In this paper, we determine for which nonnegative integers $k$, $l$ and for which homotopy $7-$sphere $Sigma$ the manifold $kS^{2}times S^{5}#lS^{3}times S^{4}#Sigma$ admits a free smooth circle action.
{"title":"Free circle actions on certain simply connected $7-$manifolds","authors":"Fupeng Xu","doi":"arxiv-2409.04938","DOIUrl":"https://doi.org/arxiv-2409.04938","url":null,"abstract":"In this paper, we determine for which nonnegative integers $k$, $l$ and for\u0000which homotopy $7-$sphere $Sigma$ the manifold $kS^{2}times\u0000S^{5}#lS^{3}times S^{4}#Sigma$ admits a free smooth circle action.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192033","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}
In this paper, we study a model for $S^1$-equivariant monopole Floer homology for rational homology three-spheres via a homological device called $mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an $mathbf{R}$-filtration on the (equivariant) complex of monopole Floer homology $HM$. This $mathbf{R}$-filtration fits $HM$ into a persistent homology theory, from which one can define a numerical quantity called the spectral invariant $rho$. The spectral invariant $rho$ is tied with the geometry of the underlying manifold. The main result of the papers shows that $rho$ provides an obstruction to the existence of positive scalar curvature metric on a ribbon homology cobordism.
{"title":"Spectral invariants and equivariant monopole Floer homology for rational homology three-spheres","authors":"Minh Lam Nguyen","doi":"arxiv-2409.04954","DOIUrl":"https://doi.org/arxiv-2409.04954","url":null,"abstract":"In this paper, we study a model for $S^1$-equivariant monopole Floer homology\u0000for rational homology three-spheres via a homological device called\u0000$mathcal{S}$-complex. Using the Chern-Simons-Dirac functional, we define an\u0000$mathbf{R}$-filtration on the (equivariant) complex of monopole Floer homology\u0000$HM$. This $mathbf{R}$-filtration fits $HM$ into a persistent homology theory,\u0000from which one can define a numerical quantity called the spectral invariant\u0000$rho$. The spectral invariant $rho$ is tied with the geometry of the\u0000underlying manifold. The main result of the papers shows that $rho$ provides\u0000an obstruction to the existence of positive scalar curvature metric on a ribbon\u0000homology cobordism.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192046","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}
This paper studies the relationship between an analytic compactification of the moduli space of flat $mathrm{SL}_2(mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of the $mathrm{SL}_2(mathbb{C})$ character variety of the fundamental group of $M$. We exhibit an explicit correspondence between $mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $mathbb{R}$-trees, as initially proposed by Taubes.
{"title":"Z/2 harmonic 1-forms, R-trees, and the Morgan-Shalen compactification","authors":"Siqi He, Richard Wentworth, Boyu Zhang","doi":"arxiv-2409.04956","DOIUrl":"https://doi.org/arxiv-2409.04956","url":null,"abstract":"This paper studies the relationship between an analytic compactification of\u0000the moduli space of flat $mathrm{SL}_2(mathbb{C})$ connections on a closed,\u0000oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen\u0000compactification of the $mathrm{SL}_2(mathbb{C})$ character variety of the\u0000fundamental group of $M$. We exhibit an explicit correspondence between\u0000$mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic\u0000maps to $mathbb{R}$-trees, as initially proposed by Taubes.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192039","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}
We study the embedded topology of certain conic-line arrangements of degree 7. Two new examples of Zariski pairs are given. Furthermore, we determine the number of connected components of the conic-line arrangements. We also calculate the fundamental groups using SageMath and the package Sirocco in the appendix.
{"title":"The realization spaces of certain conic-line arrangements of degree 7","authors":"Shinzo Bannai, Hiro-o Tokunaga, Emiko Yorisaki","doi":"arxiv-2409.05011","DOIUrl":"https://doi.org/arxiv-2409.05011","url":null,"abstract":"We study the embedded topology of certain conic-line arrangements of degree\u00007. Two new examples of Zariski pairs are given. Furthermore, we determine the\u0000number of connected components of the conic-line arrangements. We also\u0000calculate the fundamental groups using SageMath and the package Sirocco in the\u0000appendix.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"137 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192037","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}
We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving in the Hattori-Stong integrability conditions. We also give an evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers. As an application, it can be showed that the signature of a $2n$-dimensional stably almost-complex manifold whose possibly nonzero Chern numbers being $c_n$ and $c_ic_{n-i}$ is even, which particularly rules out the existence of such structure on rational projective planes. Some other related results and remarks are also discussed in this article.
{"title":"Explicit formulas for the Hattori-Stong theorem and applications","authors":"Ping Li, Wangyang Lin","doi":"arxiv-2409.05107","DOIUrl":"https://doi.org/arxiv-2409.05107","url":null,"abstract":"We employ combinatorial techniques to present an explicit formula for the\u0000coefficients in front of Chern classes involving in the Hattori-Stong\u0000integrability conditions. We also give an evenness condition for the signature\u0000of stably almost-complex manifolds in terms of Chern numbers. As an\u0000application, it can be showed that the signature of a $2n$-dimensional stably\u0000almost-complex manifold whose possibly nonzero Chern numbers being $c_n$ and\u0000$c_ic_{n-i}$ is even, which particularly rules out the existence of such\u0000structure on rational projective planes. Some other related results and remarks\u0000are also discussed in this article.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192035","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}
The $mathrm{PGL}_n(mathbb{R})$-Hitchin component of a closed oriented surface is a preferred component of the character variety consisting of homomorphisms from the fundamental group of the surface to the projective linear group $mathrm{PGL}_n(mathbb{R})$. It admits a symplectic structure, defined by the Atiyah-Bott-Goldman symplectic form. The main result of the article is an explicit computation of this symplectic form in terms of certain global coordinates for the Hitchin component. A remarkable feature of this expression is that its coefficients are constant.
{"title":"The symplectic structure of the $mathrm{PGL}_n(mathbb{R})$-Hitchin component","authors":"Francis Bonahon, Yaşar Sözen, Hatice Zeybek","doi":"arxiv-2409.04905","DOIUrl":"https://doi.org/arxiv-2409.04905","url":null,"abstract":"The $mathrm{PGL}_n(mathbb{R})$-Hitchin component of a closed oriented\u0000surface is a preferred component of the character variety consisting of\u0000homomorphisms from the fundamental group of the surface to the projective\u0000linear group $mathrm{PGL}_n(mathbb{R})$. It admits a symplectic structure,\u0000defined by the Atiyah-Bott-Goldman symplectic form. The main result of the\u0000article is an explicit computation of this symplectic form in terms of certain\u0000global coordinates for the Hitchin component. A remarkable feature of this\u0000expression is that its coefficients are constant.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192036","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}
Ian Biringer, Yassin Chandran, Tommaso Cremaschi, Jing Tao, Nicholas G. Vlamis, Mujie Wang, Brandis Whitfield
We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $mathbb{Z}/nmathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.
{"title":"Covers of surfaces","authors":"Ian Biringer, Yassin Chandran, Tommaso Cremaschi, Jing Tao, Nicholas G. Vlamis, Mujie Wang, Brandis Whitfield","doi":"arxiv-2409.03967","DOIUrl":"https://doi.org/arxiv-2409.03967","url":null,"abstract":"We study the homeomorphism types of certain covers of (always orientable)\u0000surfaces, usually of infinite-type. We show that every surface with non-abelian\u0000fundamental group is covered by every noncompact surface, we identify the\u0000universal abelian covers and the $mathbb{Z}/nmathbb{Z}$-homology covers of\u0000surfaces, and we show that non-locally finite characteristic covers of surfaces\u0000have four possible homeomorphism types.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192041","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}
We classify the finite orbits of the mapping class group action on the character variety of Deroin--Tholozan representations of punctured spheres. In particular, we prove that the action has no finite orbits if the underlying sphere has 7 punctures or more. When the sphere has six punctures, we show that there is a unique 1-parameter family of finite orbits. Our methods also recover Tykhyy's classification of finite orbits for 5-punctured spheres. The proof is inductive and uses Lisovyy--Tykhyy's classification of finite mapping class group orbits for 4-punctured spheres as the base case for the induction. Our results on Deroin--Tholozan representations cover the last missing cases to complete the proof of Tykhyy's Conjecture on finite mapping class group orbits for $mathrm{SL}_2mathbb{C}$ representations of punctured spheres, after the recent work by Lam--Landesman--Litt.
{"title":"Tykhyy's Conjecture on finite mapping class group orbits","authors":"Samuel Bronstein, Arnaud Maret","doi":"arxiv-2409.04379","DOIUrl":"https://doi.org/arxiv-2409.04379","url":null,"abstract":"We classify the finite orbits of the mapping class group action on the\u0000character variety of Deroin--Tholozan representations of punctured spheres. In\u0000particular, we prove that the action has no finite orbits if the underlying\u0000sphere has 7 punctures or more. When the sphere has six punctures, we show that\u0000there is a unique 1-parameter family of finite orbits. Our methods also recover\u0000Tykhyy's classification of finite orbits for 5-punctured spheres. The proof is\u0000inductive and uses Lisovyy--Tykhyy's classification of finite mapping class\u0000group orbits for 4-punctured spheres as the base case for the induction. Our results on Deroin--Tholozan representations cover the last missing cases\u0000to complete the proof of Tykhyy's Conjecture on finite mapping class group\u0000orbits for $mathrm{SL}_2mathbb{C}$ representations of punctured spheres,\u0000after the recent work by Lam--Landesman--Litt.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"4300 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192042","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}
In this paper, we determine those $(n-1)$-connected $(2n+1)$-manifolds with torsion free homology that admit free circle actions up to almost diffeomorphism, provided that $nequiv5,7 mod 8$.
在本文中，我们确定了那些具有无扭转同调的 $(n-1)$ 连接的 $(2n+1)$ 曼方形，只要 $nequiv5,7 mod 8$，这些曼方形就允许自由圆作用达到近乎衍射。
{"title":"Free circle actions on $(n-1)$-connected $(2n+1)$-manifolds","authors":"Yi Jiang, Yang Su","doi":"arxiv-2409.03194","DOIUrl":"https://doi.org/arxiv-2409.03194","url":null,"abstract":"In this paper, we determine those $(n-1)$-connected $(2n+1)$-manifolds with\u0000torsion free homology that admit free circle actions up to almost\u0000diffeomorphism, provided that $nequiv5,7 mod 8$.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"136 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192057","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}
Recently, Kashaev and the first author defined a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, coming from the $R$-matrix of a rank 2 Nichols algebra, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the $V_n$ polynomials for $n=1,2,3,4$ and discover applications and emerging patterns, including unexpected Conway mutations that seem undetected by the $V_n$-polynomials as well as by Heegaard Floer Homology and Knot Floer Homology.
{"title":"Patterns of the $V_2$-polynomial of knots","authors":"Stavros Garoufalidis, Shana Yunsheng Li","doi":"arxiv-2409.03557","DOIUrl":"https://doi.org/arxiv-2409.03557","url":null,"abstract":"Recently, Kashaev and the first author defined a sequence $V_n$ of 2-variable\u0000knot polynomials with integer coefficients, coming from the $R$-matrix of a\u0000rank 2 Nichols algebra, the first polynomial been identified with the\u0000Links--Gould polynomial. In this note we present the results of the computation\u0000of the $V_n$ polynomials for $n=1,2,3,4$ and discover applications and emerging\u0000patterns, including unexpected Conway mutations that seem undetected by the\u0000$V_n$-polynomials as well as by Heegaard Floer Homology and Knot Floer\u0000Homology.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192045","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}
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