Cyclic Fundamental Group Research Articles

Degree one maps on 4-manifolds with cyclic fundamental groups

Degree one maps on 4-manifolds with cyclic fundamental groups

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.

The capacity of some classes of polyhedra

K. Borsuk in 1979, in Topological Conference in Moscow, introduced the concept of the capacity of a compactum. In this paper, we compute the capacity of the product of two spheres of the same or different dimensions and the capacity of lense spaces. Also, we present an upper bound for the capacity of a $\mathbb{Z}_n$-complex, i.e., a connected finite 2-dimensional CW-complex with finite cyclic fundamental group $\mathbb{Z}_n$.

Knotted surfaces in $4$-manifolds by knot surgery and stabilization

Given a simply-connected closed 4-manifold $X$ and a smoothly embedded oriented surface $\Sigma$, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $X$ that are pairwise homeomorphic to $\Sigma$, but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $S^2\tilde{\times}S^2$. When $X$ is spin, we show in addition that any surfaces obtained by a knot surgery whose complements have cyclic fundamental group become pairwise diffeomorphic after one stabilization by $S^2\tilde{\times}S^2$.

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.

Approximating Novikov–Shubin numbers of virtually cyclic coverings

We assign real numbers to finite sheeted coverings of compact CW complexes designed as finite counterparts to the Novikov–Shubin numbers. We prove an approximation theorem in the case of virtually cyclic fundamental groups employing methods from Diophantine approximation.

Spectra of orbifolds with cyclic fundamental groups

We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamental group. The differential operators considered are Laplace-Beltrami operators twisted by characters of the corresponding fundamental group. To prove the characterization, we first give an explicit description of their spectra by using generating functions. We also include many isospectral examples.

Knotted surfaces in 4-manifolds and stabilizations

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations - analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply-connected 4-manifolds. We moreover show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.

Surfaces in 4-Manifolds: Concordance, Isotopy, and Surgery

In this paper we will show that two surfaces of the same genus and homology class in a simply connected 4-manifold are concordant. We will show they are often topologically isotopic when their complements have cyclic fundamental group. Finally, we will show that if they are 0-concordant, then surgery on one is equivalent to surgery on the other.

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.

Higher hairy graph homology

We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral homology of a related associative algebra with involution. For the operads $$\mathsf{Comm}$$ $$\mathsf{Assoc}$$ and $$\mathsf{Lie}$$ we compute this algebra explicitly, enabling us to apply known results on dihedral homology to the computation of hairy graph homology. In addition we determine the image in hairy graph homology of the trace map defined in Conant et al. (J Topology 6(1):119–153, 2013), as a symplectic representation. For the operad $$\mathsf{Lie}$$ assembling hairy graph cohomology classes yields all known non-trivial rational homology of $$Out(F_n)$$ . The hairy graph homology of $$\mathsf{Lie}$$ is also useful for constructing elements of the cokernel of the Johnson homomorphism of a once-punctured surface.

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.

ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities

Let $${N_0={\mathbb C}^2/H}$$ be an isolated quotient singularity with $${H\subset U(2)}$$ a finite subgroup. We show that for any $${\mathbb Q}$$ -Gorenstein smoothings of N 0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer’s results on hyperkahler 4-manifolds (J Differ Geom 29(3):685–697, 1989), by giving an explicit classification of the ALE Ricci-flat Kahler surfaces. We construct ALF Ricci-flat Kahler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat Kahler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.

Rigidity for Multi-Taub-NUT metrics

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkahler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard R 3 ◊ S 1 or a Multi-Taub-NUT manifold. In par- ticular, the underlying complex manifold is either C ◊ C/Z or a minimal resolution of a cyclic Kleinian singularity.

Manifolds with S 1-category 2 have cyclic fundamental groups

A closed topological n-manifold M n is of S 1-category 2 if it can be covered by two open subsets W 1, W 2 such that the inclusions W i → M n factor homotopically through maps W i → S 1. We show that for n > 3 the fundamental group of such an n-manifold is either trivial or infinite cyclic.

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.

Topological triviality of smoothly knotted surfaces in $4$-manifolds

Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we construct examples of knotted surfaces whose complements have cyclic fundamental groups.

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.