Eight papers published by Geometry & Topology Publications

Seven papers have been published by Algebraic & Geometric Topology

(1) Algebraic & Geometric Topology 11 (2011) 2237-2264
Simplicial volume and fillings of hyperbolic manifolds
by Koji Fujiwara and Jason Fox Manning
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p073.xhtml
DOI: 10.2140/agt.2011.11.2237

(2) Algebraic & Geometric Topology 11 (2011) 2265-2296
The entropy efficiency of point-push mapping classes on the punctured disk
by Philip Boyland and Jason Harrington
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p074.xhtml
DOI: 10.2140/agt.2011.11.2265

(3) Algebraic & Geometric Topology 11 (2011) 2297-2318
Bounds for fixed points and fixed subgroups on surfaces and graphs
by Boju Jiang, Shida Wang and Qiang Zhang
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p075.xhtml
DOI: 10.2140/agt.2011.11.2297

(4) Algebraic & Geometric Topology 11 (2011) 2319-2368
Families of monotone symplectic manifolds constructed via
symplectic cut and their Lagrangian submanifolds
by Agnes Gadbled
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p076.xhtml
DOI: 10.2140/agt.2011.11.2319

(5) Algebraic & Geometric Topology 11 (2011) 2369-2390
On the mapping space homotopy groups and the free loop space homology groups
by Takahito Naito
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p077.xhtml
DOI: 10.2140/agt.2011.11.2369
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(6) Algebraic & Geometric Topology 11 (2011) 2391-2436
Algebraic K-theory over the infinite dihedral group: an algebraic approach
by James F Davis, Qayum Khan and Andrew Ranicki
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p078.xhtml
DOI: 10.2140/agt.2011.11.2391

(7) Algebraic & Geometric Topology 11 (2011) 2437-2452
Free degrees of homeomorphisms on compact surfaces
by Jianchun Wu and Xuezhi Zhao
URL: http://www.msp.warwick.ac.uk/agt/2011/11-04/p079.xhtml
DOI: 10.2140/agt.2011.11.2437

One paper has been published by Geometry & Topology

(8) Geometry & Topology 15 (2011) 1545-1567
Free planar actions of the Klein bottle group
by Frederic Le Roux
URL: http://www.msp.warwick.ac.uk/gt/2011/15-03/p039.xhtml
DOI: 10.2140/gt.2011.15.1545

Abstracts follow

(1) Simplicial volume and fillings of hyperbolic manifolds
by Koji Fujiwara and Jason Fox Manning

Let M be a hyperbolic n-manifold whose cusps have torus cross-
sections. In an earlier paper, the authors constructed a variety of
nonpositively and negatively curved spaces as "2pi-fillings" of M by
replacing the cusps of M with compact "partial cones" of their
boundaries. These 2pi-fillings are closed pseudomanifolds, and so
have a fundamental class. We show that the simplicial volume of any
such 2pi-filling is positive, and bounded above by Vol(M)/v_n, where
v_n is the volume of a regular ideal hyperbolic n-simplex. This
result generalizes the fact that hyperbolic Dehn filling of a
3-manifold does not increase hyperbolic volume.

In particular, we obtain information about the simplicial volumes of
some 4-dimensional homology spheres described by Ratcliffe and
Tschantz, answering a question of Belegradek and establishing the
existence of 4-dimensional homology spheres with positive simplicial
volume.


(2) The entropy efficiency of point-push mapping classes on the punctured disk
by Philip Boyland and Jason Harrington

We study the maximal entropy per unit generator of point-push mapping
classes on the punctured disk. Our work is motivated by fluid mixing
by rods in a planar domain. If a single rod moves among N fixed
obstacles, the resulting fluid diffeomorphism is in the point-push
mapping class associated with the loop in the fundamental group of the
N-punctured disk traversed by the single stirrer. The collection of
motions where each stirrer goes around a single obstacle generate the
group of point-push mapping classes, and the entropy efficiency with
respect to these generators gives a topological measure of the mixing
per unit energy expenditure of the mapping class. We give lower and
upper bounds for Eff(N), the maximal efficiency in the presence of N
obstacles, and prove that Eff(N) approaches log(3) as N tends to
infinity. For the lower bound we compute the entropy efficiency of a
specific point-push protocol, HSP_N, which we conjecture achieves the
maximum. The entropy computation uses the action on chains in a
Z-covering space of the punctured disk which is designed for
point-push protocols. For the upper bound we estimate the exponential
growth rate of the action of the point-push mapping classes on the
fundamental group of the punctured disk using a collection of
incidence matrices and then computing the generalized spectral radius
of the collection.


(3) Bounds for fixed points and fixed subgroups on surfaces and graphs
by Boju Jiang, Shida Wang and Qiang Zhang

We consider selfmaps of hyperbolic surfaces and graphs, and give some
bounds involving the rank and the index of fixed point classes. One
consequence is a rank bound for fixed subgroups of surface group
endomorphisms, similar to the Bestvina-Handel bound (originally known
as the Scott conjecture) for free group automorphisms.

When the selfmap is homotopic to a homeomorphism, we rely on
Thurston's classification of surface automorphisms. When the surface
has boundary, we work with its spine, and Bestvina-Handel's theory of
train track maps on graphs plays an essential role.

It turns out that the control of empty fixed point classes (for
surface automorphisms) presents a special challenge. For this
purpose, an alternative definition of fixed point class is introduced,
which avoids covering spaces hence is more convenient for geometric
discussions.


(4) Families of monotone symplectic manifolds constructed via
symplectic cut and their Lagrangian submanifolds
by Agnes Gadbled

We describe families of monotone symplectic manifolds constructed via
the symplectic cutting procedure of Lerman [Math. Res. Lett. 2 (1995)
247--258] from the cotangent bundle of manifolds endowed with a free
circle action. We also give obstructions to the monotone Lagrangian
embedding of some compact manifolds in these symplectic manifolds.


(5) On the mapping space homotopy groups and the free loop space homology groups
by Takahito Naito

Let X be a Poincare duality space, Y a space and f a based map from X
to Y. We show that the rational homotopy group of the connected
component of the space of maps from X to Y containing f is contained
in the rational homology group of a space L_f Y which is the pullback
of f and the evaluation map from the free loop space LY to the space
Y. As an application of the result, when X is a closed oriented
manifold, we give a condition of a noncommutativity for the rational
loop homology algebra H_{*+d}(L_f Y;Q) defined by Gruher and Salvatore
which is the extension of the Chas-Sullivan loop homology algebra.


(6) Algebraic K-theory over the infinite dihedral group: an algebraic approach
by James F Davis, Qayum Khan and Andrew Ranicki

Two types of Nil-groups arise in the codimension 1 splitting
obstruction theory for homotopy equivalences of finite CW--complexes:
the Farrell--Bass Nil-groups in the nonseparating case when the
fundamental group is an HNN extension and the Waldhausen Nil-groups in
the separating case when the fundamental group is an amalgamated free
product. We obtain a general Nil-Nil theorem in algebraic K-theory
relating the two types of Nil-groups.

The infinite dihedral group is a free product and has an index 2
subgroup which is an HNN extension, so both cases arise if the
fundamental group surjects onto the infinite dihedral group. The
Nil-Nil theorem implies that the two types of the reduced
tilde-Nil-groups arising from such a fundamental group are isomorphic.
There is also a topological application: in the finite-index case of
an amalgamated free product, a homotopy equivalence of finite
CW-complexes is semisplit along a separating subcomplex.


(7) Free degrees of homeomorphisms on compact surfaces
by Jianchun Wu and Xuezhi Zhao

For a compact surface M, the free degree fr(M) of homeomorphisms on M
is the minimum positive integer n with property that for any self
homeomorphism xi of M, at least one of the iterates xi,xi^2,...,xi^n
has a fixed point. This is to say fr(M) is the maximum of least
periods among all periodic points of self homeomorphisms on M. We
prove that fr(F_{g,b}) is at most 24g-24 for g at least 2 and
fr(N_{g,b}) is at most 12g-24 for g at least 3.


(8) Free planar actions of the Klein bottle group
by Frederic Le Roux

We describe the structure of the free actions of the fundamental group
of the Klein bottle <a,b|aba^{-1}=b^{-1}> by orientation preserving
homeomorphisms of the plane. The main result is that a must act
properly discontinuously, while b cannot act properly discontinuously.
As a corollary, we describe some torsion free groups that may not act
freely on the plane. We also find some properties which are
reminiscent of Brouwer theory for the integers, in particular that
every free action is virtually wandering.



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