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On the KO–groups of toric manifolds 关于环面流形的ko -群
arXiv: Algebraic Topology Pub Date : 2019-03-20 DOI: 10.2140/agt.2020.20.2589
L. Cai, Suyoung Choi, Hanchul Park
{"title":"On the KO–groups of toric manifolds","authors":"L. Cai, Suyoung Choi, Hanchul Park","doi":"10.2140/agt.2020.20.2589","DOIUrl":"https://doi.org/10.2140/agt.2020.20.2589","url":null,"abstract":"In this paper we consider the real topological K-groups of a toric manifold M, which turns out to be closely related to the topology of the small cover MR, the fixed points under the canonical conjugation on M . Following the work of Bahri and Bendersky [2], we give an explicit formula for the KO-groups of toric manifolds, and then we characterize the two extreme classes of toric manifolds according to the two A(1) modules shown in [2].","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"16 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83713509","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}
引用次数: 0
Linear motion planning with controlled collisions and pure planar braids 具有控制碰撞和纯平面编织的线性运动规划
arXiv: Algebraic Topology Pub Date : 2019-02-17 DOI: 10.4310/hha.2021.v23.n1.a15
Jes'us Gonz'alez, B'arbara Guti'errez, Jos'e Luis Le'on-Medina, Christopher Roque
{"title":"Linear motion planning with controlled collisions and pure planar braids","authors":"Jes'us Gonz'alez, B'arbara Guti'errez, Jos'e Luis Le'on-Medina, Christopher Roque","doi":"10.4310/hha.2021.v23.n1.a15","DOIUrl":"https://doi.org/10.4310/hha.2021.v23.n1.a15","url":null,"abstract":"We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($TC_s$, $sgeq2$) of the \"no-$k$-equal\" configuration space Conf$_k(mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity of Khovanov's group PP$_n$ of pure planar braids on $n$ strands, which is an $mathbb{R}$-analogue of Artin's classical pure braid group on $n$ strands. Our methods can be used to describe optimal motion planners for PP$_n$ provided $n$ is small.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"150 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86332025","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}
引用次数: 18
Semi-simplicial spaces Semi-simplicial空间
arXiv: Algebraic Topology Pub Date : 2019-02-02 DOI: 10.17863/CAM.35950
Johannes Ebert, O. Randal-Williams
{"title":"Semi-simplicial spaces","authors":"Johannes Ebert, O. Randal-Williams","doi":"10.17863/CAM.35950","DOIUrl":"https://doi.org/10.17863/CAM.35950","url":null,"abstract":"J. Ebert was partially supported by the SFB 878. O. Randal-Williams was\u0000supported by EPSRC grant number EP/M027783/1.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74050843","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}
引用次数: 26
Segal spaces, spans, and semicategories 分段空间、跨度和半分类
arXiv: Algebraic Topology Pub Date : 2019-01-24 DOI: 10.1090/proc/15197
R. Haugseng
{"title":"Segal spaces, spans, and semicategories","authors":"R. Haugseng","doi":"10.1090/proc/15197","DOIUrl":"https://doi.org/10.1090/proc/15197","url":null,"abstract":"We show that Segal spaces, and more generally category objects in an $infty$-category $mathcal{C}$, can be identified with associative algebras in the double $infty$-category of spans in $mathcal{C}$. We use this observation to prove that \"having identities\" is a property of a non-unital $(infty,n)$-category.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76879764","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}
引用次数: 4
Floer homotopy theory, revisited 花同伦理论，重访
arXiv: Algebraic Topology Pub Date : 2019-01-24 DOI: 10.1201/9781351251624-10
R. Cohen
{"title":"Floer homotopy theory, revisited","authors":"R. Cohen","doi":"10.1201/9781351251624-10","DOIUrl":"https://doi.org/10.1201/9781351251624-10","url":null,"abstract":"In 1995 the author, Jones, and Segal introduced the notion of \"Floer homotopy theory\". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, \"When is the Floer homology isomorphic to the (singular) homology of a naturally occuring (pro)spectrum defined from the properties of the moduli spaces inherent in the Floer theory?\". A proposal for how to construct such a spectrum was given in terms of a \"framed flow category\", and some rather simple examples were described. Years passed before this notion found some genuine applications to symplectic geometry and low dimensional topology. However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. Here we summarize some of these exciting developments, and describe some of the new techniques that were introduced. Throughout we try to point out that this area is a very fertile ground at the interface of homotopy theory, symplectic geometry, and low dimensional topology.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"42 1","pages":"369-404"},"PeriodicalIF":0.0,"publicationDate":"2019-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73837917","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}
引用次数: 3
The cohomology rings of the unordered configuration spaces of elliptic curves 椭圆曲线无序位形空间的上同调环
arXiv: Algebraic Topology Pub Date : 2019-01-04 DOI: 10.2140/agt.2020.20.2995
Roberto Pagaria
{"title":"The cohomology rings of the unordered configuration spaces of elliptic curves","authors":"Roberto Pagaria","doi":"10.2140/agt.2020.20.2995","DOIUrl":"https://doi.org/10.2140/agt.2020.20.2995","url":null,"abstract":"We study the cohomology ring of the configuration space of unordered points in the two dimensional torus. In particular, we compute the mixed Hodge structure on the cohomology, the action of the mapping class group, the structure of the cohomology ring and we prove the formality over the rationals.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81772058","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}
引用次数: 4
Imploded cross-sections 崩溃横断面图
arXiv: Algebraic Topology Pub Date : 2018-12-11 DOI: 10.1216/RMJ.2021.51.193
Lisa Jeffrey, Sina Zabanfahm
{"title":"Imploded cross-sections","authors":"Lisa Jeffrey, Sina Zabanfahm","doi":"10.1216/RMJ.2021.51.193","DOIUrl":"https://doi.org/10.1216/RMJ.2021.51.193","url":null,"abstract":"In this survey article, we describe imploded cross-sections, which were developed in order to solve the problem that the cross-section of a Hamiltonian $K$-space is usually not symplectic. In some specific examples we contrast the intersection homology of some imploded cross-sections with their homology intersection spaces. \u0000Moreover, we compute the homology of intersection spaces associated to the open cone of a simply connected, smooth, oriented manifold and the suspension of such a manifold.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75471823","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}
引用次数: 0
On the homotopy of closed manifolds and finite CW-complexes 关于闭流形与有限cw -复形的同伦
arXiv: Algebraic Topology Pub Date : 2018-12-09 DOI: 10.1090/proc/15784
Yang Su, Xiaolei Wu
{"title":"On the homotopy of closed manifolds and finite CW-complexes","authors":"Yang Su, Xiaolei Wu","doi":"10.1090/proc/15784","DOIUrl":"https://doi.org/10.1090/proc/15784","url":null,"abstract":"We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when $X$ is a finite CW-complex of dimension $n$ and $pi_1(X)$ is virtually a Poincare duality group of dimension $geq n-1$, then $pi_i(X)$ is not finitely generated for some $i$ unless $X$ is homotopy equivalent to the Eilenberg-Maclane space $K(pi_1(X),1)$; when $M$ is an $n$-dimensional closed manifold and $pi_1(M)$ is virtually a Poincare duality group of dimension $ge n-1$, then for some $ileq [n/2]$, $pi_i(M)$ is not finitely generated, unless $M$ itself is an aspherical manifold. These generalize theorems of M. Damian from polycyclic groups to any virtually Poincare duality groups. When $pi_1(X)$ is not a virtually Poincare duality group, we also obtained similar results. As a by-product of our results, we show that if a group $G$ is of type F and $H^i(G, mathbb{Z} G)$ is finitely generated for any $i$, then $G$ is a Poincare duality group. This recovers partially a theorem of Farrell.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76579236","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}
引用次数: 0
Variations on Poincaré duality for intersection homology 交同调中poincarcarr对偶的变化
arXiv: Algebraic Topology Pub Date : 2018-12-07 DOI: 10.4171/lem/65-1/2-4
M. Saralegi-Aranguren, Daniel Tanr'e
{"title":"Variations on Poincaré duality for intersection homology","authors":"M. Saralegi-Aranguren, Daniel Tanr'e","doi":"10.4171/lem/65-1/2-4","DOIUrl":"https://doi.org/10.4171/lem/65-1/2-4","url":null,"abstract":"Intersection homology with coefficients in a field restores Poincare duality for some spaces with singularities, as pseudomanifolds. But, with coefficients in a ring, the behaviours of manifolds and pseudomanifolds are different. This work is an overview, with proofs and explicit examples, of various possible situations with their properties. \u0000We first set up a duality, defined from a cap product, between two intersection cohomologies: the first one arises from a linear dual and the second one from a simplicial blow up. Moreover, from this property, Poincare duality in intersection homology looks like the Poincare-Lefschetz duality of a manifold with boundary. Besides that, an investigation of the coincidence of the two previous cohomologies reveals that the only obstruction to the existence of a Poincare duality is the homology of a well defined complex. This recovers the case of the peripheral sheaf introduced by Goresky and Siegel for compact PL-pseudomanifolds. We also list a series of explicit computations of peripheral intersection cohomology. In particular, we observe that Poincare duality can exist in the presence of torsion in the \"critical degree\" of the intersection homology of the links of a pseudomanifold.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90966477","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}
引用次数: 1
On the delooping of (framed) embedding spaces 关于(框架)嵌入空间的展开
arXiv: Algebraic Topology Pub Date : 2018-11-29 DOI: 10.1090/TRAN/8451
J. Ducoulombier, V. Turchin, T. Willwacher
{"title":"On the delooping of (framed) embedding spaces","authors":"J. Ducoulombier, V. Turchin, T. Willwacher","doi":"10.1090/TRAN/8451","DOIUrl":"https://doi.org/10.1090/TRAN/8451","url":null,"abstract":"It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84985458","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}
引用次数: 2
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