Pub Date : 2023-11-15DOI: 10.1142/s1793525323500553
Gabriel Katz, Boris Shapiro, Volkmar Welker
In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.
{"title":"Real polynomials with constrained real divisors. i. fundamental groups","authors":"Gabriel Katz, Boris Shapiro, Volkmar Welker","doi":"10.1142/s1793525323500553","DOIUrl":"https://doi.org/10.1142/s1793525323500553","url":null,"abstract":"In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"10 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227551","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 : 2023-11-15DOI: 10.1142/s1793525323500541
Pankaj Kapari, Kashyap Rajeevsarathy
For $ggeq 2$, let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give a theoretical algorithm for computing its roots up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $text{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $text{Mod}(S_g)$, and the commutator subgroup of $text{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, realized by the roots of $T_c^{q(F)}$, where $c$ is a separating curve in $S_g$ of genus $[g/2]$ and $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Finally, for $ggeq 3$ we show that any pseudo-periodic having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $text{Mod}(S_g)$. Consequently, we establish there always exist roots of bounding pair maps and powers of Dehn twists that normally generate $text{Mod}(S_g)$.
{"title":"General primitivity in the mapping class group","authors":"Pankaj Kapari, Kashyap Rajeevsarathy","doi":"10.1142/s1793525323500541","DOIUrl":"https://doi.org/10.1142/s1793525323500541","url":null,"abstract":"For $ggeq 2$, let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give a theoretical algorithm for computing its roots up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $text{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $text{Mod}(S_g)$, and the commutator subgroup of $text{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, realized by the roots of $T_c^{q(F)}$, where $c$ is a separating curve in $S_g$ of genus $[g/2]$ and $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Finally, for $ggeq 3$ we show that any pseudo-periodic having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $text{Mod}(S_g)$. Consequently, we establish there always exist roots of bounding pair maps and powers of Dehn twists that normally generate $text{Mod}(S_g)$.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227558","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 : 2023-11-15DOI: 10.1142/s1793525323500498
Sihan Wei
{"title":"Tomiyama's K-commutative diagrams of minimal dynamical systems","authors":"Sihan Wei","doi":"10.1142/s1793525323500498","DOIUrl":"https://doi.org/10.1142/s1793525323500498","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"2 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139273905","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 : 2023-11-15DOI: 10.1142/s1793525323500462
Sreekrishna Palaparthi, Swapnendu Panda
{"title":"Reducing spheres of genus-2 Heegaard splitting of S3","authors":"Sreekrishna Palaparthi, Swapnendu Panda","doi":"10.1142/s1793525323500462","DOIUrl":"https://doi.org/10.1142/s1793525323500462","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"56 3-4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139271245","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 : 2023-11-15DOI: 10.1142/s1793525323500528
Hongbin Sun, Zhongzi Wang
{"title":"Maps from 3-manifolds to 4-manifolds that induce isomorphisms on π1","authors":"Hongbin Sun, Zhongzi Wang","doi":"10.1142/s1793525323500528","DOIUrl":"https://doi.org/10.1142/s1793525323500528","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139276145","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 : 2023-11-15DOI: 10.1142/s1793525323500486
Regina Rotman
It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 over n}$, or $tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $epsilon >0$ one can find such estimates for the length of a geodesic loop with with angle between $pi-epsilon$ and $pi$ with an explicit constant that depends both on $n$ and $epsilon$. That is, let $epsilon > 0$, and let $a = lceil{ {1 over {sin ({epsilon over 2})}}} rceil+1 $. We will prove that there exists a wide (i.e. with an angle that is wider than $pi-epsilon$) geodesic loop on $M^n$ of length at most $2n!a^nd$. We will also show that there exists a wide geodesic loop of length at most $2(n+1)!^2a^{(n+1)^3} FillRad leq 2 cdot n(n+1)!^2a^{(n+1)^3} vol^{1 over n}$. Here $FillRad$ is the Filling Radius of $M^n$.
目前尚不清楚闭合黎曼流形$M^n$上最短周期测地线的长度是否可以用$c(n) vol^{ 1 over n}$或$tilde{c}(n)d$来表示，其中$n$是$M^n$的尺寸，$vol$表示$M^n$的体积，$d$表示其直径。在本文中，我们将证明，对于每一个$epsilon >0$，我们都可以找到这样的测量回路长度的估计，其角度在$pi-epsilon$和$pi$之间，并具有一个显式常数，该常数依赖于$n$和$epsilon$。也就是说，设$epsilon > 0$，设$a = lceil{ {1 over {sin ({epsilon over 2})}}} rceil+1 $。我们将证明在$M^n$上存在一个长度不超过$2n!a^nd$的宽(即夹角大于$pi-epsilon$)测地线环。我们还将证明存在长度最多为$2(n+1)!^2a^{(n+1)^3} FillRad leq 2 cdot n(n+1)!^2a^{(n+1)^3} vol^{1 over n}$的宽测地线回路。其中$FillRad$为$M^n$的填充半径。
{"title":"Wide short geodesic loops on closed Riemannian manifolds","authors":"Regina Rotman","doi":"10.1142/s1793525323500486","DOIUrl":"https://doi.org/10.1142/s1793525323500486","url":null,"abstract":"It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 over n}$, or $tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $epsilon >0$ one can find such estimates for the length of a geodesic loop with with angle between $pi-epsilon$ and $pi$ with an explicit constant that depends both on $n$ and $epsilon$. That is, let $epsilon > 0$, and let $a = lceil{ {1 over {sin ({epsilon over 2})}}} rceil+1 $. We will prove that there exists a wide (i.e. with an angle that is wider than $pi-epsilon$) geodesic loop on $M^n$ of length at most $2n!a^nd$. We will also show that there exists a wide geodesic loop of length at most $2(n+1)!^2a^{(n+1)^3} FillRad leq 2 cdot n(n+1)!^2a^{(n+1)^3} vol^{1 over n}$. Here $FillRad$ is the Filling Radius of $M^n$.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"11 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227754","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 : 2023-11-15DOI: 10.1142/s1793525323500474
Hyungryul Baik, Dongryul M. Kim, Chenxi Wu
{"title":"Reducible normal generators for mapping class groups are abundant","authors":"Hyungryul Baik, Dongryul M. Kim, Chenxi Wu","doi":"10.1142/s1793525323500474","DOIUrl":"https://doi.org/10.1142/s1793525323500474","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"138 7","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139271090","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 : 2023-11-15DOI: 10.1142/s1793525323500577
Olga Frolkina
{"title":"New examples in dimensions N ≥ 4 related to a question of J.W. Cannon and S.G. Wayment","authors":"Olga Frolkina","doi":"10.1142/s1793525323500577","DOIUrl":"https://doi.org/10.1142/s1793525323500577","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"14 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139271130","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 : 2023-11-10DOI: 10.1142/s1793525323500383
Susumu Hirose, Genki Omori
{"title":"Finite Presentations for the Balanced Superelliptic Mapping Class Groups","authors":"Susumu Hirose, Genki Omori","doi":"10.1142/s1793525323500383","DOIUrl":"https://doi.org/10.1142/s1793525323500383","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":" April","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186914","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 : 2023-11-10DOI: 10.1142/s1793525323500413
Benedikt Hunger
Almost flat finitely generated projective Hilbert C*-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a K-homology class $etain K_*(M)$ twisted with an almost flat bundle in terms of the image of $eta$ under Lafforgue's assembly map and the almost representation associated to the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture.
{"title":"Asymptotically Flat Fredholm Bundles and Assembly","authors":"Benedikt Hunger","doi":"10.1142/s1793525323500413","DOIUrl":"https://doi.org/10.1142/s1793525323500413","url":null,"abstract":"Almost flat finitely generated projective Hilbert C*-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a K-homology class $etain K_*(M)$ twisted with an almost flat bundle in terms of the image of $eta$ under Lafforgue's assembly map and the almost representation associated to the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. ","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"84 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087571","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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