Sadayoshi Kojima

Isometry transformations of hyperbolic 3-manifolds

Abstract The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold.

Some new invariants of links

In Part I of this paper, we shall define some invariant, called the N-function, of a tame link of 2 components whose linking number is zero. The N-function is one of the extension of the A-invariant by D. Goldsmith [5], hence should be a smooth concordance invariant. The second author has obtained the formula which relates it with the polynomial of Alexanders type A(ML, i; t) defined by J. Milnor [13]. That is,

FINITE COVERS OF 3-MANIFOLDS CONTAINING ESSENTIAL SURFACES OF EULER CHARACTERISTIC = 0

We give a short proof and a slight generalization of a theorem of John Luecke, that a compact connected orientable irreducible 3-manifold containing an essential torus is finitely covered by a torus bundle or manifolds with unbounded first Betti numbers. 1. Introduction. It has been proved by Luecke in his thesis that a compact orientable irreducible 3-manifold containing an essential torus is finitely covered by either a torus bundle or manifolds with unbounded first Betti numbers (see (3)). The purpose of this paper is to reprove this result and to extend it to manifolds containing essential annuli. The argument of Luecke is based on the geometric structure which is the thesis of the uniformization theory developed by Jaco-Shalen, Johannson and Thurston. Our argument is based not on the uniformization theory but rather its topological consequence, that is, the residual finiteness of Haken manifold groups. Since we start from this fact, the argument becomes relatively short. Also we have a little advantage to extend the result for manifolds containing essential annuli. After some preliminaries, we will give our argument in the last section. I am grateful to John Luecke for sending me a copy of (3), which contains a nice exposition of the motivation of that and hence this work. 2. Preliminaries. Throughout this paper, M denotes a 3-manifold and n the fundamental group of M. We refer to Hempels book (1) for the definitions of standard terminology in 3-manifold topology. We say a properly embedded surface S C M is essential if it is incompressible and not boundary parallel. We now prepare a few lemmas which will be used later. LEMMA 1. (1) M contains a two-sided nonseparating surface if and only if bi(M) >0. (2) If M contains disjoint two-sided surfaces Si and S2 so that M — (Si U S<f) is connected, then M is finitely covered by manifolds with unbounded bi. PROOF. (I) is obvious. The assumption of (2) is equivalent to n having a representation onto a free group of rank two. A free group of rank two contains free subgroup of finite index and arbitrarily large rank. Thus n contains a subgroup of finite index which has a representation onto a free group of arbitrary rank. Since the rank of the target gives a lower bound on 61 of the finite cover of M associated to this subgroup, we are done. D

Entropy versus Volume for Pseudo-Anosovs

We discuss a comparison of the entropy of pseudo-Anosov maps and the volume of their mapping tori. Recent study of the Weil–Petersson geometry of Teichmüller space tells us that the entropy and volume admit linear inequalities for both directions under some bounded geometry condition. Based on experiments, we present various observations on the relation between minimal entropies and volumes, and on bounding constants for the entropy over the volume from below. We also provide explicit bounding constants for a punctured torus case.

Milnor's μ-invariants, Massey products and Whitney's trick in 4 dimensions

Abstract We present the failure of Whitneys lemma in dimension 4 from the homotopical and topological viewpoints. Those are detected by Massey products. The invariants for the examples represented by framed links are computed in terms of Milnors μ -invariants.

Normalized entropy versus volume for pseudo-Anosovs

Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov au- tomorphisms and the hyperbolic volumes of their mapping tori. As its corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with xed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit rst, and by Agol later, on niteness of cusped man- ifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil-Petersson translation distance of a pseudo-Anosov map (normalized by multiplying the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.

Homotopy invariants of nonorientable 4-manifolds

We define a Z 4 -quadratic function on π 2 for nonorientable 4-manifolds and show that it is a homotopy invariant. We then use it to distinguish homotopy types of certain manifolds that arose from an analysis of toral action on nonorientable 4-manifolds

Minimal dilatations of pseudo-Anosovs generated by the magic 3–manifold and their asymptotic behavior

This paper concerns the set <(M)over of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold N by Dehn filling three cusps with a mild restriction. Let N(r) be the manifold obtained from N by Dehn filling one cusp along the slope r is an element of Q. We prove that for each g (resp. g not equivalent to 0. mod 6)), the minimum among dilatations of elements ( resp. elements with orientable invariant foliations) of <(M)over defined on a closed surface Sigma(g) of genus g is achieved by the monodromy of some Sigma(g)-bundle over the circle obtained from N (3/-2) or N(1/-2) by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case g equivalent to 6 (mod12) we find a new family of pseudo-Anosovs defined on Sigma(g) with orientable invariant foliations obtained from N (-6) or N (4) by Dehn filling both cusps. We prove that if delta(+)(g) is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on Sigma(g), then

CONFIGURATION SPACES OF POINTS ON THE CIRCLE AND HYPERBOLIC DEHN FILLINGS

Abstract A purely combinatorial compactification of the configuration space of n (⩾5) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of finite volume with dimension n−3. Then, we vary weights for points. The geometrization still makes sense and yields a deformation. The effectivity of deformations arisen in this manner will be locally described in the existing deformation theory of hyperbolic structures when n−3=2, 3 .

VIRTUAL BETTI NUMBERS OF SOME HYPERBOLIC 3-MANIFOLDS

Publisher Summary This chapter discusses virtual Betti numbers of some hyperbolic 3-manifolds. The n-th virtual Betti number of a manifold means the supremum of the n-th Betti numbers of all its finite sheeted coverings. The first virtual Betti number of a hyperbolic 3-manifold is infinity. A 3-manifold with infinite fundamental group is finitely covered by a manifold with positive first Betti number. A monodromy map can be chosen to be covered by a homeomorphism on the universal cover, which commutes with the linear map via the developing map.