Publication of the Zieschang Gedenkschrift
Geometry and Topology
Geometry & Topology Publications
is delighted to announce the publication of:
The Zieschang Gedenkschrift
http://msp.warwick.ac.uk/gtm/2008/14/
A memorial volume for Heiner Zieschang (1936-2004)
Edited by Michel Boileau, Martin Scharlemann
and Richard Weidmann
The volume is dedicated to Heiner Zieschang, who has been a teacher,
mentor and friend both to the editors and to most of those that have
contributed their works. The volume comprises 24 papers on subjects
to which Heiner contributed during his lifetime and includes a paper
coauthored by Heiner himself.
Details of the papers follow:
(1) Geometry & Topology Monographs 14 (2008) 1-25
Homology of the mapping class group Gamma_2,1 for surfaces of genus
2 with a boundary curve
by Jochen Abhau, Carl-Friedrich Boedigheimer and Ralf Ehrenfried
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p001.xhtml
DOI: 10.2140/gtm.2008.14.1
(2) Geometry & Topology Monographs 14 (2008) 27-47
A classification of special 2-fold coverings
by Anne Bauval, Daciberg L Goncalves, Claude Hayat
and Maria Herminia de Paula Leite Mello
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p002.xhtml
DOI: 10.2140/gtm.2008.14.27
(3) Geometry & Topology Monographs 14 (2008) 49-62
On multiplicity of mappings between surfaces
by Semeon Bogatyi, Jan Fricke and Elena Kudryavtseva
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p003.xhtml
DOI: 10.2140/gtm.2008.14.49
(4) Geometry & Topology Monographs 14 (2008) 63-73
A Magnus theorem for some one-relator groups
by Oleg Bogopolski and Konstantin Sviridov
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p004.xhtml
DOI: 10.2140/gtm.2008.14.63
(5) Geometry & Topology Monographs 14 (2008) 75-81
Roots of torsion polynomials and dominations
by Michel Boileau, Steve Boyer and Shicheng Wang
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p005.xhtml
DOI: 10.2140/gtm.2008.14.75
(6) Geometry & Topology Monographs 14 (2008) 83-103
A characterisation of S^3 among homology spheres
by Michel Boileau, Luisa Paoluzzi and Bruno Zimmermann
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p006.xhtml
DOI: 10.2140/gtm.2008.14.83
(7) Geometry & Topology Monographs 14 (2008) 105-128
On invertible generating pairs of fundamental groups of graph manifolds
by Michel Boileau and Richard Weidmann
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p007.xhtml
DOI: 10.2140/gtm.2008.14.105
(8) Geometry & Topology Monographs 14 (2008) 129-133
A condition that prevents groups from acting nontrivially on trees
by Martin R Bridson
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p008.xhtml
DOI: 10.2140/gtm.2008.14.129
(9) Geometry & Topology Monographs 14 (2008) 135-171
Intersections of conjugates of Magnus subgroups of one-relator groups
by Donald J Collins
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p009.xhtml
DOI: 10.2140/gtm.2008.14.135
(10) Geometry & Topology Monographs 14 (2008) 173-192
Non compact Euclidean cone 3-manifolds with cone angles less than 2pi
by Daryl Cooper and Joan Porti
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p010.xhtml
DOI: 10.2140/gtm.2008.14.173
(11) Geometry & Topology Monographs 14 (2008) 193-217
Minimizing the number of Nielsen preimage classes
by Olga Frolkina
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p011.xhtml
DOI: 10.2140/gtm.2008.14.193
(12) Geometry & Topology Monographs 14 (2008) 219-294
Some quadratic equations in the free group of rank 2
by Daciberg L Goncalves, Elena Kudryavtseva and Heiner Zieschang
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p012.xhtml
DOI: 10.2140/gtm.2008.14.219
(13) Geometry & Topology Monographs 14 (2008) 295-319
Roots in 3-manifold topology
by C Hog-Angeloni and S Matveev
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p013.xhtml
DOI: 10.2140/gtm.2008.14.295
(14) Geometry & Topology Monographs 14 (2008) 321-333
Free group automorphisms with many fixed points at infinity
by Andre Jaeger and Martin Lustig
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p014.xhtml
DOI: 10.2140/gtm.2008.14.321
(15) Geometry & Topology Monographs 14 (2008) 335-351
Noncoherence of some lattices in Isom(H^n)
by Michael Kapovich, Leonid Potyagailo and Ernest Vinberg
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p015.xhtml
DOI: 10.2140/gtm.2008.14.335
(16) Geometry & Topology Monographs 14 (2008) 353-371
The first Alexander Z[Z]-modules of surface-links and of virtual links
by Akio Kawauchi
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p016.xhtml
DOI: 10.2140/gtm.2008.14.353
(17) Geometry & Topology Monographs 14 (2008) 373-391
Minimizing coincidence numbers of maps into projective spaces
by Ulrich Koschorke
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p017.xhtml
DOI: 10.2140/gtm.2008.14.373
(18) Geometry & Topology Monographs 14 (2008) 393-416
Finite groups acting on 3-manifolds and cyclic branched coverings of knots
by Mattia Mecchia
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p018.xhtml
DOI: 10.2140/gtm.2008.14.393
(19) Geometry & Topology Monographs 14 (2008) 417-450
Epimorphisms between 2-bridge link groups
by Tomotada Ohtsuki, Robert Riley and Makoto Sakuma
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p019.xhtml
DOI: 10.2140/gtm.2008.14.417
(20) Geometry & Topology Monographs 14 (2008) 451-475
Refilling meridians in a genus 2 handlebody complement
by Martin Scharlemann
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p020.xhtml
DOI: 10.2140/gtm.2008.14.451
(21) Geometry & Topology Monographs 14 (2008) 477-504
Tangential LS-category of K(pi,1)-foliations
by Wilhelm Singhof and Elmar Vogt
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p021.xhtml
DOI: 10.2140/gtm.2008.14.477
(22) Geometry & Topology Monographs 14 (2008) 505-518
The rank of the fundamental group of certain hyperbolic 3-manifolds
fibering over the circle
by Juan Souto
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p022.xhtml
DOI: 10.2140/gtm.2008.14.505
(23) Geometry & Topology Monographs 14 (2008) 519-556
Remarks on the cohomology of finite fundamental groups of 3-manifolds
by Satoshi Tomoda and Peter Zvengrowski
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p023.xhtml
DOI: 10.2140/gtm.2008.14.519
(24) Geometry & Topology Monographs 14 (2008) 557-567
On the number of optimal surfaces
by Alina Vdovina
URL: http://www.msp.warwick.ac.uk/gtm/2008/14/p024.xhtml
DOI: 10.2140/gtm.2008.14.557
Abstracts follow
(1) Homology of the mapping class group Gamma_2,1 for surfaces of genus
2 with a boundary curve
by Jochen Abhau, Carl-Friedrich Boedigheimer and Ralf Ehrenfried
We report on the computation of the integral homology of the mapping
class group Gamma_{g,1}^m of genus g surfaces with one boundary curve
and m punctures, when 2g + m < 6, in particular Gamma_{2,1}^0.
(2) A classification of special 2-fold coverings
by Anne Bauval, Daciberg L Goncalves, Claude Hayat
and Maria Herminia de Paula Leite Mello
Starting with an O(2)-principal fibration over a closed oriented surface
F_g, g>=1, a 2-fold covering of the total space is said to be special
when the monodromy sends the fiber SO(2)~S^1 to the nontrivial element of
Z_2. Adapting D Jonhson's method [Spin structures and quadratic forms on
surfaces, J London Math Soc, 22 (1980) 365-373] we define an action of
Sp(Z_2,2g), the group of symplectic isomorphisms of (H_1(F_g;Z_2),.),
on the set of special 2-fold coverings which has two orbits, one with
2^{g-1}(2^g+1) elements and one with 2^{g-1}(2^g-1) elements. These two
orbits are obtained by considering Arf-invariants and some congruence of
the derived matrices coming from Fox Calculus. Sp(Z_2,2g) is described
as the union of conjugacy classes of two subgroups, each of them fixing
a special 2-fold covering. Generators of these two subgroups are made
explicit.
(3) On multiplicity of mappings between surfaces
by Semeon Bogatyi, Jan Fricke and Elena Kudryavtseva
Let M and N be two closed (not necessarily orientable) surfaces,
and f a continuous map from M to N. By definition, the minimal
multiplicity MMR[f] of the map f denotes the minimal integer k
having the following property: f can be deformed into a map g such
that the number |g^{-1}(c)| of preimages of any point c in N under
g is at most k. We calculate MMR[f] for any map f of positive
absolute degree A(f). The answer is formulated in terms of A(f),
[pi_1(N):f_#(pi_1(M))], and the Euler characteristics of M and N.
For a map f with A(f)=0, we prove the inequalities 2 <= MMR[f] <= 4.
(4) A Magnus theorem for some one-relator groups
by Oleg Bogopolski and Konstantin Sviridov
We will say that a group G possesses the Magnus property if for any
two elements u,v in G with the same normal closure, u is conjugate
to v or v^{-1}. We prove that some one-relator groups, including
the fundamental groups of closed nonorientable surfaces of genus g>3
possess this property. The analogous result for orientable surfaces of
any finite genus was obtained by the first author [Geometric methods
in group theory, Contemp. Math, 372 (2005) 59-69].
(5) Roots of torsion polynomials and dominations
by Michel Boileau, Steve Boyer and Shicheng Wang
We show that the nonzero roots of the torsion polynomials associated
to the infinite cyclic covers of a given compact, connected, orientable
3-manifold M are contained in a compact part of C* a priori determined
by M. This result is applied to prove that when M is closed, it
dominates at most finitely many Sol manifolds.
(6) A characterisation of S^3 among homology spheres
by Michel Boileau, Luisa Paoluzzi and Bruno Zimmermann
We prove that an integral homology 3-sphere is S^3 if and only if it
admits four periodic diffeomorphisms of odd prime orders whose space of
orbits is S^3. As an application we show that an irreducible integral
homology sphere which is not S^3 is the cyclic branched cover of odd
prime order of at most four knots in S^3. A result on the structure
of finite groups of odd order acting on integral homology spheres is
also obtained.
(7) On invertible generating pairs of fundamental groups of graph manifolds
by Michel Boileau and Richard Weidmann
We study invertible genrating pairs of fundamental groups of graph
manifolds, that is, pairs of elements (g,h) for which the map gmapsto
g^-1, hmapsto h^-1 extends to an automorphism. We show in particular
that a graph manifold is of Heegaard genus 2 if and only if its
fundamental group has an invertible generating pair.
(8) A condition that prevents groups from acting nontrivially on trees
by Martin R Bridson
We describe a simple criterion for showing that a group has Serre's
property FA. By exhibiting a certain pattern of finite subgroups,
we show that this criterion is satisfied by Aut(F_n) and SL(n,Z)
when n>=3.
(9) Intersections of conjugates of Magnus subgroups of one-relator groups
by Donald J Collins
In the theory of one-relator groups, Magnus subgroups, which are free
subgroups obtained by omitting a generator that occurs in the given
relator, play an essential structural role. In a previous article,
the author proved that if two distinct Magnus subgroups M and N of
a one-relator group, with free bases S and T are given, then the
intersection of M and N is either the free subgroup P generated by
the intersection of S and T or the free product of P with an infinite
cyclic group.
The main result of this article is that if M and N are Magnus subgroups
(not necessarily distinct) of a one-relator group G and g and h are
elements of G, then either the intersection of gMg^{-1} and hNh^{-1} is
cyclic (and possibly trivial), or gh^{-1} is an element of NM in which
case the intersection is a conjugate of the intersection of M and N.
(10) Non compact Euclidean cone 3-manifolds with cone angles less than 2pi
by Daryl Cooper and Joan Porti
We describe some properties of noncompact Euclidean cone manifolds with
cone angles less than c < 2pi and singular locus a submanifold. More
precisely, we describe its structure outside a compact set. As a
corollary we classify those with cone angles < 3pi/2 and those with
all cone angles = 3pi/2.
(11) Minimizing the number of Nielsen preimage classes
by Olga Frolkina
We find conditions on topological spaces X, Y and nonempty subset B
of Y which guarantee that for each continuous map f from X to Y there
exists a map g homotopic to f such that Nielsen preimage classes of
g^{-1}(B) are all topologically essential.
(12) Some quadratic equations in the free group of rank 2
by Daciberg L Goncalves, Elena Kudryavtseva and Heiner Zieschang
For a given quadratic equation with any number of unknowns in any
free group F, with right-hand side an arbitrary element of F, an
algorithm for solving the problem of the existence of a solution
was given by Culler [Topology 20 (1981) 133--145] using a surface
method and generalizing a result of Wicks [J. London Math. Soc. 37
(1962) 433--444]. Based on different techniques, the problem has
been studied by the authors [Manuscripta Math. 107 (2002) 311--341
and Atti Sem. Mat. Fis. Univ. Modena 49 (2001) 339--400] for
parametric families of quadratic equations arising from continuous
maps between closed surfaces, with certain conjugation factors as
the parameters running through the group F. In particular, for a
one-parameter family of quadratic equations in the free group F_2 of
rank 2, corresponding to maps of absolute degree 2 between closed
surfaces of Euler characteristic 0, the problem of the existence
of faithful solutions has been solved in terms of the value of
the self-intersection index mu: F_2 --> Z[F_2] on the conjugation
parameter. The present paper investigates the existence of faithful,
or non-faithful, solutions of similar families of quadratic equations
corresponding to maps of absolute degree 0. The existence results
are proved by constructing solutions. The non-existence results
are based on studying two equations in Z[pi] and in its quotient Q,
respectively, which are derived from the original equation and are
easier to work with, where pi is the fundamental group of the target
surface, and Q is the quotient of the abelian group Z[pi - {1}] by the
system of relations g ~ -g^{-1}, g in pi - {1}. Unknown variables
of the first and second derived equations belong to pi, Z[pi], Q,
while the parameters of these equations are the projections of the
conjugation parameter to pi and Q, respectively. In terms of these
projections, sufficient conditions for the existence, or non-existence,
of solutions of the quadratic equations in F_2 are obtained.
(13) Roots in 3-manifold topology
by C Hog-Angeloni and S Matveev
Let C be some class of objects equipped with a set of simplifying
moves. When we apply these to a given object M in C as long as
possible, we get a root of M.
Our main result is that under certain conditions the root of
any object exists and is unique. We apply this result to different
situations and get several new results and new proofs of known
results. Among them there are a new proof of the Kneser-Milnor prime
decomposition theorem for 3-manifolds and different versions of this
theorem for cobordisms, knotted graphs, and orbifolds.
(14) Free group automorphisms with many fixed points at infinity
by Andre Jaeger and Martin Lustig
A concrete family of automorphisms alpha_n of the free group F_n is
exhibited, for any n > 2, and the following properties are proved:
alpha_n is irreducible with irreducible powers, has trivial fixed
subgroup, and has 2n-1 attractive as well as 2n repelling fixed points
at bdry F_n. As a consequence of a recent result of V Guirardel
there can not be more fixed points on bdry F_n, so that this family
provides the answer to a question posed by G Levitt.
(15) Noncoherence of some lattices in Isom(H^n)
by Michael Kapovich, Leonid Potyagailo and Ernest Vinberg
We prove noncoherence of certain families of lattices in the isometry
group of the hyperbolic n-space for n greater than 3. For instance,
every nonuniform arithmetic lattice in SO(n,1) is noncoherent,
provided that n is at least 6.
(16) The first Alexander Z[Z]-modules of surface-links and of virtual links
by Akio Kawauchi
We characterize the first Alexander Z[Z]-modules of ribbon
surface-links in the 4-sphere fixing the number of components and the
total genus, and then the first Alexander Z[Z]-modules of surface-links
in the 4-sphere fixing the number of components. Using the result
of ribbon torus-links, we also characterize the first Alexander
Z[Z]-modules of virtual links fixing the number of components. For a
general surface-link, an estimate of the total genus is given in terms
of the first Alexander Z[Z]-module. We show a graded structure on the
first Alexander Z[Z]-modules of all surface-links and then a graded
structure on the first Alexander Z[Z]-modules of classical links,
surface-links and higher-dimensional manifold-links.
(17) Minimizing coincidence numbers of maps into projective spaces
by Ulrich Koschorke
In this paper we continue to study (`strong') Nielsen coincidence
numbers (which were introduced recently for pairs of maps between
manifolds of arbitrary dimensions) and the corresponding minimum
numbers of coincidence points and pathcomponents. We explore
compatibilities with fibrations and, more specifically, with covering
maps, paying special attention to selfcoincidence questions. As
a sample application we calculate each of these numbers for all
maps from spheres to (real, complex, or quaternionic) projective
spaces. Our results turn out to be intimately related to recent work of
D Goncalves and D Randall concerning maps which can be deformed away
from themselves but not by small deformations; in particular, there
are close connections to the Strong Kervaire Invariant One Problem.
(18) Finite groups acting on 3-manifolds and cyclic branched coverings of knots
by Mattia Mecchia
We are interested in finite groups acting orientation-preservingly on
3-manifolds (arbitrary actions, ie not necessarily free actions). In
particular we consider finite groups which contain an involution with
nonempty connected fixed point set. This condition is satisfied
by the isometry group of any hyperbolic cyclic branched covering
of a strongly invertible knot as well as by the isometry group of
any hyperbolic 2-fold branched covering of a knot in the 3-sphere.
In the paper we give a characterization of nonsolvable groups
of this type. Then we consider some possible applications to the
study of cyclic branched coverings of knots and of hyperelliptic
diffeomorphisms of 3-manifolds. In particular we analyze the basic
case of two distinct knots with the same cyclic branched covering.
(19) Epimorphisms between 2-bridge link groups
by Tomotada Ohtsuki, Robert Riley and Makoto Sakuma
We give a systematic construction of epimorphisms between 2-bridge
link groups. Moreover, we show that 2-bridge links having such an
epimorphism between their link groups are related by a map between the
ambient spaces which only have a certain specific kind of singularity.
We show applications of these epimorphisms to the character varieties
for 2-bridge links and pi_1-dominating maps among 3-manifolds.
(20) Refilling meridians in a genus 2 handlebody complement
by Martin Scharlemann
Suppose a genus two handlebody is removed from a 3-manifold M and
then a single meridian of the handlebody is restored. The result
is a knot or link complement in M and it is natural to ask whether
geometric properties of the link complement say something about the
meridian that was restored. Here we consider what the relation must
be between two not necessarily disjoint meridians so that restoring
each of them gives a trivial knot or a split link.
(21) Tangential LS-category of K(pi,1)-foliations
by Wilhelm Singhof and Elmar Vogt
A K(pi,1)-foliation is one for which the universal covers of all
leaves are contractible (thus all leaves are K(pi,1)'s for some
pi). In the first part of the paper we show that the tangential
Lusternik--Schnirelmann category cat F of a K(pi,1)-foliation F on a
manifold M is bounded from below by t-codim F for any t with H_t(M;A)
nonzero for some coefficient group A. Since for any C^2-foliation
F one has cat F <= dim F by our Theorem 5.2 of [Topology 42 (2003)
603-627], this implies that cat F = dim F for K(pi,1)-foliations of
class C^2 on closed manifolds.
For K(pi,1)-foliations on open manifolds the above estimate is far
from optimal, so one might hope for some other homological lower bound
for cat F. In the second part we see that foliated cohomology will
not work. For we show that the p-th foliated cohomology group of
a p-dimensional foliation of positive codimension is an infinite
dimensional vector space, if the foliation is obtained from a
foliation of a manifold by removing an appropriate closed set, for
example a point. But there are simple examples of K(pi,1)-foliations
of this type with cat F < dim F. Other, more interesting examples of
K(pi,1)-foliations on open manifolds are provided by the finitely
punctured Reeb foliations on lens spaces whose tangential category
we calculate.
In the final section we show that C^1-foliations of tangential
category at most 1 on closed manifolds are locally trivial homotopy
sphere bundles. Thus among 2-dimensional C^2-foliations on closed
manifolds the only ones whose tangential category is still unknown
are those which are 2-sphere bundles which do not admit sections.
(22) The rank of the fundamental group of certain hyperbolic 3-manifolds
fibering over the circle
by Juan Souto
We determine the rank of the fundamental group of those hyperbolic
3-manifolds fibering over the circle whose monodromy is a sufficiently
high power of a pseudo-Anosov map. Moreover, we show that any two
generating sets with minimal cardinality are Nielsen equivalent.
(23) Remarks on the cohomology of finite fundamental groups of 3-manifolds
by Satoshi Tomoda and Peter Zvengrowski
Computations based on explicit 4-periodic resolutions are given for
the cohomology of the finite groups G known to act freely on S^3, as
well as the cohomology rings of the associated 3-manifolds (spherical
space forms) M = S^3/G. Chain approximations to the diagonal are
constructed, and explicit contracting homotopies also constructed for
the cases G is a generalized quaternion group, the binary tetrahedral
group, or the binary octahedral group. Some applications are briefly
discussed.
(24) On the number of optimal surfaces
by Alina Vdovina
Let X be a closed oriented Riemann surface of genus > 1 of constant
negative curvature -1. A surface containing a disk of maximal radius
is an optimal surface. This paper gives exact formulae for the number
of optimal surfaces of genus > 3 up to orientation-preserving isometry.
We show that the automorphism group of such a surface is always cyclic
of order 1,2,3 or 6. We also describe a combinatorial structure of
nonorientable hyperbolic optimal surfaces.