The contraction of the image of the Johnson homomorphism is called the Chillingworth class. In this paper, we derive a combinatorial description of the Chillingworth class for Putman's subsurface Torelli groups. We also prove the naturality and uniqueness properties of the map whose image is the dual of the Chillingworth classes of the subsurface Torelli groups. Moreover, we relate the Chillingworth class of the subsurface Torelli group to the partitioned Johnson homomorphism.
{"title":"Generalized Chillingworth Classes on Subsurface Torelli Groups","authors":"H. Eroğlu","doi":"10.18910/79425","DOIUrl":"https://doi.org/10.18910/79425","url":null,"abstract":"The contraction of the image of the Johnson homomorphism is called the Chillingworth class. In this paper, we derive a combinatorial description of the Chillingworth class for Putman's subsurface Torelli groups. We also prove the naturality and uniqueness properties of the map whose image is the dual of the Chillingworth classes of the subsurface Torelli groups. Moreover, we relate the Chillingworth class of the subsurface Torelli group to the partitioned Johnson homomorphism.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82516547","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 investigate a family of graphs associated to collections of arcs on surfaces. These {it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the "obvious way". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.
{"title":"Geometric simplicial embeddings of arc-type graphs","authors":"H. Parlier, Ashley Weber","doi":"10.4134/JKMS.J190407","DOIUrl":"https://doi.org/10.4134/JKMS.J190407","url":null,"abstract":"In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the \"obvious way\". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79180692","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}
It is shown that the diffeomorphism type of the complement to a real space line arrangement in any dimensional affine ambient space is determined only by the number of lines and the data on multiple points.
证明了在任意维仿射环境空间中，实空间线排列补的微分同态类型仅由线的数目和多点上的数据决定。
{"title":"Topology of complements to real affine space line arrangements","authors":"G. Ishikawa, Motoki Oyama","doi":"10.5427/jsing.2020.22v","DOIUrl":"https://doi.org/10.5427/jsing.2020.22v","url":null,"abstract":"It is shown that the diffeomorphism type of the complement to a real space line arrangement in any dimensional affine ambient space is determined only by the number of lines and the data on multiple points.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75334236","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}
Pub Date : 2019-02-20DOI: 10.1142/s0218216520430038
Haimiao Chen
Suppose $Gamma$ is a discrete group, and $alphain Z^3(BGamma;A)$, with $A$ an abelian group. Given a representation $rho:pi_1(M)toGamma$, with $M$ a closed 3-manifold, put $F(M,rho)=langle(Brho)^ast[alpha],[M]rangle$, where $Brho:Mto BGamma$ is a continuous map inducing $rho$ which is unique up to homotopy, and $langle-,-rangle:H^3(M;A)times H_3(M;mathbb{Z})to A$ is the pairing. We present a practical method for computing $F(M,rho)$ when $M$ is given by a surgery along a link $Lsubset S^3$. In particular, the Chern-Simons invariant can be computed this way.
{"title":"Cohomological invariants of representations of 3-manifold groups","authors":"Haimiao Chen","doi":"10.1142/s0218216520430038","DOIUrl":"https://doi.org/10.1142/s0218216520430038","url":null,"abstract":"Suppose $Gamma$ is a discrete group, and $alphain Z^3(BGamma;A)$, with $A$ an abelian group. Given a representation $rho:pi_1(M)toGamma$, with $M$ a closed 3-manifold, put $F(M,rho)=langle(Brho)^ast[alpha],[M]rangle$, where $Brho:Mto BGamma$ is a continuous map inducing $rho$ which is unique up to homotopy, and $langle-,-rangle:H^3(M;A)times H_3(M;mathbb{Z})to A$ is the pairing. We present a practical method for computing $F(M,rho)$ when $M$ is given by a surgery along a link $Lsubset S^3$. In particular, the Chern-Simons invariant can be computed this way.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85144785","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}
Pub Date : 2019-02-19DOI: 10.1142/S0218216519500949
M. Freedman, J. Hillman
We extend the classical definition of {it width} to higher dimensional, smooth codimension 2 knots and show in each dimension there are knots of arbitrarily large width.
我们将{it宽度}的经典定义扩展到更高维度，光滑余维2节，并显示在每个维度中都有任意大宽度的节。
{"title":"Width of codimension two knots","authors":"M. Freedman, J. Hillman","doi":"10.1142/S0218216519500949","DOIUrl":"https://doi.org/10.1142/S0218216519500949","url":null,"abstract":"We extend the classical definition of {it width} to higher dimensional, smooth codimension 2 knots and show in each dimension there are knots of arbitrarily large width.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87240288","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}
Pub Date : 2019-02-18DOI: 10.1142/S021821652050087X
A. Bosman
We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake r-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly r-shake slice and strongly r shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for r=0 we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor mu bar invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.
{"title":"Shake slice and shake concordant links","authors":"A. Bosman","doi":"10.1142/S021821652050087X","DOIUrl":"https://doi.org/10.1142/S021821652050087X","url":null,"abstract":"We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake r-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly r-shake slice and strongly r shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for r=0 we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor mu bar invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"180 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78725112","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}
For any positive integer $g$, we completely determine the minimal genus function for $Sigma_{g}times T^{2}$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_{2}(Sigma_{g}times T^{2})$. However, we construct a suitable embedded surface for each class and we have exact values of minimal genus functions.
{"title":"Minimal genus problem for T2–bundles over\u0000surfaces","authors":"R. Nakashima","doi":"10.2140/AGT.2021.21.893","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.893","url":null,"abstract":"For any positive integer $g$, we completely determine the minimal genus function for $Sigma_{g}times T^{2}$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_{2}(Sigma_{g}times T^{2})$. However, we construct a suitable embedded surface for each class and we have exact values of minimal genus functions.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76450926","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}
Let $Sigma$ be a compact oriented surface. The Dehn twist along every simple closed curve $gamma subset Sigma$ induces an automorphism of the fundamental group $pi$ of $Sigma$. There are two possible ways to generalize such automorphisms if the curve $gamma$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $gamma$: an automorphism of the Malcev completion of $pi$ whose definition involves intersection operations and only depends on the homotopy class $[gamma]in pi$ of $gamma$. Another way is to choose in the usual cylinder $U:=Sigma times [-1,+1]$ a knot $L$ projecting onto $gamma$, to perform a surgery along $L$ so as to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotent quotient $pi/Gamma_{j} pi$ of $pi$ (where $Gamma_jpi$ denotes the subgroup of $pi$ generated by commutators of length $j$). In this paper, assuming that $[gamma]$ is in $Gamma_k pi$ for some $kgeq 2$, we prove that (whatever the choice of $L$ is) the automorphism of $pi/Gamma_{2k+1} pi$ induced by $U_L$ agrees with the generalized Dehn twist along $gamma$ and we explicitly compute this automorphism in terms of $[gamma]$ modulo ${Gamma_{k+2}}pi$. As applications, we obtain new formulas for certain evaluations of the Johnson homomorphisms showing, in particular, how to realize any element of their targets by some explicit homology cylinders and/or generalized Dehn twists.
设$Sigma$为紧致定向曲面。沿每条简单闭曲线$gamma subset Sigma$的Dehn扭转引起$Sigma$的基本群$pi$的自同构。如果允许曲线$gamma$具有自交，则有两种可能的方法来推广这种自同构。一种方法是考虑$gamma$上的“广义Dehn扭转”:$pi$的Malcev补全的自同构，其定义涉及交操作并且仅依赖于$gamma$的同伦类$[gamma]in pi$。另一种方法是在通常的$U:=Sigma times [-1,+1]$柱面上选择一个结点$L$投影到$gamma$上，沿着$L$进行手术，得到一个同源柱面$U_L$，并让$U_L$作用于$pi$的每一个幂零商$pi/Gamma_{j} pi$(其中$Gamma_jpi$表示长度为$j$的对易子产生的$pi$子群)。本文假设$[gamma]$对于某个$kgeq 2$在$Gamma_k pi$中，证明了(无论$L$的选择是什么)由$U_L$引起的$pi/Gamma_{2k+1} pi$的自同构符合沿$gamma$的广义Dehn扭曲，并明确地计算了$[gamma]$模${Gamma_{k+2}}pi$的自同构。作为应用，我们得到了Johnson同态的某些评价的新公式，特别是如何通过一些显式同态柱面和/或广义Dehn扭转来实现其目标的任何元素。
{"title":"Generalized Dehn twists on surfaces and homology cylinders","authors":"Y. Kuno, G. Massuyeau","doi":"10.2140/AGT.2021.21.697","DOIUrl":"https://doi.org/10.2140/AGT.2021.21.697","url":null,"abstract":"Let $Sigma$ be a compact oriented surface. The Dehn twist along every simple closed curve $gamma subset Sigma$ induces an automorphism of the fundamental group $pi$ of $Sigma$. There are two possible ways to generalize such automorphisms if the curve $gamma$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $gamma$: an automorphism of the Malcev completion of $pi$ whose definition involves intersection operations and only depends on the homotopy class $[gamma]in pi$ of $gamma$. Another way is to choose in the usual cylinder $U:=Sigma times [-1,+1]$ a knot $L$ projecting onto $gamma$, to perform a surgery along $L$ so as to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotent quotient $pi/Gamma_{j} pi$ of $pi$ (where $Gamma_jpi$ denotes the subgroup of $pi$ generated by commutators of length $j$). In this paper, assuming that $[gamma]$ is in $Gamma_k pi$ for some $kgeq 2$, we prove that (whatever the choice of $L$ is) the automorphism of $pi/Gamma_{2k+1} pi$ induced by $U_L$ agrees with the generalized Dehn twist along $gamma$ and we explicitly compute this automorphism in terms of $[gamma]$ modulo ${Gamma_{k+2}}pi$. As applications, we obtain new formulas for certain evaluations of the Johnson homomorphisms showing, in particular, how to realize any element of their targets by some explicit homology cylinders and/or generalized Dehn twists.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82823478","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 give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.
{"title":"On 3-manifolds that are boundaries of exotic 4-manifolds","authors":"John B. Etnyre, Hyunki Min, Anubhav Mukherjee","doi":"10.1090/tran/8586","DOIUrl":"https://doi.org/10.1090/tran/8586","url":null,"abstract":"We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"22 35","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91418529","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}
Pub Date : 2019-01-11DOI: 10.1017/CBO9781139106993.004
Cyril Lecuire
The Masur domain is a subset of the space of projective measured geodesic laminations on the boundary of a 3-manifold M. This domain plays an important role in the study of the hyperbolic structures on the interior of M. In this paper, we define an extension of the Masur domain and explain that it shares a lot of properties with the Masur domain.
{"title":"Spaces of Kleinian Groups: An extension of the Masur domain","authors":"Cyril Lecuire","doi":"10.1017/CBO9781139106993.004","DOIUrl":"https://doi.org/10.1017/CBO9781139106993.004","url":null,"abstract":"The Masur domain is a subset of the space of projective measured geodesic laminations on the boundary of a 3-manifold M. This domain plays an important role in the study of the hyperbolic structures on the interior of M. In this paper, we define an extension of the Masur domain and explain that it shares a lot of properties with the Masur domain.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76140855","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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