Isomorphic Fundamental Groups Research Articles

Homotopy type of planar Peano continua

We prove that the homotopy equivalence class of a map from a Peano continuum into a planar or one-dimensional Peano continuum is determined by the induced homomorphism of fundamental groups. This provides a new proof that planar sets are aspherical and is used to prove that two planar or one-dimensional Peano continua are homotopy equivalent if and only if they have isomorphic fundamental groups.

Recovering algebraic curves from L-functions of Hilbert class fields

In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new proof of a result of Mochizuki and Tamagawa that two such curves with isomorphic fundamental groups are themselves isomorphic.

On special subgroups of fundamental group

Suppose ? is a nonzero cardinal number, I is an ideal on arc connected Topological space X, and B?I(X) is the subgroup of ?1(X) (the first fundamental group of X) generated by homotopy classes of ?_I loops. The main aim of this text is to study B?I(X)s and compare them. Most interest is in ? ? {?,c} and I ? {Pfin(X), {?}}, where Pfin(X) denotes the collection of all finite subsets of X. We denote B?{?}(X) with B?(X). We prove the following statements: for arc connected topological spaces X and Y if B?(X) is isomorphic to B?(Y) for all infinite cardinal number ?, then ?1(X) is isomorphic to ?1(Y); there are arc connected topological spaces X and Y such that ?1(X) is isomorphic to ?1(Y) but B?(X) is not isomorphic to B?(Y); for arc connected topological space X we have B?(X) ? Bc(X) ? ?1(X); for Hawaiian earring X, the sets B?(X), Bc(X), and ?1(X) are pairwise distinct. So B?(X)s and B?I(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

The fundamental group as a topological group

This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type. It is known that viewing the fundamental group as the quotient of the loop space often fails to result in a topological group; we use free topological groups to construct a topology which promotes the fundamental group of any space to topological group structure. The resulting invariant, denoted π1τ, takes values in the category of topological groups, can distinguish spaces with isomorphic fundamental groups, and agrees with the quotient fundamental group precisely when the quotient topology yields a topological group. Most importantly, this choice of topology allows us to naturally realize free topological groups and pushouts of topological groups as fundamental groups via topological analogues of classical results in algebraic topology.

GENERALIZED KNOT GROUPS DISTINGUISH THE SQUARE AND GRANNY KNOTS (WITH AN APPENDIX BY DAVID SAVITT)

Given a knot K we may construct a group Gn(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These "generalized knot groups" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well-known example of a pair of distinct knots with isomorphic fundamental groups. We show that Gn(SK) and Gn(GK) are non-isomorphic for all n ≥ 2. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of Gn(K), n ≥ 2, carries more information about K than the isomorphism type of the fundamental group. The appendix contains some results on representations of the trefoil group in PSL(2, p) that are needed for the proof.

Homology and derived p -series of groups

We give homological conditions that ensure that a group homomorphism induces an isomorphism modulo any term of the derived p − series, in analogy to Stallings's 1963 result for the p-lower central series. In fact, we prove a stronger theorem that is analogous to Dwyer's extensions of Stallings’ results. It follows that spaces that are ℤp-homology equivalent have isomorphic fundamental groups modulo any term of their p-derived series. Various authors have related the ranks of the successive quotients of the p-lower central series and of the derived p-series of the fundamental group of a 3-manifold M to the volume of M, to whether certain subgroups of π1(M) are free, to whether finite index subgroups of π1(M) map onto non-abelian free groups, and to whether finite covers of M are ‘large’ in various other senses.

The fundamental group of the clique graph

Given a finite connected bipartite graph B = ( X , Y ) we consider the simplicial complexes of complete subgraphs of the square B 2 of B and of its induced subgraphs B 2 [ X ] and B 2 [ Y ] . We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K ( G ) , the line graph L ( G ) and the total graph T ( G ) .

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.

Formula omitted]-COMPACTIFICATIONS OF OPEN MANIFOLDS

Suppose an open n-manifold M n may be compactified to an ANR M n so that M n −M n is a Z -set in M n . It is shown that (when n⩾5) the double of M n along its “ Z -boundary” is an n-manifold. More generally, if M n and N n each admit compactifications with homeomorphic Z -boundaries, then their union along this common boundary is an n-manifold. This result is used to show that in many cases Z -compactifiable manifolds are determined by their Z -boundaries. For example, contractible open n-manifolds with homeomorphic Z -boundaries are homeomorphic. As an application, some special cases of a weak Borel conjecture are verified. Specifically, it is shown that closed aspherical n-manifolds ( n≠4) having isomorphic fundamental groups which are either word hyperbolic or CAT(0) have homeomorphic universal covers.

Rigidity of Einstein manifolds of nonpositive curvature

We give sufficient conditions for a compact Einstein manifold of nonpositive sectional curvature to be isometric to a hyperbolic manifold if they have isomorphic fundamental groups. We prove a finiteness result for Einstein metrics on a compact manifold of negative sectional curvature. Spectral rigidity of locally symmetric spaces of negative curvature is considered.

Superrigidity of lattices in solvable Lie groups

Let Γ be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that AdG Γ has the same Zariski closure as AdG. If α: Γ → GLn(ℝ) is any finite-dimensional representation of Γ, we show that α virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if Γ is isomorphic to a closed, cocompact subgroup Γ′ of another simply connected, solvable Lie groupG′, then any isomorphism from Γ to Γ′ extends to a crossed isomorphism fromG toG′. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.

Homotopy equivalence and hemeomorphism of 3-manifolds

IN THIS paper we extend the class of 3-manifolds which are determined up to homeomorphism by their fundamental groups to the class of closed orientable irreducible 3-manifolds containing a singular surface satisfying two properties, the l-line-intersection property and the 4-plane property. A basic problem in the classification of 3-dimensional manifolds is to decide to what extent the homotopy type of a closed manifold determines the manifold up to homeomorphism. In the case of 3-manifolds with finite fundamental groups, it is known that there are homotopy equivalent manifolds which are not homeomorphic, but there are no known examples of closed orientable irreducible 3-manifolds with isomorphic infinite fundamental groups which are not homcomorphic. Waldhausen [30] and Heil Cl23 proved that if M and M’ are Haken 3-manifolds which have isomorphic fundamental groups then they are homcomorphic. If M and M’ arc hyperbolic then the Mostow rigidity theorem implies the same result [19]. but if only M is assumed to be hyperbolic then it is unknown. Boehme [3] extended Waldhausen’s theorem to certain non-Haken Seifert fiber spaces. Scott [27] showed that a closed orientable irreducible 3-manifold which is homotopy equivalent to a Seifcrt fiber space with infinite fundamental group is homeomorphic to that Seifert fiber space. Many of these Scifert fiber spaces are non-Haken. To date however, Seifert fiber spaces have provided the only examples of non-Haken 3-manifolds which are known to be determined up to homeomorphism by their fundamental groups. For manifolds with boundary there are simple examples of non-homemorphic homotopy equivalent Haken manifolds, such as the product with the circle of a thricepunctured sphere and the product with the circle of a once-punctured torus. Such examples are well understood in terms of the characteristic decomposition of the 3-manifold [17] [16]. Another possible source of troublesome examples comes by taking the connected sum of a 3-manifold with a fake homotopy 3-sphere, resulting in a homotopy equivalent non-homeomorphic 3-manifold. This possibility would be ruled out by a successful solution to the Poincari conjecture. To get around this potential problem we work with irreducible 3-manifolds, in which any 2-sphere bounds a bail. Since non-orientable P2-irreducible 3-manifolds are always Haken, we restrict our attention to the orientable case. For simplicity of notation, we sometimes follow the convention of not distinguishing between a map of a surface into a manifold M and the image of the map. When it is

Finite graphs of groups with isomorphic fundamental groups

In this paper the answer is given to the following question: How are the finite graphs of groups, the fundamental groups of which are isomorphic, while the edge and vertex groups satisfy Restrictions 1 and 2 cited below, connected? i) The vertex groups are ~ -groups, that is, nonrepresentable in the form of a nontrivial free product with union or a free extension. 2) The edge groups are co-Hopf, that is not isomorphic to their proper subgroups. An attempt to answer this question under those same conditions was made in [4], however, the proof of the basic theorem of this paper contains mistakes, and its conclusion is refuted by the counterexamples constructed below. One can, it is true, note that under a stronger condition than 1 the requirement that the vertex groups are Abelian the theorem from [4] is true. As a corollary of the basic theorem, we obtain the decidability of the isomorphism problem for the fundamental groups of finite graphs of finite groups, proved previously in [5].

Compact negatively curved manifolds (of dim [unk] 3,4) are topologically rigid

Let M be a complete (connected) Riemannian manifold having finite volume and whose sectional curvatures lie in the interval [c(1), c(2)] with -infinity < c(1)[unk]c(2) < 0. Then any proper homotopy equivalence h:N --> M from a topological manifold N is properly homotopic to a homeomorphism, provided the dimension of M is >5. In particular, if M and N are both compact (connected) negatively curved Riemannian manifolds with isomorphic fundamental groups, then M and N are homeomorphic provided dim M [unk] 3 and 4. {If both are locally symmetric, this is a consequence of Mostow's rigidity theorem [Mostow, G. D. (1967) Publ. Inst. Haut. Etud. Sci. 34, 53-104].} When M has infinite volume we can still calculate the surgery L-groups of pi(1)M, even when dim M = 3, 4, or 5, provided M is locally symmetric. An identification of the weak homotopy type of the homeomorphism group of (finite volume) M is also made through a stable range. We have previously announced these results for the special case that c(1) = c(2) = -1.

The oriented homotopy type of spun 3-manifolds

We show that, bar unexpected developments in 3-manifold theory, the fundamental group and the choice of framing determine the oriented homotopy type of spun 3-manifolds. 1. The object of this note is to classify spun 3-manifolds up to oriented homotopy type. The notion of spinning was introduced by Artin [1] in the context of knots. The asphericity of classical knots implies that spun knots with isomorphic fundamental groups have homotopy equivalent complements. What we do is extend this to closed manifolds. Let M3 be a closed, oriented 3-manifold, and M be M with an open 3-ball removed. Gordon [4] defines the spin of M to be the closed, oriented, smooth 4-manifold s(M) = d(M X D2). Note that s(M) is obtained by gluing M X Sι to S2 X D2 via id52x5i. There is one other possible choice of gluing map, the Gluck twist r: ((0,φ), ψ)-> ({θ + ψ,φ), ψ) corresponding to 77^80(3)) = Z2. The resulting manifold s\M) = M X Sι U τ S2 X D2 is called the twisted spin of M [9]. The two spins of M have the same fundamental group as M. In fact, they have identical 3-skeleta, but different attaching maps for the 4-cell. If M admits a circle action with fixed points (e.g. M is a lens space), then s(M) = s'(M)9 but if M is aspherical s(M) * s\M), as shown by Plotnick [11]. Every closed, oriented M3 admits a (unique up to order) connected sum decomposition M$M2$ %Mn, with prime factors Mi either aspherical, spherical, or S2 X Sι (see e.g. [6]). The spherical factors are of the form Σ3/7r, with Σ3 a homotopy 3-sphere and π a finite group acting freely on Σ 3. Consider only manifolds M3 satisfying the condition All spherical factors are either homotopy 3-spheres or (1.1) spherical Clifford-Klein manifolds (i.e. S3/π, π acting linearly).

A Topological Interpretation for the Bias Invariant

The bias invariant has been used to distinguish between the homotopy types of 2-complexes. In this note we show that two finite, connected 2-complexes X and Y with isomorphic fundamental groups and the same Euler characteristic have the same bias invariant if and only if there is a map f: X -> Y which is a homology equivalence (w,/ and H2f are isomorphisms).

A topological interpretation for the bias invariant

The bias invariant has been used to distinguish between the homotopy types of 2 2 -complexes. In this note we show that two finite, connected 2 2 -complexes X X and Y Y with isomorphic fundamental groups and the same Euler characteristic have the same bias invariant if and only if there is a map f : X → Y f:X \to Y which is a homology equivalence π 1 f {\pi _1}f and H 2 f {H_2}f are isomorphisms).

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.

Semistability at the End of A Group Extension

A $1$-ended ${\text {CW}}$-complex, $Q$, is semistable at $\infty$ if all proper maps $r:\ [0,\infty ) \to Q$ are properly homotopic. If ${X_1}$ and ${X_2}$ are finite ${\text {CW}}$-complexes with isomorphic fundamental groups, then the universal cover of ${X_1}$ is semistable at $\infty$ if and only if the universal cover of ${X_2}$ is semistable at $\infty$. Hence, the notion of a finitely presented group being semistable at $\infty$ is well defined. We prove Main Theorem. Let $1 \to H \to G \to K \to 1$ be a short exact sequence of finitely generated infinite groups. If $G$ is finitely presented, then $G$ is semistable at $\infty$. Theorem. If $A$ and $B$ are locally compact, connected noncompact $CW$-complexes, then $A \times B$ is semistable at $\infty$. Theorem. $\langle \;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle$ is semistable at $\infty$. The proofs are geometrical in nature and the main tool is covering space theory.