove that $(p,q)$-homogeneous functors are fully determined by orthogonal spectra with a genuine action of $C_2$ and a naive action of the orthogonal group $O(p,q)$.
{"title":"$C_2$-Equivariant Orthogonal Calculus","authors":"Emel Yavuz","doi":"arxiv-2408.15891","DOIUrl":"https://doi.org/arxiv-2408.15891","url":null,"abstract":"In this thesis, we construct a new version of orthogonal calculus for\u0000functors $F$ from $C_2$-representations to $C_2$-spaces, where $C_2$ is the\u0000cyclic group of order 2. For example, the functor $BO(-)$, which sends a\u0000$C_2$-representation $V$ to the classifying space of its orthogonal group\u0000$BO(V)$. We obtain a bigraded sequence of approximations to $F$, called the\u0000strongly $(p,q)$-polynomial approximations $T_{p,q}F$. The bigrading arises\u0000from the bigrading on $C_2$-representations. The homotopy fibre $D_{p,q}F$ of\u0000the map from $T_{p+1,q}T_{p,q+1}F$ to $T_{p,q}F$ is such that the approximation\u0000$T_{p+1,q}T_{p,q+1}D_{p,q}F$ is equivalent to the functor $D_{p,q}F$ itself and\u0000the approximation $T_{p,q}D_{p,q}F$ is trivial. A functor with these properties\u0000is called $(p,q)$-homogeneous. Via a zig-zag of Quillen equivalences, we prove\u0000that $(p,q)$-homogeneous functors are fully determined by orthogonal spectra\u0000with a genuine action of $C_2$ and a naive action of the orthogonal group\u0000$O(p,q)$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195572","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 define digital $m-$homotopic distance and its higher version. We also mention related notions such as $m-$category in the sense of Lusternik-Schnirelmann and $m-$complexity in topological robotics. Later, we examine the homotopy invariance or $m-$homotopy invariance property of these concepts.
{"title":"$m-$homotopic Distances in Digital Images","authors":"Melih İs, İsmet Karaca","doi":"arxiv-2408.15596","DOIUrl":"https://doi.org/arxiv-2408.15596","url":null,"abstract":"We define digital $m-$homotopic distance and its higher version. We also\u0000mention related notions such as $m-$category in the sense of\u0000Lusternik-Schnirelmann and $m-$complexity in topological robotics. Later, we\u0000examine the homotopy invariance or $m-$homotopy invariance property of these\u0000concepts.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227736","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 $P_k$ be the graded polynomial algebra $mathbb F_2[x_1,x_2,ldots ,x_k]$ over the prime field $mathbb F_2$ with two elements and the degree of each variable $x_i$ being 1, and let $GL_k$ be the general linear group over $mathbb F_2$ which acts on $P_k$ as the usual manner. The algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra $mathcal A$. In 1989, Singer [22] defined the $k$-th homological algebraic transfer, which is a homomorphism $$varphi_k :{rm Tor}^{mathcal A}_{k,k+d} (mathbb F_2,mathbb F_2) to (mathbb F_2otimes_{mathcal A}P_k)_d^{GL_k}$$ from the homological group of the mod-2 Steenrod algebra $mbox{Tor}^{mathcal A}_{k,k+d} (mathbb F_2,mathbb F_2)$ to the subspace $(mathbb F_2otimes_{mathcal A}P_k)_d^{GL_k}$ of $mathbb F_2{otimes}_{mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $d$. In this paper, by using the results of the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer of rank five is an isomorphism in the internal degrees $d= 20$ and $d = 30$. Our result refutes the proof for the case of $d=20$ in Ph'uc [17].
{"title":"Determination of the fifth Singer algebraic transfer in some degrees","authors":"Nguyen Sum","doi":"arxiv-2408.15120","DOIUrl":"https://doi.org/arxiv-2408.15120","url":null,"abstract":"Let $P_k$ be the graded polynomial algebra $mathbb F_2[x_1,x_2,ldots ,x_k]$\u0000over the prime field $mathbb F_2$ with two elements and the degree of each\u0000variable $x_i$ being 1, and let $GL_k$ be the general linear group over\u0000$mathbb F_2$ which acts on $P_k$ as the usual manner. The algebra $P_k$ is\u0000considered as a module over the mod-2 Steenrod algebra $mathcal A$. In 1989,\u0000Singer [22] defined the $k$-th homological algebraic transfer, which is a\u0000homomorphism $$varphi_k :{rm Tor}^{mathcal A}_{k,k+d} (mathbb F_2,mathbb\u0000F_2) to (mathbb F_2otimes_{mathcal A}P_k)_d^{GL_k}$$ from the homological\u0000group of the mod-2 Steenrod algebra $mbox{Tor}^{mathcal A}_{k,k+d} (mathbb\u0000F_2,mathbb F_2)$ to the subspace $(mathbb F_2otimes_{mathcal\u0000A}P_k)_d^{GL_k}$ of $mathbb F_2{otimes}_{mathcal A}P_k$ consisting of all\u0000the $GL_k$-invariant classes of degree $d$. In this paper, by using the results of the Peterson hit problem we present\u0000the proof of the fact that the Singer algebraic transfer of rank five is an\u0000isomorphism in the internal degrees $d= 20$ and $d = 30$. Our result refutes\u0000the proof for the case of $d=20$ in Ph'uc [17].","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195575","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}
In algebraic topology, we usually represent surfaces by mean of cellular complexes. This representation is intrinsic, but requires to identify some points through an equivalence relation. On the other hand, embedding a surface in a Euclidean space is not intrinsic but does not require to identify points. In the present paper, we are interested in the M"obius strip, the torus, and the real projective plane. More precisely, we construct explicit homeomorphisms, as well as their inverses, from cellular complexes to surfaces of 3-dimensional (for the M"obius strip and the torus) and 4-dimensional (for the projective plane) Euclidean spaces. All the embeddings were already known, but we are not aware if explicit formulas for their inverses exist.
{"title":"Cellular complexes and embeddings into Euclidean spaces: Möbius strip, torus, and projective plane","authors":"Anthony Fraga","doi":"arxiv-2408.14882","DOIUrl":"https://doi.org/arxiv-2408.14882","url":null,"abstract":"In algebraic topology, we usually represent surfaces by mean of cellular\u0000complexes. This representation is intrinsic, but requires to identify some\u0000points through an equivalence relation. On the other hand, embedding a surface\u0000in a Euclidean space is not intrinsic but does not require to identify points.\u0000In the present paper, we are interested in the M\"obius strip, the torus, and\u0000the real projective plane. More precisely, we construct explicit\u0000homeomorphisms, as well as their inverses, from cellular complexes to surfaces\u0000of 3-dimensional (for the M\"obius strip and the torus) and 4-dimensional (for\u0000the projective plane) Euclidean spaces. All the embeddings were already known,\u0000but we are not aware if explicit formulas for their inverses exist.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195578","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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