Eleven papers published by Geometry & Topology Publications

[Geometry and Topology]

Seven papers have been published by Algebraic & Geometric Topology
opening Issue 3 of Volume 13

(1) Algebraic & Geometric Topology 13 (2013) 1243-1271
Context-free manifold calculus and the Fulton-MacPherson operad
by Victor Turchin
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p041.xhtml
DOI: 10.2140/agt.2013.13.1243

(2) Algebraic & Geometric Topology 13 (2013) 1273-1298
Restricting the topology of 1-cusped arithmetic 3-manifolds
by Mark D Baker and Alan W Reid
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p042.xhtml
DOI: 10.2140/agt.2013.13.1273

(3) Algebraic & Geometric Topology 13 (2013) 1299-1367
The simplicial boundary of a CAT(0) cube complex
by Mark F Hagen
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p043.xhtml
DOI: 10.2140/agt.2013.13.1299

(4) Algebraic & Geometric Topology 13 (2013) 1369-1412
Lipschitz minimality of Hopf fibrations and Hopf vector fields
by Dennis DeTurck, Herman Gluck and Peter Storm
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p044.xhtml
DOI: 10.2140/agt.2013.13.1369

(5) Algebraic & Geometric Topology 13 (2013) 1413-1463
The classification of rational subtangle
replacements between rational tangles
by Kenneth L Baker and Dorothy Buck
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p045.xhtml
DOI: 10.2140/agt.2013.13.1413

(6) Algebraic & Geometric Topology 13 (2013) 1465-1488
Odd Khovanov homology
by Peter S Ozsvath, Jacob Rasmussen and Zoltann Szabo
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p046.xhtml
DOI: 10.2140/agt.2013.13.1465

(7) Algebraic & Geometric Topology 13 (2013) 1489-1511
Faithful simple objects, orders and gradings of fusion categories
by Sonia Natale
URL: http://www.msp.warwick.ac.uk/agt/2013/13-03/p047.xhtml
DOI: 10.2140/agt.2013.13.1489

Four papers have been published by Geometry & Topology

(8) Geometry & Topology 17 (2013) 839-904
Saturated fusion systems as idempotents in the double Burnside ring
by Kari Ragnarsson and Radu Stancu
URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p020.xhtml
DOI: 10.2140/gt.2013.17.839

(9) Geometry & Topology 17 (2013) 905-924
On the number of ends of rank one locally symmetric spaces
by Matthew Stover
URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p021.xhtml
DOI: 10.2140/gt.2013.17.905

(10) Geometry & Topology 17 (2013) 925-974
On the equivalence of Legendrian and transverse invariants
in knot Floer homology
by John A Baldwin, David Shea Vela-Vick and Vera Vertesi
URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p022.xhtml
DOI: 10.2140/gt.2013.17.925

(11) Geometry & Topology 17 (2013) 975-1112
Knot contact homology
by Tobias Ekholm, John B Etnyre, Lenhard Ng and Michael G Sullivan
URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p023.xhtml
DOI: 10.2140/gt.2013.17.975

Abstracts follow

(1) Context-free manifold calculus and the Fulton-MacPherson operad
by Victor Turchin

The paper gives an explicit description of the Weiss embedding tower
in terms of spaces of maps of truncated modules over the framed
Fulton-MacPherson operad.


(2) Restricting the topology of 1-cusped arithmetic 3-manifolds
by Mark D Baker and Alan W Reid

This paper makes progress on classifying those closed orientable
3-manifolds M that contain knots K so that M \ K is arithmetic.


(3) The simplicial boundary of a CAT(0) cube complex
by Mark F Hagen

For a CAT(0) cube complex X, we define a simplicial flag complex
d_\triangle X, called the simplicial boundary, which is a natural
setting for studying nonhyperbolic behavior of X. We compare
d_\triangle X to the Roller, visual and Tits boundaries of X, give
conditions under which the natural CAT(1) metric on d_\triangle X
makes it isometric to the Tits boundary, and prove a more general
statement relating the simplicial and Tits boundaries. The simplicial
flag complex d_\triangle X allows us to interpolate between studying
geodesic rays in X and the geometry of its contact graph \Gamma X,
which is known to be quasi-isometric to a tree, and we characterize
essential cube complexes for which the contact graph is bounded.
Using related techniques, we study divergence of combinatorial
geodesics in X using d_\triangle X. Finally, we rephrase the
rank-rigidity theorem of Caprace and Sageev in terms of group actions
on \Gamma X and d_\triangle X and state characterizations of cubulated
groups with linear divergence in terms of \Gamma X and d_\triangle X.


(4) Lipschitz minimality of Hopf fibrations and Hopf vector fields
by Dennis DeTurck, Herman Gluck and Peter Storm

Given a Hopf fibration of a round sphere by parallel great subspheres,
we prove that the projection map to the base space is, up to
isometries of domain and range, the unique Lipschitz constant
minimizer in its homotopy class.
Similarly, given a Hopf fibration of a round sphere by parallel great
circles, we view a unit vector field tangent to the fibers as a
cross-section of the unit tangent bundle of the sphere, and prove that
it is, up to isometries of domain and range, the unique Lipschitz
constant minimizer in its homotopy class.
Previous attempts to find a mathematical sense in which Hopf
fibrations and Hopf vector fields are optimal have met with limited
success.


(5) The classification of rational subtangle
replacements between rational tangles
by Kenneth L Baker and Dorothy Buck

A natural generalization of a crossing change is a rational subtangle
replacement (RSR). We characterize the fundamental situation of the
rational tangles obtained from a given rational tangle via RSR,
building on work of Berge and Gabai, and determine the sites where
these RSR may occur. In addition we also determine the sites for RSR
distance at least two between 2-bridge links. These proofs depend on
the geometry of the branched double cover. Furthermore, we classify
all knots in lens spaces whose exteriors are generalized Seifert
fibered spaces and their lens space surgeries, extending work of
Darcy-Sumners. This work is in part motivated by the common
biological situation of proteins cutting, rearranging and resealing
DNA segments, effectively performing RSR on DNA `tangles'.


(6) Odd Khovanov homology
by Peter S Ozsvath, Jacob Rasmussen and Zoltan Szabo

We describe an invariant of links in S^3 which is closely related to
Khovanov's Jones polynomial homology. Our construction replaces the
symmetric algebra appearing in Khovanov's definition with an exterior
algebra. The two invariants have the same reduction modulo 2, but
differ over Q. There is a reduced version which is a link invariant
whose graded Euler characteristic is the normalized Jones polynomial.


(7) Faithful simple objects, orders and gradings of fusion categories
by Sonia Natale

We establish some relations between the orders of simple objects in a
fusion category and the structure of its universal grading group. We
consider fusion categories that have a faithful simple object and show
that their universal grading groups must be cyclic. As for the
converse, we prove that a braided nilpotent fusion category with
cyclic universal grading group always has a faithful simple object.
We study the universal grading of fusion categories with generalized
Tambara-Yamagami fusion rules. As an application, we classify modular
categories in this class and describe the modularizations of braided
Tambara-Yamagami fusion categories.


(8) Saturated fusion systems as idempotents in the double Burnside ring
by Kari Ragnarsson and Radu Stancu

We give a new characterization of saturated fusion systems on a
p-group S in terms of idempotents in the p-local double Burnside
ring of S that satisfy a Frobenius reciprocity relation. Interpreting
our results in stable homotopy, we characterize the stable summands of
the classifying space of a finite p-group that have the homotopy type
of the classifying spectrum of a saturated fusion system, and prove an
invariant theorem for double Burnside modules analogous to the
Adams-Wilkerson criterion for rings of invariants in the cohomology
of an elementary abelian p-group. This work is partly motivated by a
conjecture of Haynes Miller that proposes p-tract groups as a purely
homotopy-theoretical model for p-local finite groups. We show that a
p-tract group gives rise to a p-local finite group when two
technical assumptions are made, thus reducing the conjecture to
proving those two assumptions.


(9) On the number of ends of rank one locally symmetric spaces
by Matthew Stover

Let Y be a noncompact rank one locally symmetric space of finite
volume. Then Y has a finite number e(Y) > 0 of topological ends. In
this paper, we show that for any natural number n, the Y with e(Y) at
most n that are arithmetic fall into finitely many commensurability
classes. In particular, there is a constant c_n such that n--cusped
arithmetic orbifolds do not exist in dimension greater than c_n. We
make this explicit for one-cusped arithmetic hyperbolic n-orbifolds
and prove that none exist for n at least 30.


(10) On the equivalence of Legendrian and transverse invariants
in knot Floer homology
by John A Baldwin, David Shea Vela-Vick and Vera Vertesi

Using the grid diagram formulation of knot Floer homology, Ozsvath,
Szabo and Thurston defined an invariant of transverse knots in the
tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz
and Szabo defined an invariant of transverse knots in arbitrary
contact 3-manifolds using open book decompositions. It has been
conjectured that these invariants agree where they are both
defined. We prove this fact by defining yet another invariant of
transverse knots, showing that this third invariant agrees with the
two mentioned above.


(11) Knot contact homology
by Tobias Ekholm, John B Etnyre, Lenhard Ng and Michael G Sullivan

The conormal lift of a link K in R^3 is a Legendrian submanifold
Lambda_K in the unit cotangent bundle U^* R^3 of R^3 with contact
structure equal to the kernel of the Liouville form. Knot contact
homology, a topological link invariant of K, is defined as the
Legendrian homology of Lambda_K, the homology of a differential graded
algebra generated by Reeb chords whose differential counts holomorphic
disks in the symplectization R x U^*R^3 with Lagrangian boundary
condition R x Lambda_K.
We perform an explicit and complete computation of the Legendrian
homology of Lambda_K for arbitrary links K in terms of a braid
presentation of K, confirming a conjecture that this invariant agrees
with a previously defined combinatorial version of knot contact
homology. The computation uses a double degeneration: the braid
degenerates toward a multiple cover of the unknot, which in turn
degenerates to a point. Under the first degeneration, holomorphic
disks converge to gradient flow trees with quantum corrections. The
combined degenerations give rise to a new generalization of flow trees
called multiscale flow trees. The theory of multiscale flow trees is
the key tool in our computation and is already proving to be useful
for other computations as well.



Geometry & Topology Publications is an imprint of
Mathematical Sciences Publishers

[Geometry and Topology]

Nine papers have been published by Algebraic & Geometric Topology,
completing Issue 2 of Volume 13

(1) Algebraic & Geometric Topology 13 (2013) 1049-1051
A streamlined proof of Goodwillie's n-excisive approximation
by Charles Rezk
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p032.xhtml
DOI: 10.2140/agt.2013.13.1049

(2) Algebraic & Geometric Topology 13 (2013) 1053-1070
Unstable splittings for real spectra
by Nitu Kitchloo and W Stephen Wilson
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p033.xhtml
DOI: 10.2140/agt.2013.13.1053

(3) Algebraic & Geometric Topology 13 (2013) 1071-1087
On the geometric realization and subdivisions of dihedral sets
by Sho Saito
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p034.xhtml
DOI: 10.2140/agt.2013.13.1071

(4) Algebraic & Geometric Topology 13 (2013) 1089-1124
On the construction of functorial factorizations for model categories
by Tobias Barthel and Emily Riehl
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p035.xhtml
DOI: 10.2140/agt.2013.13.1089

(5) Algebraic & Geometric Topology 13 (2013) 1125-1141
Bridge number and tangle products
by Ryan Blair
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p036.xhtml
DOI: 10.2140/agt.2013.13.1125

(6) Algebraic & Geometric Topology 13 (2013) 1143-1159
Nonseparating spheres and twisted Heegaard Floer homology
by Yi Ni
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p037.xhtml
DOI: 10.2140/agt.2013.13.1143

(7) Algebraic & Geometric Topology 13 (2013) 1161-1182
Cosimplicial models for the limit of the Goodwillie tower
by Rosona Eldred
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p038.xhtml
DOI: 10.2140/agt.2013.13.1161

(8) Algebraic & Geometric Topology 13 (2013) 1183-1224
Homology of moduli spaces of linkages in high-dimensional Euclidean space
by Dirk Schuetz
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p039.xhtml
DOI: 10.2140/agt.2013.13.1183

(9) Algebraic & Geometric Topology 13 (2013) 1225-1241
The Kunneth Theorem in equivariant K-theory
for actions of a cyclic group of order 2
by Jonathan Rosenberg
URL: http://www.msp.warwick.ac.uk/agt/2013/13-02/p040.xhtml
DOI: 10.2140/agt.2013.13.1225

Two papers have been published by Geometry & Topology

(10) Geometry & Topology 17 (2013) 639-731
Parametrized ring-spectra and the nearby Lagrangian conjecture
by Thomas Kragh
Appendix: Mohammed Abouzaid
URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p018.xhtml
DOI: 10.2140/gt.2013.17.639

(11) Geometry & Topology 17 (2013) 733-838
A universal characterization of higher algebraic K-theory
by Andrew Blumberg, David Gepner and Goncalo Tabuada
URL: http://www.msp.warwick.ac.uk/gt/2013/17-02/p019.xhtml
DOI: 10.2140/gt.2013.17.733


Abstracts follow

(1) A streamlined proof of Goodwillie's n-excisive approximation
by Charles Rezk

We give a shorter proof of Goodwillie's, [Geom. Topol. 7 (2003)
645--711; Lemma 1.9], which is the key step in proving that the
construction P_n F gives an n-excisive functor.


(2) Unstable splittings for real spectra
by Nitu Kitchloo and W Stephen Wilson

We show that the unstable splittings of the spaces in the Omega
spectra representing BP, BP<n> and E(n) from [Amer. J. Math. 97 (1975)
101--123] may be obtained for the real analogs of these spectra using
techniques similar to those in [Progr. Math. 196 (2001) 35--45].
Explicit calculations for ER(2) are given.


(3) On the geometric realization and subdivisions of dihedral sets
by Sho Saito

We extend to dihedral sets Drinfeld's filtered-colimit expressions of
the geometric realization of simplicial and cyclic sets. We prove
that the group of homeomorphisms of the circle continuously act on the
geometric realization of a dihedral set. We also see how these
expressions of geometric realization clarify subdivision operations on
simplicial, cyclic and dihedral sets defined by Boekstedt, Hsiang and
Madsen, and Spalinski.


(4) On the construction of functorial factorizations for model categories
by Tobias Barthel and Emily Riehl

We present general techniques for constructing functorial
factorizations appropriate for model structures that are not known to
be cofibrantly generated. Our methods use `algebraic'
characterizations of fibrations to produce factorizations that have
the desired lifting properties in a completely categorical fashion. We
illustrate these methods in the case of categories enriched, tensored
and cotensored in spaces, proving the existence of Hurewicz-type model
structures, thereby correcting an error in earlier attempts by others.
Examples include the categories of (based) spaces, (based) G-spaces
and diagram spectra among others.


(5) Bridge number and tangle products
by Ryan Blair

We show that essential punctured spheres in the complement of links
with distance three or greater bridge spheres have bounded complexity.
We define the operation of tangle product, a generalization of both
connected sum and Conway product. Finally, we use the bounded
complexity of essential punctured spheres to show that the bridge
number of a tangle product is at least the sum of the bridge numbers
of the two factor links up to a constant error.


(6) Nonseparating spheres and twisted Heegaard Floer homology
by Yi Ni

If a 3-manifold Y contains a nonseparating sphere, then some twisted
Heegaard Floer homology of Y is zero. This simple fact allows us to
prove several results about Dehn surgery on knots in such manifolds.
Similar results have been proved for knots in L-spaces.


(7) Cosimplicial models for the limit of the Goodwillie tower
by Rosona Eldred

We call attention to the intermediate constructions T_n F in
Goodwillie's Calculus of homotopy functors, giving a new model which
naturally gives rise to a family of towers filtering the Taylor tower
of a functor. We also establish a surprising equivalence between the
homotopy inverse limits of these towers and the homotopy inverse
limits of certain cosimplicial resolutions. This equivalence gives a
greatly simplified construction for the homotopy inverse limit of the
Taylor tower of a functor F under general assumptions.


(8) Homology of moduli spaces of linkages in high-dimensional Euclidean space
by Dirk Schutz

We study the topology of moduli spaces of closed linkages in R^d
depending on a length vector l in R^n. In particular, we use
equivariant Morse theory to obtain information on the homology groups
of these spaces, which works best for odd d. In the case d = 5 we
calculate the Poincare polynomial in terms of combinatorial
information encoded in the length vector.


(9) The Kunneth Theorem in equivariant K-theory
for actions of a cyclic group of order 2
by Jonathan Rosenberg

The Kuenneth Theorem for equivariant (complex) K-theory K^*_G, in the
form developed by Hodgkin and others, fails dramatically when G is a
finite group, and even when G is cyclic of order 2. We remedy this
situation in this very simplest case G=Z/2 by using the power of
RO(G)-graded equivariant K-theory.


(10) Parametrized ring-spectra and the nearby Lagrangian conjecture
by Thomas Kragh
Appendix: Mohammed Abouzaid

Let L be an embedded closed connected exact Lagrangian sub-manifold in
a connected cotangent bundle T*N. In this paper we prove that such an
embedding is, up to a finite covering space lift of T*N, a homology
equivalence. We prove this by constructing a fibrant parametrized
family of ring spectra FL parametrized by the manifold N. The homology
of FL will be (twisted) symplectic cohomology of T*L. The fibrancy
property will imply that there is a Serre spectral sequence converging
to the homology of FL. The fiber-wise ring structure combined with the
intersection product on N induces a product on this spectral
sequence. This product structure and its relation to the intersection
product on L is then used to obtain the result. Combining this result
with work of Abouzaid we arrive at the conclusion that L --> N is
always a homotopy equivalence.


(11) A universal characterization of higher algebraic K-theory
by Andrew Blumberg, David Gepner and Gonalo Tabuada

In this paper we establish a universal characterization of higher
algebraic K-theory in the setting of small stable infinity-categories.
Specifically, we prove that connective algebraic K-theory is the
universal additive invariant, ie the universal functor with values in
spectra which inverts Morita equivalences, preserves filtered
colimits, and satisfies Waldhausen's additivity theorem. Similarly,
we prove that nonconnective algebraic K-theory is the universal
localizing invariant, ie the universal functor that moreover satisfies
the Thomason-Trobaugh-Neeman Localization Theorem.

To prove these results, we construct and study two stable
infinity-categories of `noncommutative motives'; one associated to
additivity and another to localization. In these stable
infinity-categories, Waldhausen's S_dot-construction corresponds to
the suspension functor and connective and nonconnective algebraic
K-theory spectra become corepresentable by the noncommutative motive
of the sphere spectrum. In particular, the algebraic K-theory of every
scheme, stack, and ring spectrum can be recovered from these
categories of noncommutative motives. In the case of a connective ring
spectrum R, we prove moreover that its negative K-groups are
isomorphic to the negative K-groups of the ordinary ring pi_0(R).

In order to work with these categories of noncommutative motives, we
establish comparison theorems between the category of spectral
categories localized at the Morita equivalences and the category of
small idempotent-complete stable infinity-categories. We also explain
in detail the comparison between the infinity-categorical version of
Waldhausen K-theory and the classical definition.

As an application of our theory, we obtain a complete classification
of the natural transformations from higher algebraic K-theory to
topological Hochschild homology (THH) and topological cyclic
homology (TC). Notably, we obtain an elegant conceptual description
of the cyclotomic trace map.