scholarly journals The fundamental group of an algebra with a strongly simply connected Galois covering

2019 ◽  
Vol 18 (03) ◽  
pp. 1950052
Author(s):  
Diane Castonguay ◽  
Claudia Chaio ◽  
Sonia Trepode
Keyword(s):  
Fundamental Group ◽  
Galois Covering ◽  
Triangular Algebra ◽  
Simply Connected

In this work, we prove that if a triangular algebra [Formula: see text] admits a strongly simply connected universal Galois covering for a given presentation, then the fundamental group associated to this presentation is free.

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

Non-cocompact Group Actions and -Semistability at Infinity

2019 ◽  
Vol 72 (5) ◽  
pp. 1275-1303 ◽  
Author(s):  
Ross Geoghegan ◽  
Craig Guilbault ◽  
Michael Mihalik
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Fundamental Property ◽  
Companion Paper ◽  
Infinite Index ◽  
Locally Compact ◽  
Finitely Presented Groups ◽  
Simply Connected ◽  
Open Question ◽  
Finitely Presented

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

scholarly journals On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

2020 ◽  
Author(s):  
Michael Wiemeler
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Positive Scalar Curvature ◽  
Spin Manifold ◽  
Homotopy Groups ◽  
Positive Scalar ◽  
Simply Connected ◽  
Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals On the fundamental group of a Lie semigroup

1992 ◽  
Vol 34 (3) ◽  
pp. 379-394 ◽  
Author(s):  
Karl-Hermann Neeb
Keyword(s):  
Fundamental Group ◽  
Lie Group ◽  
Unit Group ◽  
Universal Covering ◽  
Covering Group ◽  
Lie Semigroup ◽  
Lie Semigroups ◽  
Inclusion Mapping ◽  
Image Position ◽  
Simply Connected

The simplest type of Lie semigroups are closed convex cones in finite dimensional vector spaces. In general one defines a Lie semigroup to be a closed subsemigroup of a Lie group which is generated by one-parameter semigroups. If W is a closed convex cone in a vector space V, then W is convex and therefore simply connected. A similar statement for Lie semigroups is false in general. There exist generating Lie semigroups in simply connected Lie groups which are not simply connected (Example 1.15). To find criteria for cases when this is true, one has to consider the homomorphisminduced by the inclusion mapping i:S→G, where S is a generating Lie semigroup in the Lie group G. Our main results concern the description of the image and the kernel of this mapping. We show that the image is the fundamental group of the largest covering group of G, into which S lifts, and that the kernel is the fundamental group of the inverse image of 5 in the universal covering group G. To get these results we construct a universal covering semigroup S of S. If j: H(S): = S ∩ S-1 →S is the inclusion mapping of the unit group of S into S, then it turns out that the kernel of the induced mappingmay be identfied with the fundamental group of the unit group H(S)of S and that its image corresponds to the intersection H(S)0 ⋂π1(S), where π1(s) is identified with a central subgroup of S.

FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

2000 ◽  
Vol 02 (01) ◽  
pp. 75-86 ◽  
Author(s):  
FUQUAN FANG ◽  
XIAOCHUN RONG
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Fundamental Groups ◽  
Vanishing Theorem ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Finite Fundamental Group ◽  
Finiteness Theorems ◽  
Point Set ◽  
Simply Connected

We prove a vanishing theorem of certain cohomology classes for an 2n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any Tk-action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ ≤ sec ≤ Λ, and diameter, diam ; ≤ d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n ≤ 6) with λ ≤ sec ≤ Λ and diam ≤ d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d.

scholarly journals On the formality and strong Lefschetz property of symplectic manifolds – Erratum

2009 ◽  
Vol 147 (1) ◽  
pp. 255-255
Author(s):  
Taek Kyu Hwang ◽  
Jin Hong Kim
Keyword(s):  
Fundamental Group ◽  
Symplectic Manifold ◽  
Symplectic Manifolds ◽  
Finite Covering ◽  
Massey Product ◽  
Lefschetz Property ◽  
The Stability ◽  
Simply Connected ◽  
Open Question ◽  
Strong Lefschetz Property

Professor Vicente Muñoz kindly informed us that there is an inaccuracy in Lemma 3.5 of [1]. The correct statement of Lemma 3.5 is now that the fundamental group π1(X′) of the manifold X′ is Z, since the monodromy coming from φ8 does not imply that g4 = g4−1. Therefore, what we have actually constructed in Section 3 of [1] is a closed non-formal 8-dimensional symplectic manifold with π1 = Z whose triple Massey product is non-zero, so that the simply-connectedness in Theorem 1.1 should be dropped. As far as we know, the existence of a simply connected closed non-formal 8-dimensional symplectic manifold whose triple Massey product is non-zero still remains an open question. All other main results, especially Theorem 1.2 and Corollary 1.3, in [1] are not affected by this mistake. Furthermore, the stability of the non-formality under a finite covering as in Subsection 3.3 holds in general. We want to thank Professor Muñoz for his careful reading.

scholarly journals THE FUNDAMENTAL GROUP OF G-MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1250056 ◽  
Author(s):  
HUI LI
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Lie Group ◽  
Maximal Torus ◽  
Moment Map ◽  
Compact Lie Group ◽  
Identity Component ◽  
Connected Subgroup ◽  
Stabilizer Group ◽  
Simply Connected

Let G be a connected compact Lie group, and let M be a connected Hamiltonian G-manifold with equivariant moment map ϕ. We prove that if there is a simply connected orbit G ⋅ x, then π1(M) ≅ π1(M/G); if additionally ϕ is proper, then π1(M) ≅ π1 (ϕ-1(G⋅a)), where a = ϕ(x). We also prove that if a maximal torus of G has a fixed point x, then π1(M) ≅ π1(M/K), where K is any connected subgroup of G; if additionally ϕ is proper, then π1(M) ≅ π1(ϕ-1(G⋅a)) ≅ π1(ϕ-1(a)), where a = ϕ(x). Furthermore, we prove that if ϕ is proper, then [Formula: see text] for all a ∈ ϕ(M), where [Formula: see text] is any connected subgroup of G which contains the identity component of each stabilizer group; in particular, π1(M/G) ≅ π1(ϕ-1(G⋅a)/G) for all a ∈ ϕ(M).

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