Finite Fundamental Group Research Articles

Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups

Abstract We show that for an oriented 4-dimensional Poincaré complex X with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type of X is determined by its quadratic 2-type.

Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group

Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group

Geometric topology of CR-warped products in six-dimensional sphere

<p>In this paper, we prove that a three-dimensional CR-warped product submanifold of a nearly Kaehler six-dimensional sphere, under some restrictions, contains finite fundamental groups. Using another approach from the theory of eigenvalues and also the gradient of Ricci curvature, we obtain similar results.</p>

Low-dimensional solenoidal manifolds

In this paper we survey n -dimensional solenoidal manifolds for n=1,2 and 3, and present new results about them. Solenoidal manifolds of dimension n are metric spaces locally modeled on the product of a Cantor set and an open n -dimensional disk. Therefore, they can be “laminated” (or “foliated”) by n -dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids, i.e., are obtained as the inverse limit of an increasing tower of finite, regular covering spaces of a compact manifold with an infinite and residually finite fundamental group. In this case, their structure is very rich since they are principal Cantor-group bundles over a compact manifold and behave like “laminated” versions of compact manifolds, thus they share many of their properties.

Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Given a closed smooth manifold M of even dimension 2n≥6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n\\ge 6$$\\end{document} with finite fundamental group, we show that the classifying space BDiff(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\ extrm{Diff}(M)$$\\end{document} of the diffeomorphism group of M is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold N⊂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\subset M$$\\end{document} of arbitrary codimension into M has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.

Characterizing slopes for torus knots, II

A slope [Formula: see text] is called a characterizing slope for a given knot [Formula: see text] if whenever the [Formula: see text]-surgery on a knot [Formula: see text] is homeomorphic to the [Formula: see text]-surgery on [Formula: see text] via an orientation preserving homeomorphism, then [Formula: see text]. In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot [Formula: see text]. More explicitly besides all negative integer slopes there are [Formula: see text] slopes which were unknown to be characterizing for [Formula: see text], including [Formula: see text] nontrivial [Formula: see text]-space slopes. Applying recent work of Baldwin–Hu–Sivek, we improve our result by showing that a nontrivial slope [Formula: see text] is a characterizing slope for [Formula: see text] if [Formula: see text] and [Formula: see text]. In particular every nontrivial [Formula: see text]-space slope of [Formula: see text] is characterizing for [Formula: see text]. More explicitly this work yields [Formula: see text] new characterizing slopes for [Formula: see text]. Another interesting consequence of this work is that if a nontrivial [Formula: see text]-surgery on a non-torus knot in [Formula: see text] yields a manifold of finite fundamental group, then [Formula: see text].

Minimal Euler characteristics for even-dimensional manifolds with finite fundamental group

Abstract We consider the Euler characteristics $\chi (M)$ of closed, orientable, topological $2n$ -manifolds with $(n-1)$ -connected universal cover and a given fundamental group G of type $F_n$ . We define $q_{2n}(G)$ , a generalised version of the Hausmann-Weinberger invariant [19] for 4–manifolds, as the minimal value of $(-1)^n\chi (M)$ . For all $n\geq 2$ , we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of G. As an application, we obtain new restrictions for nonabelian finite groups arising as fundamental groups of rational homology 4–spheres.

Finite torsors on projective schemes defined over a discrete valuation ring

Given a Henselian and Japanese discrete valuation ring $A$ and a flat and projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce a full subcategory of coherent modules on $X$ which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler $A$-module) and that it classifies finite torsors $Q\to X$. This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor $Q\to X$ has a reduction of structure group to a finite one.

A global Weinstein splitting theorem for holomorphic Poisson manifolds

We prove that if a compact K\"ahler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite \'etale cover, it decomposes as the product of the universal cover of the leaf and some other Poisson manifold. As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact K\"ahler manifolds with split tangent bundle.

Homotopy classification of 4–manifolds whose fundamental group is dihedral

We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.

Complete shrinking Ricci–Bourguignon harmonic solitons

Studying self-similar solutions of geometric flows on manifolds plays an important role in understanding geometrical and topological properties of underlying manifolds. In this paper, we prove that a complete shrinking Ricci–Bourguignon harmonic flow soliton [Formula: see text] is compact if and only if [Formula: see text] is bounded on [Formula: see text]. Also, we show that a complete shrinking Ricci–Bourguignon harmonic flow soliton has finite fundamental group.

Moduli space of nonnegatively curved metrics on manifolds of dimension 4k + 1

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

Automaton groups and complete square complexes

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and \operatorname{CAT}(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete \mathcal{VH} complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. For the class of complete directed \mathcal{VH} complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise. Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete \mathcal{VH} complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F_2 .

The conformal group of a compact simply connected Lorentzian manifold

We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D'Ambra proved in 1988 that the isometry group of such a manifold is compact. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group. Third version includes corrections and clarifications, particularly in section 6.

On the Genus and Area of Constant Mean Curvature Surfaces with Bounded Index

Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh et al. (Invent Math 209(3):617–664, [1]) we show that in any closed Riemannian 3-manifold (M, g), the genus of an embedded CMC surface can be bounded only in terms of its index and area, independently of the value of its mean curvature. We also show that if M has finite fundamental group, the genus and area of any non-minimal embedded CMC surface can be bounded in term of its index and a lower bound for its mean curvature.

ℂℙ2-stable classification of 4-manifolds with finite fundamental group

We show that two closed, connected $4$-manifolds with finite fundamental groups are $\mathbb{CP}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.

Positive scalar curvature on manifolds with odd order abelian fundamental groups

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimension at least five which have odd order abelian fundamental groups, are nonspin and atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.

On macroscopic dimension of non-spin 4-manifolds

We prove that for 4-manifolds [Formula: see text] with residually finite fundamental group and non-spin universal covering [Formula: see text], the inequality [Formula: see text] implies the inequality [Formula: see text]. This allows us to complete the proof of Gromov’s Conjecture for 4-manifolds with abelian fundamental group.

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy R P 5 \mathbb {R} P^5 has infinitely many path components. We also show that in each dimension 4 k + 1 4k+1 there are at least 2 2 k 2^{2k} homotopy R P 4 k + 1 \mathbb {R} P^{4k+1} ’s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4 k + 3 ≥ 7 4k+3\geq 7 .

О некоторых аспектах геометрии почти C(λ)-многообразий

In this paper almost C(λ)-manifolds are considered. The local structure of Ricci-flat almost C(λ)-manifolds is obtained. On the space of the adjoint G-structure, necessary and sufficient conditions are obtained under which the al-most C(λ)-manifolds are manifolds of constant curvature and the structure of the Riemannian curvature tensor of an almost C(λ)-manifold of constant curvature is obtained. Relations are obtained that characterize the Einstein almost C(λ)-manifolds. It is proved that a complete almost C(λ)-Einstein manifold is either holomorphically isometrically covered by the product of a real line by a Ricciflat Kähler manifold, or is compact and has a finite fundamental group. For almost C(λ)-manifolds that are -Einstein, analytic expressions for the functions  and  characterizing these manifolds are obtained. It is shown that an almost C(λ)-manifold has an Ф-invariant Ricci tensor. We study also almost C(λ)-manifolds of pointwise constant Ф-holomorphic sectional curvature.