Pub Date : 2021-11-29DOI: 10.1142/s1793525321500588
Paul Seidel
We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fiber. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.
{"title":"Fukaya A∞-structures associated to Lefschetz fibrations. V","authors":"Paul Seidel","doi":"10.1142/s1793525321500588","DOIUrl":"https://doi.org/10.1142/s1793525321500588","url":null,"abstract":"We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fiber. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"26 50","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495688","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 : 2021-11-29DOI: 10.1142/s1793525321500710
A. Milivojević
{"title":"The weak form of Hirzebruch's prize question via rational surgery","authors":"A. Milivojević","doi":"10.1142/s1793525321500710","DOIUrl":"https://doi.org/10.1142/s1793525321500710","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89450148","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 : 2021-11-23DOI: 10.1142/s1793525323500358
C. Ketterer
We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.
{"title":"Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds","authors":"C. Ketterer","doi":"10.1142/s1793525323500358","DOIUrl":"https://doi.org/10.1142/s1793525323500358","url":null,"abstract":"We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72804073","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 : 2021-11-15DOI: 10.1142/s179352532150062x
Tyrone Crisp
By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.
{"title":"Commutativity of the Haagerup tensor product and base change for operator modules","authors":"Tyrone Crisp","doi":"10.1142/s179352532150062x","DOIUrl":"https://doi.org/10.1142/s179352532150062x","url":null,"abstract":"By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"276 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79155764","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 : 2021-11-09DOI: 10.1142/s1793525323500218
Sheng-Fu Chiu
Inspired by recent developments of quantum speed limit we introduce a categorical energy of sheaves in the derived category over a manifold relative to a microlocal projector. We utilize the tool of algebraic microlocal analysis to show that with regard to the microsupports of sheaves, our categorical energy gives a lower bound of the Hofer displacement energy. We also prove that on the other hand our categorical energy obeys a relative energy-capacity type inequality. As a by-product this provides a sheaf-theoretic proof of the positivity of the Hofer displacement energy for disjointing the zero section $L$ from an open subset $O$ in $T^*L$ , given that $L cap O neq emptyset$.
受量子速度极限最新发展的启发，我们引入了相对于微局部投影仪的流形上派生范畴的束的范畴能量。我们利用代数微局部分析的工具来证明，对于轴系微支承，我们的分类能量给出了霍弗位移能量的下界。另一方面，我们也证明了我们的分类能量服从一个相对能量-容量型不等式。作为一个副产品，这提供了一个轴理论的证明，证明了从$T^*L$中的开放子集$O$中分离零段$L$的Hofer位移能量的正性，假设$L cap O neq emptyset$。
{"title":"Quantum Speed Limit and Categorical Energy relative to Microlocal Projector","authors":"Sheng-Fu Chiu","doi":"10.1142/s1793525323500218","DOIUrl":"https://doi.org/10.1142/s1793525323500218","url":null,"abstract":"Inspired by recent developments of quantum speed limit we introduce a categorical energy of sheaves in the derived category over a manifold relative to a microlocal projector. We utilize the tool of algebraic microlocal analysis to show that with regard to the microsupports of sheaves, our categorical energy gives a lower bound of the Hofer displacement energy. We also prove that on the other hand our categorical energy obeys a relative energy-capacity type inequality. As a by-product this provides a sheaf-theoretic proof of the positivity of the Hofer displacement energy for disjointing the zero section $L$ from an open subset $O$ in $T^*L$ , given that $L cap O neq emptyset$.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72996014","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 : 2021-10-29DOI: 10.1142/s1793525323500073
Liang Guo, Zheng Luo, Qin Wang, Yazhou Zhang
Let X be a discrete metric space with bounded geometry. In this paper, we show that if X admits an "A-by-CE coarse fibration", then the canonical quotient map λ : C max(X) → C (X) from the maximal Roe algebra to the Roe algebra of X, and the canonical quotient map λ : C u,max(X) → C ∗ u(X) from the maximal uniform Roe algebra to the uniform Roe algebra of X, induce isomorphisms on Ktheory. A typical example of such a space arises from a sequence of group extensions {1 → Nn → Gn → Qn → 1} such that the sequence {Nn} has Yu’s property A, and the sequence {Qn} admits a coarse embedding into Hilbert space. This extends an early result of J. Špakula and R. Willett [24] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum-Connes conjecture holds for a large class of metric spaces which may not admit a fibred coarse embedding into Hilbert space.
设X是一个几何有界的离散度量空间。本文证明了如果X允许“a - byce粗纤化”，则从最大Roe代数到X的Roe代数的正则商映射λ: C max(X)→C (X)，以及从最大一致Roe代数到X的一致Roe代数的正则商映射λ: C u,max(X)→C * u(X)在k论上诱导同构。这种空间的一个典型例子来自于一个群扩展序列{1→Nn→Gn→Qn→1}，使得序列{Nn}具有Yu的性质A，并且序列{Qn}允许粗嵌入到Hilbert空间中。这将J. Špakula和R. Willett[24]的早期结果扩展到度量空间的情况，度量空间可能不允许粗嵌入到Hilbert空间中。此外，它还表明极大粗Baum-Connes猜想对于不允许纤维粗嵌入到希尔伯特空间的度量空间是成立的。
{"title":"K-theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations","authors":"Liang Guo, Zheng Luo, Qin Wang, Yazhou Zhang","doi":"10.1142/s1793525323500073","DOIUrl":"https://doi.org/10.1142/s1793525323500073","url":null,"abstract":"Let X be a discrete metric space with bounded geometry. In this paper, we show that if X admits an \"A-by-CE coarse fibration\", then the canonical quotient map λ : C max(X) → C (X) from the maximal Roe algebra to the Roe algebra of X, and the canonical quotient map λ : C u,max(X) → C ∗ u(X) from the maximal uniform Roe algebra to the uniform Roe algebra of X, induce isomorphisms on Ktheory. A typical example of such a space arises from a sequence of group extensions {1 → Nn → Gn → Qn → 1} such that the sequence {Nn} has Yu’s property A, and the sequence {Qn} admits a coarse embedding into Hilbert space. This extends an early result of J. Špakula and R. Willett [24] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum-Connes conjecture holds for a large class of metric spaces which may not admit a fibred coarse embedding into Hilbert space.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87166414","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 : 2021-10-28DOI: 10.1142/s1793525323500176
R. Guzman, P. Shalen
Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) $pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or (b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2, 3$ or $4$, then $text{dim}, H_1(M;F_p)<157.763 cdot text{vol}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that $pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g = 2, dots,8$. These results should be compared with those of our previous paper $The ratio of homology rank to hyperbolic volume, I$, in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of $pi_1(M)$ in terms of $text{vol},M$, assuming that either $pi_1(M)$ is $9$-free, or $M$ is closed and $pi_1(M)$ is $5$-free.
在温和的拓扑限制下，我们得到了有限体积可定向双曲$3$流形$M$的模$p$同调(对于任意素数$p$)的维数的新的线性上界。本文论点的一个令人惊讶的特点是它们需要应用四色定理。如果$M$是封闭的，并且(a) $pi_1(M)$没有子群同构于$2, 3$或$4$的封闭可定向曲面的基群，或者(b) $p = 2$，并且$M$不包含$2, 3$或$4$的(嵌入的，双面的)不可压缩曲面，则$text{dim}, H_1(M;F_p)<157.763 cdot text{vol}(M)$。如果$M$有一个或多个顶点，我们得到一个非常相似的界，假设$pi_1(M)$没有子群同构于$g = 2, dots,8$属$g$的一个闭合可定向曲面的基群。这些结果应该与我们之前的论文$The ratio of homology rank to hyperbolic volume, I$的结果进行比较，在该论文中，我们得到了一个系数在$168$而不是$158$范围内的界，没有对曲面子群或不可压缩曲面的限制。在未来的论文中，我们期望使用一个更复杂的论证，得到与本文给出的边界接近的边界，而不受这样的限制。参数还给出了以$text{vol},M$表示$pi_1(M)$的秩的新的线性上界(带有常数项)，假设$pi_1(M)$是$9$自由的，或者$M$是封闭的，$pi_1(M)$是$5$自由的。
{"title":"The ratio of homology rank to hyperbolic volume","authors":"R. Guzman, P. Shalen","doi":"10.1142/s1793525323500176","DOIUrl":"https://doi.org/10.1142/s1793525323500176","url":null,"abstract":"Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) $pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or (b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2, 3$ or $4$, then $text{dim}, H_1(M;F_p)<157.763 cdot text{vol}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that $pi_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g = 2, dots,8$. These results should be compared with those of our previous paper $The ratio of homology rank to hyperbolic volume, I$, in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of $pi_1(M)$ in terms of $text{vol},M$, assuming that either $pi_1(M)$ is $9$-free, or $M$ is closed and $pi_1(M)$ is $5$-free.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"4 1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90799648","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 : 2021-10-25DOI: 10.1142/s1793525321500576
Yong Fang
A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.
{"title":"On geodesically reversible Finsler manifolds","authors":"Yong Fang","doi":"10.1142/s1793525321500576","DOIUrl":"https://doi.org/10.1142/s1793525321500576","url":null,"abstract":"A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73529913","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 : 2021-10-22DOI: 10.1142/s1793525321500552
J. C. Álvarez Paiva, J. Barbosa Gomes
It is shown that a possibly irreversible [Formula: see text] Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed [Formula: see text]-form. This is used to prove that if [Formula: see text] is a compact Riemannian symmetric space of rank greater than one and [Formula: see text] is a reversible [Formula: see text] Finsler metric on [Formula: see text] whose unparametrized geodesics coincide with those of [Formula: see text], then [Formula: see text] is a Finsler symmetric space.
{"title":"Periodic solutions of Hilbert’s fourth problem","authors":"J. C. Álvarez Paiva, J. Barbosa Gomes","doi":"10.1142/s1793525321500552","DOIUrl":"https://doi.org/10.1142/s1793525321500552","url":null,"abstract":"It is shown that a possibly irreversible [Formula: see text] Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed [Formula: see text]-form. This is used to prove that if [Formula: see text] is a compact Riemannian symmetric space of rank greater than one and [Formula: see text] is a reversible [Formula: see text] Finsler metric on [Formula: see text] whose unparametrized geodesics coincide with those of [Formula: see text], then [Formula: see text] is a Finsler symmetric space.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"90 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80477408","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 : 2021-10-12DOI: 10.1142/s1793525323500346
Dong Zhang
Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian $Delta_p$, which allows us to analyse and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue $lambda(Delta_p)$, the function $pmapsto p(2lambda(Delta_p))^{frac1p}$ is locally increasing, while the function $pmapsto 2^{-p}lambda(Delta_p)$ is locally decreasing. As a special class of homological eigenvalues, the min-max eigenvalues $lambda_1(Delta_p)$, $cdots$, $lambda_k(Delta_p)$, $cdots$, are locally Lipschitz continuous with respect to $pin[1,+infty)$. We also establish the monotonicity of $p(2lambda_k(Delta_p))^{frac1p}$ and $2^{-p}lambda_k(Delta_p)$ with respect to $pin[1,+infty)$. These results systematically establish a refined analysis of $Delta_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of $p$-Laplacian with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min-max form; (3) refine the higher order Cheeger inequalities for graph $p$-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the $p$-Laplacian case. Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min-max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors' work on discrete Morse theory.
{"title":"Homological Eigenvalues of graph p-Laplacians","authors":"Dong Zhang","doi":"10.1142/s1793525323500346","DOIUrl":"https://doi.org/10.1142/s1793525323500346","url":null,"abstract":"Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian $Delta_p$, which allows us to analyse and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue $lambda(Delta_p)$, the function $pmapsto p(2lambda(Delta_p))^{frac1p}$ is locally increasing, while the function $pmapsto 2^{-p}lambda(Delta_p)$ is locally decreasing. As a special class of homological eigenvalues, the min-max eigenvalues $lambda_1(Delta_p)$, $cdots$, $lambda_k(Delta_p)$, $cdots$, are locally Lipschitz continuous with respect to $pin[1,+infty)$. We also establish the monotonicity of $p(2lambda_k(Delta_p))^{frac1p}$ and $2^{-p}lambda_k(Delta_p)$ with respect to $pin[1,+infty)$. These results systematically establish a refined analysis of $Delta_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of $p$-Laplacian with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min-max form; (3) refine the higher order Cheeger inequalities for graph $p$-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the $p$-Laplacian case. Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min-max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors' work on discrete Morse theory.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"58 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91295862","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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