Pub Date : 2021-04-13DOI: 10.1142/s1793525322500078
Kang Li, Ján Špakula, Jiawen Zhang
By measured graphs, we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe algebras.
{"title":"Measured expanders","authors":"Kang Li, Ján Špakula, Jiawen Zhang","doi":"10.1142/s1793525322500078","DOIUrl":"https://doi.org/10.1142/s1793525322500078","url":null,"abstract":"By measured graphs, we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe algebras.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79264646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-13DOI: 10.1142/s1793525323500139
Sinem Onaran, Ferit Ozturk
We classify the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on $S^3$ and real lens spaces $L(p,pm 1)$. We prove that there is a unique real tight $S^3$ and $mathbb{R}P^3$. We show there is at most one real tight $L(p,pm 1)$ with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an $L(p,p-1)$ which cannot be made convex equivariantly.
{"title":"Real tight contact structures on lens spaces and surface singularities","authors":"Sinem Onaran, Ferit Ozturk","doi":"10.1142/s1793525323500139","DOIUrl":"https://doi.org/10.1142/s1793525323500139","url":null,"abstract":"We classify the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on $S^3$ and real lens spaces $L(p,pm 1)$. We prove that there is a unique real tight $S^3$ and $mathbb{R}P^3$. We show there is at most one real tight $L(p,pm 1)$ with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an $L(p,p-1)$ which cannot be made convex equivariantly.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82129035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-07DOI: 10.1142/S1793525321500357
Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya
We study the homotopy type of the space of the unitary group [Formula: see text] of the uniform Roe algebra [Formula: see text] of [Formula: see text]. We show that the stabilizing map [Formula: see text] is a homotopy equivalence. Moreover, when [Formula: see text], we determine the homotopy type of [Formula: see text], which is the product of the unitary group [Formula: see text] (having the homotopy type of [Formula: see text] or [Formula: see text] depending on the parity of [Formula: see text]) of the Roe algebra [Formula: see text] and rational Eilenberg–MacLane spaces.
{"title":"Homotopy type of the unitary group of the uniform Roe algebra on ℤn","authors":"Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya","doi":"10.1142/S1793525321500357","DOIUrl":"https://doi.org/10.1142/S1793525321500357","url":null,"abstract":"We study the homotopy type of the space of the unitary group [Formula: see text] of the uniform Roe algebra [Formula: see text] of [Formula: see text]. We show that the stabilizing map [Formula: see text] is a homotopy equivalence. Moreover, when [Formula: see text], we determine the homotopy type of [Formula: see text], which is the product of the unitary group [Formula: see text] (having the homotopy type of [Formula: see text] or [Formula: see text] depending on the parity of [Formula: see text]) of the Roe algebra [Formula: see text] and rational Eilenberg–MacLane spaces.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73475913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-31DOI: 10.1142/s1793525323500280
Markus Zeggel
In this article we study a coarse version of the $K$-theoretic Farrell--Jones conjecture we call coarse or bounded isomorphism conjecture. Using controlled category theory we are able to translate this conjecture for asymptotically faithful covers into a more familiar form. This allows us to prove the conjecture for box spaces of residually finite groups whose Farrell--Jones assembly map with coefficients is an isomorphism.
{"title":"The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups","authors":"Markus Zeggel","doi":"10.1142/s1793525323500280","DOIUrl":"https://doi.org/10.1142/s1793525323500280","url":null,"abstract":"In this article we study a coarse version of the $K$-theoretic Farrell--Jones conjecture we call coarse or bounded isomorphism conjecture. Using controlled category theory we are able to translate this conjecture for asymptotically faithful covers into a more familiar form. This allows us to prove the conjecture for box spaces of residually finite groups whose Farrell--Jones assembly map with coefficients is an isomorphism.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77398451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-23DOI: 10.1142/s1793525322500029
Geunho Lim
We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by its boundary. This result can be regarded as an algebraic topological analogue of Gromov’s conjecture for quantitative topology. The author hopes for applications to various fields including the smooth knot concordance group, quantitative topology and complexity theory.
{"title":"Enhanced Bounds for rho-invariants for both general and spherical 3-manifolds","authors":"Geunho Lim","doi":"10.1142/s1793525322500029","DOIUrl":"https://doi.org/10.1142/s1793525322500029","url":null,"abstract":"We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by its boundary. This result can be regarded as an algebraic topological analogue of Gromov’s conjecture for quantitative topology. The author hopes for applications to various fields including the smooth knot concordance group, quantitative topology and complexity theory.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"45 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80000278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-22DOI: 10.1142/s1793525322500042
Ruxi Shi, M. Tsukamoto
When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of [Formula: see text]-periodic points admits a natural free action of [Formula: see text] for each prime number [Formula: see text]. We are interested in the growth of its index and coindex as [Formula: see text]. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by M. Tsukamoto, M. Tsutaya and M. Yoshinaga, [Formula: see text]-index, topological dynamics and marker property, preprint (2020), arXiv: 2012.15372.
当有限群自由作用于拓扑空间时，我们可以定义它的索引和协索引。它们大致衡量给定动作的大小。我们探讨了该指标理论与拓扑动力学之间的相互作用。给定一个不动点自由动力系统，[公式:见文]-周期点的集合对于每个素数[公式:见文]承认[公式:见文]的自然自由作用。我们感兴趣的是它的指数和协指数的增长[公式:见文本]。我们的主要结果表明存在一个具有发散协指数序列的不动点自由动力系统。本文解决了M. Tsukamoto, M. Tsutaya和M. Yoshinaga提出的问题，[公式:见文本]-索引，拓扑动力学和标记性质，预印本(2020)，arXiv: 2012.15372。
{"title":"Divergent coindex sequence for dynamical systems","authors":"Ruxi Shi, M. Tsukamoto","doi":"10.1142/s1793525322500042","DOIUrl":"https://doi.org/10.1142/s1793525322500042","url":null,"abstract":"When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of [Formula: see text]-periodic points admits a natural free action of [Formula: see text] for each prime number [Formula: see text]. We are interested in the growth of its index and coindex as [Formula: see text]. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by M. Tsukamoto, M. Tsutaya and M. Yoshinaga, [Formula: see text]-index, topological dynamics and marker property, preprint (2020), arXiv: 2012.15372.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86428055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-18DOI: 10.1142/s1793525321500631
J. García-Calcines
The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fiberwise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally, we give sufficient conditions so that both notions agree.
{"title":"Formal aspects of parametrized topological complexity and its pointed version","authors":"J. García-Calcines","doi":"10.1142/s1793525321500631","DOIUrl":"https://doi.org/10.1142/s1793525321500631","url":null,"abstract":"The notion of parametrized topological complexity, introduced by Cohen, Farber and Weinberger, is extended to fiberwise spaces which are not necessarily Hurewicz fibrations. After exploring some formal properties of this extension we also introduce the pointed version of parametrized topological complexity. Finally, we give sufficient conditions so that both notions agree.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"230 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86030299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-01DOI: 10.1142/S1793525321500291
J. Boissonnat, Siddharth Pritam, Divyansh Pareek
In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.
{"title":"Strong collapse and persistent homology","authors":"J. Boissonnat, Siddharth Pritam, Divyansh Pareek","doi":"10.1142/S1793525321500291","DOIUrl":"https://doi.org/10.1142/S1793525321500291","url":null,"abstract":"In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"10 1","pages":"1-29"},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74720112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-25DOI: 10.1142/s179352532250008x
J. Hedicke
An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.
洛伦兹距离函数是分析洛伦兹流形因果结构的一个重要工具。根据接触形式的选择，在闭合接触流形的接触同构群上定义了一类洛伦兹距离函数。这些距离函数相对于Shelukhin定义的接触同构的Hofer范数是连续的[接触同构的Hofer范数，J. simplectic Geom. 15(2017) 1173-1208]，并且当且仅当接触同构群是有序的。为了证明这一点，我们证明了由正关系定义的区间相对于由Hofer范数诱导的拓扑是开放的。对于有序的Legendrian同位素类，我们证明了在[D]中定义的chekanov型度量。张俊，Chekanov在接触拓扑中的二分法，数学。Res. Lett. 27(2020) 1165-1194]是非简并的。在这种情况下，类似的结果适用于洛伦兹距离函数在Legendrian同位素类上。这导致了与全局双曲洛伦兹流形相关联的一类自然度量，使得其柯西超曲面具有具有有序同位素类纤维的单位共切束。
{"title":"Lorentzian distance functions in contact geometry","authors":"J. Hedicke","doi":"10.1142/s179352532250008x","DOIUrl":"https://doi.org/10.1142/s179352532250008x","url":null,"abstract":"An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84482711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-14DOI: 10.1142/s1793525321500618
Natalia Cadavid-Aguilar, Jes'us Gonz'alez, B'arbara Guti'errez, Cesar A. Ipanaque-Zapata
We introduce the effectual topological complexity (ETC) of a [Formula: see text]-space [Formula: see text]. This is a [Formula: see text]-equivariant homotopy invariant sitting in between the effective topological complexity of the pair [Formula: see text] and the (regular) topological complexity of the orbit space [Formula: see text]. We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the nontrivial obstruction responsible for the fact that the topological complexity of the Klein bottle is [Formula: see text]. In addition, this gives a counterexample to the possibility — suggested in Pavešić’s work on the topological complexity of a map — that ETC of [Formula: see text] would agree with Farber’s [Formula: see text] whenever the projection map [Formula: see text] is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners.
{"title":"Effectual topological complexity","authors":"Natalia Cadavid-Aguilar, Jes'us Gonz'alez, B'arbara Guti'errez, Cesar A. Ipanaque-Zapata","doi":"10.1142/s1793525321500618","DOIUrl":"https://doi.org/10.1142/s1793525321500618","url":null,"abstract":"We introduce the effectual topological complexity (ETC) of a [Formula: see text]-space [Formula: see text]. This is a [Formula: see text]-equivariant homotopy invariant sitting in between the effective topological complexity of the pair [Formula: see text] and the (regular) topological complexity of the orbit space [Formula: see text]. We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the nontrivial obstruction responsible for the fact that the topological complexity of the Klein bottle is [Formula: see text]. In addition, this gives a counterexample to the possibility — suggested in Pavešić’s work on the topological complexity of a map — that ETC of [Formula: see text] would agree with Farber’s [Formula: see text] whenever the projection map [Formula: see text] is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"43 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80225254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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