Infinite Cyclic Fundamental Group Research Articles

Handle decompositions for a class of closed orientable PL 4-manifolds

In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{{Z}}$, we have constructed a handle decomposition for $M$ as one of the following types: $(1)$ one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, $(2)$ one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients.

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

There is a question asking whether a handle-irreducible summand of every stable-ribbon surface-link is a unique ribbon surface-link. This question for the case of a trivial surface-link is affirmatively answered. That is, a handle-irreducible summand of every stably trivial surface-link is only a trivial 2-link. By combining this result with an old result of F. Hosowaka and the author that every surface-knot with infinite cyclic fundamental group is a stably trivial surface-knot, it is concluded that every surface-knot with infinite cyclic fundamental group is a trivial (i.e., an unknotted) surface-knot.

On classical upper bounds for slice genera

We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We characterize the algebraic genus in terms of cobordisms in three-space, and explore the connections to other knot invariants related to the Seifert form, the Blanchfield form, knot genera and unknotting. Employing Casson-Gordon invariants, we discuss the algebraic genus as a candidate for the optimal upper bound for the topological slice genus that is determined by the S-equivalence class of Seifert matrices.

On the Lee classes of locally conformally symplectic complex surfaces

We prove that the deRham cohomology classes of Lee forms of locally conformally symplectic structures taming the complex structure of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of H 1 dR (S, R), or a single point. In the latter case, we show that S must be biholomorphic to a blow-up of an Inoue-Bombieri surface. Similarly, the deRham co-homology classes of Lee forms of locally conformally Kahler structures of a compact complex surface S with first Betti number equal to 1 is either a non-empty open subset of H 1 dR (S, R), a single point or the empty set. We give a characterization of Enoki surfaces in terms of the existence of a special foliation, and obtain a vanishing result for the Lichnerowicz-Novikov cohomology groups on the class VII compact complex surfaces with infinite cyclic fundamental group.

Splitting a 4-manifold with infinite cyclic fundamental group, revised in a definite case

A sufficient condition that a closed connected definite 4-manifold with infinite cyclic fundamental group is TOP-split is given. By this condition, it is shown that every closed connected definite smooth 4-manifold with infinite cyclic fundamental group is TOP-split. By combining with an earlier result, it is confirmed that every closed connected oriented smooth 4-manifold with infinite cyclic fundamental group is TOP-split. This also implies that every smooth sphere-knot in a closed simply connected smooth 4-manifold is topologically unknotted if the fundamental group of the complement is infinite cyclic.

SPLITTING A 4-MANIFOLD WITH INFINITE CYCLIC FUNDAMENTAL GROUP, REVISED

This article is a revised version of the author's earlier paper on a TOP-splitting of a closed connected oriented 4-manifold with infinite cyclic fundamental group. We show that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if it is virtually TOP-split. As a consequence, we see that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if the intersection form is indefinite. This also implies that every closed connected oriented smooth spin 4-manifold with infinite cyclic fundamental group is TOP-split.

Constructions of small symplectic 4-manifolds using Luttinger surgery

In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to 3CP^2#5(-CP^2) which has two essential Lagrangian tori with simply connected complement. These manifolds can be used to replace E(1) in many known theorems and constructions. Examples in this article include the smallest known minimal symplectic manifolds with abelian fundamental groups including symplectic manifolds with finite and infinite cyclic fundamental group and Euler characteristic 6.

Non-smoothable Four-manifolds with Infinite Cyclic Fundamental Group

In [11], two of us constructed a closed oriented 4-dimensional manifold with fundamental group Z that does not split off S1 × S3. In this note we show that this 4-manifold, and various others derived from it, do not admit smooth structures. Moreover, we find an infinite family of 4-manifolds with exactly the same properties. As a corollary, we obtain topologically slice knots that are not smoothly slice in any rational homology ball.

Planar groups and the Seifert conjecture

We describe a number of characterisations of virtual surface groups which are based on the following result. Let G be a group and F be a field. We show that if G is FP2 over F and if H2ðG; FÞ, thought of as a k-vector space, contains a 1-dimensional Ginvariant subspace, then G is a virtual surface group (i.e. contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane). In particular, this applies to rational Poincare´ duality groups. We also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group. We recover the result of Mess which characterises such groups as groups which are quasiisometric to complete riemannian planes. We also give a cohomological version of the Seifert conjecture, from which the topological Seifert conjecture (proven by Tukia, Mess, Gabai, Casson and Jungreis) can be recovered via work of Zieschang and Scott.

A Proof of the Finite Filling Conjecture

Let M be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus. We show that there are at most five slopes on ∂M whose associated Dehn fillings have either a finite or an infinite cyclic fundamental group. Furthermore, the distance between two slopes yielding such manifolds is no more than three, and there is at most one pair of slopes which realize the distance three. Each of these bounds is realized when M is taken to be the exterior of the figure-8 sister knot.

Self-homeomorphisms of 4-manifolds with fundamental group [formula omitted

In this paper we study the classification of self-homeomorphisms of closed, connected, oriented 4-manifolds with infinite cyclic fundamental group up to pseudoisotopy, or equivalently up to homotopy. We find that for manifolds with even intersection form homeomorphisms are classified up to pseudoisotopy by their action on π 1 , π 2 and the set of spin structures on the manifold. For manifolds with odd intersection form they are classified by the action on π 1 and π 2 and an additional Z/2 Z . As a consequence we complete the classification program for closed, connected, oriented 4-manifolds with infinite cyclic fundamental group, begun by Freedman, Quinn and Wang.

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an infinite cyclic fundamental group.

Classification of closed nonorientable 4-manifolds with infinite cyclic fundamental group

Closed, connected, nonorientable, topological 4-manifolds with infinite cyclic fundamental group are classified. The classification is an extension of results of Freedman and Quinn and of Kreck. The stable classification of such 4-manifolds is also obtained.

Splitting a 4-manifold with infinite cyclic fundamental group

The idea of the proof is to investigate when a homology cobordism between closed 4-rnanifolds with infinite cyclic fundamental groups is an h-cobordism which is always a product cobordism by Freedman [2], for we can construct a homology cobordism between and S x *S3fl:M1 by a method similar to Kervaire's surgery argument [11]. Freedman showed in [1] that any two closed oriented simply connected 4-manifolods Mly M are orientation-preservingly homeomorphic if and only if the intersection forms on H2(M1'yZ), H2(M\\Z) are isomorphic and the KirbySiebenmann invariants ks(7W\), ks(7kf'1)(eZ2) are equal, and in this case there is an orientation-preserving homeomorphism M±~M inducing the isomorphism of the intersection forms. By combining this classification of Freedman with the above splitting theorem, we have a similar characterization for closed oriented 4-manifolds with infinite cyclic fundamental groups:

Some Non-Fibred 3-Knots

Bulletin of the London Mathematical SocietyVolume 15, Issue 4 p. 365-367 Articles Some Non-Fibred 3-Knots C. Kearton, C. Kearton Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, EnglandSearch for more papers by this author C. Kearton, C. Kearton Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, EnglandSearch for more papers by this author First published: July 1983 https://doi.org/10.1112/blms/15.4.365Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume15, Issue4July 1983Pages 365-367 RelatedInformation

Actions of the torus on 4-manifolds—II

IN THE first part [5] of this investigation of closed orientable 4-manifolds admitting an action of the 2-dimensional torus group, G = T2, we obtained an equivariant classification of actions with no finite isotropy groups, provided the action was not free. The main tool there was a cross-sectioning theorem [5, 1.12 and 4.41. Our first result in this sequel is to obtain an equivariant classification of all actions. This is accomplished by specifying a cross-section on the boundary of a tubular neighborhood of each orbit with finite isotropy group (E-orbit) and considering the problem of extending this partial cross-section to the rest of the orbit space. This approach is analogous to the equivariant classification of 3-manifolds with S’-action [7] and in principle it goes back to Seifert [8]. We have to consider two cases. If there are points with infinite isotropy group, F u C # @, then there is no obstruction to extending this cross-section and the orbit data form a complete set of invariants. If theaction has only finite isotropy groups, F u C = @, then M is a Seifert manifold and an additional invariant appears, representing the obstruction to extending the cross-section. The remaining sections contain topological results. Since we have an equivariant classification, we may think of our manifolds given in terms of an action and ask when two such manifolds are homeomorphic (diffeomorphic). This problem was solved for simply connected manifolds in our first paper [5; $51. In $2 we investigate the Seifert case, F u C = @, using the techniques of Conner and Raymond [l-4]. In “almost all ” cases we see that two manifolds are homeomorphic if and only if they are equivariantly homeomorphic. This result was obtained independently by H. Zieschang [lo] using different methods. We also observe that M fibers over S’ with fiber a 3-dimensional Seifert manifold. In 93 we show that in the presence of fixed points, F# 0, M can be represented as an equivariant connected sum of “ elementary ” 4-manifolds with G-action. Each elementary manifold has cyclic fundamental group. There are two unfortunate aspects of this result, ’ however. First the mutual homeomorphism relationships of these elementary manifolds are not known. This difficulty resembles the classification problem of lens spaces. Next, the decomposition is not unique and in the light of [5; $5.81 this may turn out to be a rather serious obstacle. $4 is a partial answer to the first problem for manifolds with infinite cyclic fundamental group, i.e. for manifolds with no finite isotropy, E = $3. We call M

Correction to "Factors of Automorphy and Other Formal Cohomology Groups for Lie Groups"

It has been brought to my attention that a few words of clarification should be appended to the discussion of the example in Section 6 of my paper Factors of Automorphy *, these Annals, p. 314. The functions jp(T, Z) and >(T, Z) introduced there are factors of automorphy for the matrix group G, which is not the full group of analytic automorphisms in the sense of E. Cartan, since it does not act effectively. The subgroup of G consisting of the matrices of determinant 1, which is isomorphic to the quotient group of G by the subgroup of scalar matrices (representing the transformations leaving all points fixed), is the effective trans orination group involved, and it has an infinite cyclic fundamental group; indeed, all of the Cartan groups proper have infinite cyclic fundamental groups. Hence there can be but a single class of factors of automorphy, the integral powers of the complex Jacobians. The only possibility for more varied factors of automorphy is that there exist an extension of the Cartan group which also acts as a group of analytic transformations in M, but not effectively of course; this is the case in the example under discussion, where a group extension of the Cartan group by the circle group introduces a second non-contractible path and hence a second family of factors of automorphy. The same observations also arise purely formally. The equations defining the two basic factors of automorphy were chosen with the relation at the top of page 325 in mind; a more elegant definition could be given using the relation p( T, Z) >(T, Z) = det T, and thus showing directly that these two factors belong to the same family upon restriction to the subgroup of matrices T of determinant 1.