Pub Date : 2023-11-22DOI: 10.4310/hha.2023.v25.n2.a14
Valentina Grazian, Ettore Marmo
We prove that the Díaz–Park sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group $mathrm{G}_2 (p)$, for $p geqslant 5$.
{"title":"Sharpness of saturated fusion systems on a Sylow $p$-subgroup of $mathrm{G}_2 (p)$","authors":"Valentina Grazian, Ettore Marmo","doi":"10.4310/hha.2023.v25.n2.a14","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a14","url":null,"abstract":"We prove that the Díaz–Park sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group $mathrm{G}_2 (p)$, for $p geqslant 5$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"53 2","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520629","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 : 2023-11-22DOI: 10.4310/hha.2023.v25.n2.a16
Kensuke Arakawa
We prove that the classifying space of a simplicial group is modeled by its homotopy coherent nerve. We will also show that the claim remains valid for simplicial groupoids.
证明了一个简单群的分类空间是由它的同伦相干神经来建模的。我们还将证明该声明对简单群类群仍然有效。
{"title":"Classifying space via homotopy coherent nerve","authors":"Kensuke Arakawa","doi":"10.4310/hha.2023.v25.n2.a16","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a16","url":null,"abstract":"We prove that the classifying space of a simplicial group is modeled by its homotopy coherent nerve. We will also show that the claim remains valid for simplicial groupoids.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"12 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520611","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 : 2023-11-22DOI: 10.4310/hha.2023.v25.n2.a17
Sudeep Podder, Parameswaran Sankaran
Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $mathbb{R}^n$. We compute the complex $K$-ring of $G_{n,k}:$, up to a small indeterminacy, for all values of $n,k$ where $2 leqslant k leqslant n - 2$. When $n equiv 0 (operatorname{mod} 4), k equiv 1 (operatorname{mod} 2)$, we use the Hodgkin spectral sequence to determine the $K$-ring completely.
设$G_{n,k}$表示$mathbb{R}^n$的$k$维向量子空间的实Grassmann流形。我们计算了$G_{n,k}:$的复杂$K$ -环，直到一个小的不确定性，对于$n,k$的所有值，其中$2 leqslant k leqslant n - 2$。当$n equiv 0 (operatorname{mod} 4), k equiv 1 (operatorname{mod} 2)$时，我们使用霍奇金谱序列完全确定$K$ -环。
{"title":"$K$-theory of real Grassmann manifolds","authors":"Sudeep Podder, Parameswaran Sankaran","doi":"10.4310/hha.2023.v25.n2.a17","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a17","url":null,"abstract":"Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $mathbb{R}^n$. We compute the complex $K$-ring of $G_{n,k}:$, up to a small indeterminacy, for all values of $n,k$ where $2 leqslant k leqslant n - 2$. When $n equiv 0 (operatorname{mod} 4), k equiv 1 (operatorname{mod} 2)$, we use the Hodgkin spectral sequence to determine the $K$-ring completely.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"7 11","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520624","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 : 2023-11-01DOI: 10.4310/hha.2023.v25.n2.a10
Jonathan Beardsley
This paper establishes several results for coalgebraic structure in $infty$-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor product over the underlying algebra or coalgebra of the bialgebra). We give two examples of higher coalgebraic structure: first, following Hess we show that for a map of $mathbb{E}_n$-ring spectra $varphi : A to B$, the associated $infty$-category of descent data is equivalent to the $infty$-category of comodules over $B otimes_A B$, the so-called descent coring; secondly, we show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the $infty$-categorical Thom diagonal of Ando, Blumberg, Gepner, Hopkins and Rezk (which we describe explicitly) and that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors.
本文建立了$infty$ -范畴中共代数结构的几个结果，特别是与共数论的谱非交换几何的联系。我们证明了双代数上的模和模的范畴总是允许适当结构的单面结构，其中张量积在周围范畴中(相对于在双代数的基础代数或协代数上的相对(co)张量积)。我们给出了两个更高共代数结构的例子:首先，在Hess之后，我们证明了对于$mathbb{E}_n$ -环谱$varphi : A to B$的映射，相关的$infty$ -类下降数据等价于$B otimes_A B$上的$infty$ -类模，即所谓的下降取心;其次，我们证明了Thom谱通常具有一个高度结构化的模结构，该结构相当于Ando, Blumberg, Gepner, Hopkins和Rezk的$infty$ -分类Thom对角线(我们明确描述了)，并且这个高度结构化的对角线以预期的方式分解了定向Thom谱的Thom同构，表明Thom谱是谱非交换环量的好例子。
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Pub Date : 2023-11-01DOI: 10.4310/hha.2023.v25.n2.a11
Ryan Grady, Anna Schenfisch
Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig‑zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig‑zag modules and the combinatorial entrance path category on stratified $mathbb{R}$. Finally, we compute the algebraic $K$-theory of generalized zig‑zag modules and describe connections to both Euler curves and $K_0$ of the monoid of persistence diagrams as described by Bubenik and Elchesen.
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Pub Date : 2023-10-11DOI: 10.4310/hha.2023.v25.n2.a7
Philippe Kupper
We consider the space $Lambda M := H^1 (S^1, M)$ of loops of Sobolev class $H^1$ of a compact smooth manifold $M$, the so-called free loop space of $M$. We take quotients $Lambda M / G$ where $G$ is a finite subgroup of $O(2)$ acting by linear reparametrization of $S^1$. We use the existence of transfer maps $operatorname{tr} : H_ast (Lambda M / G) to H_ast (Lambda M)$ to define a homology product on $Lambda M / G$ via the Chas–Sullivan loop product. We call this product $P_G$ the transfer product. The involution $vartheta : Lambda M to Lambda M$ which reverses orientation, $vartheta ( gamma (t) := gamma (1-t)$, is of particular interest to us. We compute $H_ast (Lambda S^n / vartheta ; mathbb{Q}), n gt 2$, and the product[P_vartheta : H_i (Lambda S^n / vartheta ; mathbb{Q}) times H_j (Lambda S^n / vartheta ; mathbb{Q)} to H_{i+j-n} (Lambda Sn/vartheta ; mathbb{Q})]associated to orientation reversal. Rationally Pvartheta can be realized “geometrically” using the concatenation of equivalence classes of loops. There is a qualitative difference between the homology of $Lambda S^n / vartheta$ and the homology of $Lambda S^n / G$ when $G subset S^1 subset O(2)$ does not “contain” the orientation reversal. This might be interesting with respect to possible differences in the number of closed geodesics between non-reversible and reversible Finsler metrics on $S^n$, the latter might always be infinite.
我们考虑紧致光滑流形$M$的Sobolev类循环$H^1$的空间$Lambda M := H^1 (S^1, M)$，即$M$的自由循环空间。我们取商$Lambda M / G$，其中$G$是$O(2)$的一个有限子群，由$S^1$的线性重参数化作用。我们利用迁移映射$operatorname{tr} : H_ast (Lambda M / G) to H_ast (Lambda M)$的存在性，通过查斯-苏利文环积在$Lambda M / G$上定义了一个同源积。我们称这个产品为$P_G$传递产品。反转方向的对合$vartheta : Lambda M to Lambda M$$vartheta ( gamma (t) := gamma (1-t)$对我们来说特别有趣。我们计算$H_ast (Lambda S^n / vartheta ; mathbb{Q}), n gt 2$和与方向反转相关的乘积[P_vartheta : H_i (Lambda S^n / vartheta ; mathbb{Q}) times H_j (Lambda S^n / vartheta ; mathbb{Q)} to H_{i+j-n} (Lambda Sn/vartheta ; mathbb{Q})]。合理地，P vartheta可以通过循环等价类的串联“几何地”实现。当$G subset S^1 subset O(2)$不“包含”取向反转时，$Lambda S^n / vartheta$的同源性与$Lambda S^n / G$的同源性有质的区别。对于在$S^n$上的不可逆芬斯勒度量和可逆芬斯勒度量之间的封闭测地线数目的可能差异，这可能很有趣，后者可能总是无限的。
{"title":"Homology transfer products on free loop spaces: orientation reversal on spheres","authors":"Philippe Kupper","doi":"10.4310/hha.2023.v25.n2.a7","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a7","url":null,"abstract":"We consider the space $Lambda M := H^1 (S^1, M)$ of loops of Sobolev class $H^1$ of a compact smooth manifold $M$, the so-called free loop space of $M$. We take quotients $Lambda M / G$ where $G$ is a finite subgroup of $O(2)$ acting by linear reparametrization of $S^1$. We use the existence of transfer maps $operatorname{tr} : H_ast (Lambda M / G) to H_ast (Lambda M)$ to define a homology product on $Lambda M / G$ via the Chas–Sullivan loop product. We call this product $P_G$ the transfer product. The involution $vartheta : Lambda M to Lambda M$ which reverses orientation, $vartheta ( gamma (t) := gamma (1-t)$, is of particular interest to us. We compute $H_ast (Lambda S^n / vartheta ; mathbb{Q}), n gt 2$, and the product[P_vartheta : H_i (Lambda S^n / vartheta ; mathbb{Q}) times H_j (Lambda S^n / vartheta ; mathbb{Q)} to H_{i+j-n} (Lambda Sn/vartheta ; mathbb{Q})]associated to orientation reversal. Rationally Pvartheta can be realized “geometrically” using the concatenation of equivalence classes of loops. There is a qualitative difference between the homology of $Lambda S^n / vartheta$ and the homology of $Lambda S^n / G$ when $G subset S^1 subset O(2)$ does not “contain” the orientation reversal. This might be interesting with respect to possible differences in the number of closed geodesics between non-reversible and reversible Finsler metrics on $S^n$, the latter might always be infinite.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"9 4","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520608","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 : 2023-10-04DOI: 10.4310/hha.2023.v25.n2.a4
Neeti Gauniyal
$defEmb{overline{Emb}}$We show that for the spaces of spherical embeddings modulo immersions $Emb (S^n, S^{n+q})$ and long embeddings modulo immersions $Emb_partial (D^n, D^{n+q})$, the set of connected components is isomorphic to $pi_{n+1} (SG, SG_q)$ for $q geqslant 3$. As a consequence, we show that all the terms of the long exact sequence of the triad $(SG; SO, SG_q)$ have a geometric meaning relating to spherical embeddings and immersions.
{"title":"Haefliger’s approach for spherical knots modulo immersions","authors":"Neeti Gauniyal","doi":"10.4310/hha.2023.v25.n2.a4","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a4","url":null,"abstract":"$defEmb{overline{Emb}}$We show that for the spaces of spherical embeddings modulo immersions $Emb (S^n, S^{n+q})$ and long embeddings modulo immersions $Emb_partial (D^n, D^{n+q})$, the set of connected components is isomorphic to $pi_{n+1} (SG, SG_q)$ for $q geqslant 3$. As a consequence, we show that all the terms of the long exact sequence of the triad $(SG; SO, SG_q)$ have a geometric meaning relating to spherical embeddings and immersions.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"46 6","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520630","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 : 2023-08-23DOI: 10.4310/HHA.2007.v9.n2.a16
W. Mannan
We work over an arbitrary ring R. Given two truncated projective resolutions of equal length for the same module, we consider their underlying chain complexes. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type
{"title":"Homotopy types of truncated projective resolutions","authors":"W. Mannan","doi":"10.4310/HHA.2007.v9.n2.a16","DOIUrl":"https://doi.org/10.4310/HHA.2007.v9.n2.a16","url":null,"abstract":"We work over an arbitrary ring R. Given two truncated projective\u0000resolutions of equal length for the same module, we consider\u0000their underlying chain complexes. We show they may be\u0000stabilized by projective modules to obtain a pair of complexes\u0000of the same homotopy type","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"9 1","pages":"445-449"},"PeriodicalIF":0.5,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42199405","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 : 2023-08-23DOI: 10.4310/HHA.2017.v19.n1.a9
W. Mannan
We show that the homological properties of a 5-manifold M with fundamental group G are encapsulated in a G-invariant stable form on the dual of the third syzygy of Z. In this notation one may express an even stronger version of Poincare duality for M. However we find an obstruction to this duality.
{"title":"Duality in the homology of 5-manifolds","authors":"W. Mannan","doi":"10.4310/HHA.2017.v19.n1.a9","DOIUrl":"https://doi.org/10.4310/HHA.2017.v19.n1.a9","url":null,"abstract":"We show that the homological properties of a 5-manifold M with fundamental group G are encapsulated in a G-invariant stable form on the dual of the third syzygy of Z. In this notation one may express an even stronger version of Poincare duality for M. However we find an obstruction to this duality.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"19 1","pages":"171-179"},"PeriodicalIF":0.5,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41756829","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.a11
Sajjad Mohammadi
{"title":"The homotopy types of $Sp(n)$-gauge groups over $mathbb{C}P^2$","authors":"Sajjad Mohammadi","doi":"10.4310/hha.2023.v25.n1.a11","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a11","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435079","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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