Nilpotent Fundamental Groups Research Articles

Nilpotent Aspherical Sasakian Manifolds

Abstract We show that every compact aspherical Sasakian manifold with nilpotent fundamental group is diffeomorphic to a Heisenberg nilmanifold.

On the topology of leaves of singular Riemannian foliations

In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation (M,\mathcal{F}) . If M is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If M has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well.

Topological invariants of groups and Koszul modules

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K⊆∧2V as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, and the degree of growth and virtual nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves, (2) nilpotent fundamental groups of compact Kähler manifolds, and (3) the Torelli group of a free group.

3-Manifolds with nilpotent embeddings in S4

We consider embeddings of 3-manifolds [Formula: see text] in [Formula: see text] such that the two complementary regions [Formula: see text] and [Formula: see text] each have nilpotent fundamental group. If [Formula: see text] is odd then these groups are abelian and [Formula: see text]. In general [Formula: see text] and [Formula: see text] have 3-generator presentations, and [Formula: see text]. We give two examples illustrating our results.

Fixed points of diffeomorphisms on nilmanifolds with a free nilpotent fundamental group

Let $M$ be a nilmanifold with a fundamental group which is free $2$-step nilpotent on at least 4 generators. We will show that for any nonnegative integer $n$ there exists a self-diffeomorphism $h_n$ of $M$ such that $h_n$ has exactly $n$ fixed points and any self-map $f$ of $M$ which is homotopic to $h_n$ has at least $n$ fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes.

Rough index theory on spaces of polynomial growth and contractibility

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K -theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get higher-codimensional index obstructions to metrics of positive scalar curvature on closed manifolds with virtually nilpotent fundamental groups. We will give a computation of the homology of (a dense, smooth subalgebra of) the uniform Roe algebra of manifolds of polynomial volume growth.

Residually nilpotent fundamental groups of compact Sol-3-manifolds

We give criteria for the fundamental groups of compact Sol-3-manifolds to be residually nilpotent and residually finite p-groups.

Homotopy decompositions of polyhedra into products with the second factor [formula omitted

As the main results of this paper we prove that for every polyhedron P with abelian or torsion-free nilpotent fundamental group there are only finitely many different homotopy types of X i such that X i × S 1 ≃ P . The same holds for any finite K ( G , 1 ) with nilpotent fundamental group in place of S 1 . The problem, if there exists a polyhedron with infinitely many direct factors of different homotopy types (K. Borsuk, 1970) [2] is still unsolved, even if we assume the second factor to be S 1 .

Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

Let M be a complete Riemannian metric of sectional curvature within [−a 2,−1] whose fundamental group contains a k-step nilpotent subgroup of finite index. We prove that a ≥ k answering a question of M. Gromov. Furthermore, we show that for any $$\epsilon > 0,$$ the manifold M admits a complete Riemannian metric of sectional curvature within $$[ - (k + \epsilon )^2 , - 1].$$

Polyhedra dominating finitely many different homotopy types

In 1968 K. Borsuk asked: Is it true that every finite polyhedron dominates only finitely many different shapes? In this question the notions of shape and shape domination can be replaced by the notions of homotopy type and homotopy domination. We obtained earlier a negative answer to the Borsuk question and next results that the examples of such polyhedra are not rare. In particular, there exist polyhedra with nilpotent fundamental groups dominating infinitely many different homotopy types. On the other hand, we proved that every polyhedron with finite fundamental group dominates only finitely many different homotopy types. Here we obtain next positive results that the same is true for some classes of polyhedra with Abelian fundamental groups and for nilpotent polyhedra. Therefore we also get that every finitely generated, nilpotent torsion-free group has only finitely many r-images up to isomorphism.

Preserving asphericity of virtually nilpotent spaces under P-localization

For a set of prime numbers P, we study when the P-localization of an Eilenberg–Mac Lane space with virtually nilpotent fundamental group is again aspherical. While investigating this problem, we devote special attention to infra-nilmanifolds. We prove that the P-localization of an orientable infra-nilmanifold is aspherical if and only if its holonomy group is P-torsion. The same holds for non-orientable infra-nilmanifolds if 2 is in P. We also develop computational techniques to check preservation of asphericity. These are explicitly applied to show that the P-localization of a non-orientable infra-nilmanifold of dimension ⩽3 is always aspherical. We point out that this is no longer true from dimension 4 onwards.

Nilpotent quotients of fundamental groups of special 3-manifolds with boundary

We use a Betti number estimate of Freedman-Hain-Teichner to show that the maximal torsion-free nilpotent quotient of the fundamental group of a 3-manifold with boundary is either Z or Z ⊕ Z. In particular we reobtain the Evans-Moser classification of 3-manifolds with boundary which have nilpotent fundamental groups.

Nilpotent groups and universal coverings of smooth projective varieties

In this paper we prove that the universal cover of a smooth projective variety with nilpotent fundamental group is holomorphically convex.

Maximal Nilpotent Quotients of $3$-Manifold Groups

We show that if the lower central series of the fundamental group of a closed oriented 3-manifold stabilizes then the maximal nilpotent quotient is a cyclic group, a quaternion 2-group cross an odd order cyclic group, or a Heisenberg group. These groups are well known to be precisely the nilpotent fundamental groups of closed oriented 3-manifolds. There are many different approaches to the study of 3-manifolds. There is a flourishing combinatorial school from Dehn, Papakryiakopoulos, Haken and Waldhausen to Gordon and Luecke, which has shown that many 3-manifold questions can be reduced to questions about the fundamental group. A co-equal off-shoot of the combinatorial school folds in dynamics and complex analysis. This is Thurston's program on geometrizing certain characteristic and simple pieces of 3-manifolds by showing that each is modelled locally on one of the eight 3-dimensional geometries (of which only the hyperbolic case is not fully understood). A third perspective on 3-manifolds is through quantuum field theory. The ideas of Witten, Jones, Vassiliev and many others have inspired tremendous activity and, in time, may contribute substantially to the topological understanding of 3-manifolds. This paper takes a fourth perspective by looking at a 3-manifold through nilpotent eyes, observing only the tower of nilpotent quotients of the fundamental group, but never the group itself. This point of view has a long history in the study of link complements and it arises naturally if one studies 3- and 4- dimensional manifolds together. For example, Stallings proved that for a link in S 3 certain nilpotent quotients of the fundamental group of the link complement are invariants of the topological concordance class of the link. These quotients contain the same information as Milnor's ¯ µ-invariants which are generalized linking numbers. For precise references about this area of research and the most recent applications to 4-manifolds see (5). Turaev (11) seems to have been the first to consider nilpotent quotients of closed 3-manifold rather than link complements. Much earlier, the nilpotent fundamental groups of closed 3-manifolds were classified. Thomas (10) showed in particular that statements (1) and (2) in the following Theorem 1 are equivalent.

Minimal Geodesics and Nilpotent Fundamental Groups

Hedlund [18] constructed Riemannian metrics on n-tori, n ≥ 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics [4] are optimal for nilpotent fundamental groups.

Aspherical Manifolds with Virtually 3-Step Nilpotent Fundamental Group

A closed aspherical manifold of dimension ? 3, 4 with virtually 3-step nilpotent fundamental group supports a complete affinely flat structure. Topological and Seifert Bundle classifications are given for a special case called Seifert relatives of flat Riemannian manifolds. We use the Seifert fiber space construction as a main tool. 0. Introduction. It is well-known that given a torsion-free virtually poly-Z group ir, there exists a closed smooth K(-,r, 1) manifold (eg. a double coset space of a Lie group). A question we are interested in is whether such a manifold is homeomorphic to a quotient of R' by an action of a discrete group of affine motions. The best known result seems to be J. Scheuneman's [18], stating that the answer is yes if ir is 3-step nilpotent. In this paper we generalize this to virtually 3-step nilpotent groups (in dimension ? 3, 4). The main idea is to refine the Seifert fiber space construction of P. Conner and F. Raymond [4]. This gives rise to an embedding of such ir into the group of affine motions so that Rm/Ir is compact. Then we apply T. Farrell and W. C. Hsiang's theorem [6] stating that any two closed aspherical manifolds with isomorphic fundamental groups which are virtually nilpotent are homeomorphic in dimension ?3, 4. This paper is organized as follows. In section 1, we refine the Seifert fiber space construction by restricting the coefficient module to some submodules of M(W, Rk). The main result is that, if the first coboundary homomorphism 6 of certain cohomology exact sequence is surjective, then one can convert the algebraic object (group) into a geometric one (group action). Manuscript received November 23, 1981. Manuscript revised May 14, 1981. *This is a part of the author's Ph.D. Thesis at the University of Michigan, 1981. The second part (geometric realization) will appear as a separate article elsewhere. The author wishes to express his sincere appreciation to Professor Frank Raymond for his guidance in this work.

Geometric realization of a finite subgroup of 𝜋₀𝜖(𝑀). II

Let M M be a closed aspherical manifold with a virtually 2 2 -step nilpotent fundamental group. Then any finite group G G of homotopy classes of self-homotopy equivalences of M M can be realized as an effective group of self-homeomorphisms of M M if and only if there exists a group extension E E of π \pi by G G realizing G → Out π 1 M G \to {\operatorname {Out }}{\pi _1}M so that C E ( π ) {C_E}(\pi ) , the centralizer of π \pi in E E , is torsion-free. If this is the case, the action ( G , M ) (G,M) is equivalent to an affine action ( G , M ′ ) (G,M’) on a complete affinely flat manifold homeomorphic to M M . This generalizes the same result for flat Riemannian manifolds.

Locally Affine Spaces with Nilpotent Fundamental Groups

Locally Affine Spaces with Nilpotent Fundamental Groups

Locally affine spaces with nilpotent fundamental groups

Locally affine spaces with nilpotent fundamental groups