Let M be a compact n-manifold of positive sectional curvature.We will review classical results on the fundamental group of M,a motivation on the c(n)-cyclic conjecture that the fundamental group of M contains a cyclic...Let M be a compact n-manifold of positive sectional curvature.We will review classical results on the fundamental group of M,a motivation on the c(n)-cyclic conjecture that the fundamental group of M contains a cyclic subgroup of index bounded above by c(n),a constant depending only on n,and we will survey partial results(up to date)on the c(n)-cyclic conjecture.展开更多
Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positiv...Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.展开更多
We call a group A-simple,if it has no non-trivial normal abelian subgroup.We will present finiteness results in controlled topology via geometry on manifolds whose fundamental groups are A-simple.
We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form<h...We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form<h>×D,where h has infinite order and D is finite.展开更多
In this paper , depending on the theory of buildings, we obtain a kind of method completely different from past ones, to compute topological fundamental groups of some triangle geometries. This method will enable us...In this paper , depending on the theory of buildings, we obtain a kind of method completely different from past ones, to compute topological fundamental groups of some triangle geometries. This method will enable us to easily compulte topological fundametal groups of infinitely finite triangle geometries.展开更多
In this paper,we study the growth of fundamental groups of Finsler manifolds.Some relationships between the growth of fundamental groups and the volume growth of universal covers of Finsler manifolds are found.Some es...In this paper,we study the growth of fundamental groups of Finsler manifolds.Some relationships between the growth of fundamental groups and the volume growth of universal covers of Finsler manifolds are found.Some estimates of entropies and the number of generators of fundamental groups of Finsler manifolds are given.Moreover,the quasi-isometry and the geometric norm in Finsler geometry are considered.展开更多
In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this present...In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this presentation,we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.展开更多
In this article,we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth.We prove that on each e...In this article,we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth.We prove that on each energy level E(x,v)=k with k>c(L),where c(L)is Mane’s critical value,the EulerLagrange flow has positive topological entropy.This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.展开更多
Using the theory of harmonic maps the authors discuss theproperties of the fundamental group of a complete nonpositivelycurved Riemannian manifold, and prove that the finitely generatedvirtual solvable subgroup of fun...Using the theory of harmonic maps the authors discuss theproperties of the fundamental group of a complete nonpositivelycurved Riemannian manifold, and prove that the finitely generatedvirtual solvable subgroup of fundamental group of a completenonpositively curved Riemannian manifold either is a peripheralsubgroup of fundamental group or can be realized by animmersed totall geodesic closed flat manifold. It generalizessome results of Gromoll-Wolf, Lawson-Yan and Schoen-Yau.展开更多
It is shown that the generalized Baumslag-Solitar graphs which have soluble fundamental groups can be constructed from four simple types of graph by a process of decoration. The method of proof is elementary and emplo...It is shown that the generalized Baumslag-Solitar graphs which have soluble fundamental groups can be constructed from four simple types of graph by a process of decoration. The method of proof is elementary and employs the operations of pinching and contracting edges of graphs.展开更多
In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each oth...In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.展开更多
In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifol...In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifold fundamental group is isomorphic to the deck translation group.展开更多
The aim of this study is to establish that, the equivalent class <img src="Edit_d35dd794-39a5-4ce4-992b-5130559b3c82.png" width="70" height="22" alt="" /> which is made up...The aim of this study is to establish that, the equivalent class <img src="Edit_d35dd794-39a5-4ce4-992b-5130559b3c82.png" width="70" height="22" alt="" /> which is made up of homotopic loops is a group with respect to <img src="Edit_3577ec7c-e6f5-4d71-8bd5-c63ea8fdb24f.png" width="30" height="15" alt="" /> in the general interval <span style="white-space:nowrap;">[m,n]. The study proved from homotopical point of view that <img src="Edit_4cb511c3-e469-47e3-bd9c-e971594f939c.png" width="70" height="22" alt="" /> is associative, has an identity and inverse function. The study established with proof that <img src="Edit_39497a4b-b0e9-40d9-8f31-49816e760d6a.png" width="70" height="22" alt="" /> is a fundamental group in <span style="white-space:nowrap;">[m,n] ,<img src="Edit_077b19f1-afb3-41f5-8d39-df073165c9dc.png" width="75" height="18" alt="" />.展开更多
In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.
The authors establish some uniform estimates for the distance to halfway points of minimalgeodesics in terms of the distantce to end points on some types of Riemannian manifolds, andthen prove some theorems about the ...The authors establish some uniform estimates for the distance to halfway points of minimalgeodesics in terms of the distantce to end points on some types of Riemannian manifolds, andthen prove some theorems about the finite generation of fundamental group of Riemannianmanifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.展开更多
Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including al...Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including all those not containing two-sided RP2’s,we determineχ_(4)(π).We address when p(π)equalsχ_(4)(π)and when q^(*)(π)equalsχ_(4)(π),and answer a question raised by Hillman(2002).展开更多
Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is ...Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.展开更多
The author constructs the sheaf of generalized fundamental group of a topological space Xand obtains the relationship between its lst cohomology set and all covering spaces Of X undersuitable conditions imposed upon X.
We introduce a representation via(n+l)-colored graphs of compact n-manifolds with(possibly empty)boundary,which appears to be very convenient for computer aided study and tabulation.Our construction is a generalizatio...We introduce a representation via(n+l)-colored graphs of compact n-manifolds with(possibly empty)boundary,which appears to be very convenient for computer aided study and tabulation.Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three,and it is dual to the one given by Pezzana in the 1970s.In this context we establish some results concerning the topology of the represented manifolds:suspensions,fundamental groups,connected sums and moves between graphs representing the same manifold.Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained,together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.展开更多
文摘Let M be a compact n-manifold of positive sectional curvature.We will review classical results on the fundamental group of M,a motivation on the c(n)-cyclic conjecture that the fundamental group of M contains a cyclic subgroup of index bounded above by c(n),a constant depending only on n,and we will survey partial results(up to date)on the c(n)-cyclic conjecture.
文摘Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.
基金the author Rong at Capital Normal University,which was partially supported by NSFC Grant 11821101,Beijing Natural Science Foundation Z19003,and a research fund from Capital Normal University.
文摘We call a group A-simple,if it has no non-trivial normal abelian subgroup.We will present finiteness results in controlled topology via geometry on manifolds whose fundamental groups are A-simple.
文摘We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form<h>×D,where h has infinite order and D is finite.
文摘In this paper , depending on the theory of buildings, we obtain a kind of method completely different from past ones, to compute topological fundamental groups of some triangle geometries. This method will enable us to easily compulte topological fundametal groups of infinitely finite triangle geometries.
基金supported by National Natural Science Foundation of China (Grant No.10871171)
文摘In this paper,we study the growth of fundamental groups of Finsler manifolds.Some relationships between the growth of fundamental groups and the volume growth of universal covers of Finsler manifolds are found.Some estimates of entropies and the number of generators of fundamental groups of Finsler manifolds are given.Moreover,the quasi-isometry and the geometric norm in Finsler geometry are considered.
基金Partially supported by the NSFC(Grant No.11971112)the China Scholarship Council(Grant No.202106100095)。
文摘In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this presentation,we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.
基金supported by National Natural Science Foundation of China(Grant Nos.11301305 and 11571207)supported by the State Scholarship Fund from China Scholarship Council(CSC)+2 种基金supported by National Natural Science Foundation of China(Grant No.11701559)the Fundamental Research Funds for the Central Universities(Grant No.2018QC054)supported by National Natural Science Foundation of China(Grant No.11571387)。
文摘In this article,we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth.We prove that on each energy level E(x,v)=k with k>c(L),where c(L)is Mane’s critical value,the EulerLagrange flow has positive topological entropy.This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.
文摘Using the theory of harmonic maps the authors discuss theproperties of the fundamental group of a complete nonpositivelycurved Riemannian manifold, and prove that the finitely generatedvirtual solvable subgroup of fundamental group of a completenonpositively curved Riemannian manifold either is a peripheralsubgroup of fundamental group or can be realized by animmersed totall geodesic closed flat manifold. It generalizessome results of Gromoll-Wolf, Lawson-Yan and Schoen-Yau.
文摘It is shown that the generalized Baumslag-Solitar graphs which have soluble fundamental groups can be constructed from four simple types of graph by a process of decoration. The method of proof is elementary and employs the operations of pinching and contracting edges of graphs.
基金The NSF(11329101,11431009,11301135 and 11471245) of China
文摘In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.
基金Supported by General Project of Science Research of Guangzhou(Grant No.2017070126)National Natural Science Foundation of China(Grant No.11226034).
文摘In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifold fundamental group is isomorphic to the deck translation group.
文摘The aim of this study is to establish that, the equivalent class <img src="Edit_d35dd794-39a5-4ce4-992b-5130559b3c82.png" width="70" height="22" alt="" /> which is made up of homotopic loops is a group with respect to <img src="Edit_3577ec7c-e6f5-4d71-8bd5-c63ea8fdb24f.png" width="30" height="15" alt="" /> in the general interval <span style="white-space:nowrap;">[m,n]. The study proved from homotopical point of view that <img src="Edit_4cb511c3-e469-47e3-bd9c-e971594f939c.png" width="70" height="22" alt="" /> is associative, has an identity and inverse function. The study established with proof that <img src="Edit_39497a4b-b0e9-40d9-8f31-49816e760d6a.png" width="70" height="22" alt="" /> is a fundamental group in <span style="white-space:nowrap;">[m,n] ,<img src="Edit_077b19f1-afb3-41f5-8d39-df073165c9dc.png" width="75" height="18" alt="" />.
基金Supported by the National Natural Science Foundation of China (10701032)Natural Science Foundation of Hebei Province (A2008000132)the Doctoral Foundation of Hebei Normal University (L2005B02)
文摘In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.
基金the National Natural Science Foundation of China(No.19971081).
文摘The authors establish some uniform estimates for the distance to halfway points of minimalgeodesics in terms of the distantce to end points on some types of Riemannian manifolds, andthen prove some theorems about the finite generation of fundamental group of Riemannianmanifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.
基金supported by Simons Collaborations in Mathematics and the Physical Sciences(Grant No.615229).
文摘Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including all those not containing two-sided RP2’s,we determineχ_(4)(π).We address when p(π)equalsχ_(4)(π)and when q^(*)(π)equalsχ_(4)(π),and answer a question raised by Hillman(2002).
基金supported by DAADEU-network HPRN-CT-2009-00099(EAGER)+2 种基金The Emmy Noether Research Institute for Mathematicsthe Minerva Foundation of GermanyThe Israel Science Foun dation grant #8008/02-3 (Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties")
文摘Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.
文摘The author constructs the sheaf of generalized fundamental group of a topological space Xand obtains the relationship between its lst cohomology set and all covering spaces Of X undersuitable conditions imposed upon X.
文摘We introduce a representation via(n+l)-colored graphs of compact n-manifolds with(possibly empty)boundary,which appears to be very convenient for computer aided study and tabulation.Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three,and it is dual to the one given by Pezzana in the 1970s.In this context we establish some results concerning the topology of the represented manifolds:suspensions,fundamental groups,connected sums and moves between graphs representing the same manifold.Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained,together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.
