The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative Lefschetz fixed point theorem and its converse.
(2) The intersecting kernels of Heegaard splittings by Fengchun Lei and Jie Wu
The paper is principally concerned with the intersecting kernel of a Heegaard splitting for a closed 3-manifold M. By observing that the second homotopy group of M is a canonical quotient group of the intersecting kernel, our first result is to algebraically give an exact sequence of the modules over pi_1(M) concerning pi_2(M), which is derived from the Fox derivatives related to the gluing data of the Heegaard splitting. As a consequence, we give an algebraic criterion on the irreducibility of 3-manifolds from Heegaard splittings. Our second main result describes the intersecting kernel of the Heegaard splittings for connected sums.
(3) Planar open books with four binding components by Yanki Lekili
We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of the three-sphere supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsvath--Szabo contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez and Matic from [J. Differential Geom. 83 (2009) 289--311].
(4) Infinite generation of non-cocompact lattices on right-angled buildings by Anne Thomas and Kevin Wortman
Let Gamma be a non-cocompact lattice on a locally finite regular right-angled building X. We prove that if Gamma has a strict fundamental domain then Gamma is not finitely generated. We use the separation properties of subcomplexes of X called tree-walls.
(5) Homology of E_n ring spectra and iterated THH by Maria Basterra and Michael A Mandell
We describe an iterable construction of THH for an E_n ring spectrum. The reduced version is an iterable bar construction and its nth iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented E_n algebra.
(6) Stable systolic category of the product of spheres by Hoil Ryu
The stable systolic category of a closed manifold M indicates the complexity in the sense of volume. This is a homotopy invariant, even though it is defined by some relations between homological volumes on M. We show an equality of the stable systolic category and the real cup-length for the product of arbitrary finite dimensional real homology spheres. Also we prove the invariance of the stable systolic category under the rational equivalences for orientable 0-universal manifolds.
(7) The embedded contact homology of sutured solid tori by Roman Golovko
We calculate the relative versions of embedded contact homology, contact homology and cylindrical contact homology of the sutured solid torus (S^1 x D^2, Gamma), where Gamma consists of 2n parallel longitudinal sutures.
(8) Configuration-like spaces and coincidences of maps on orbits by R N Karasev and A Yu Volovikov
In this paper we study the spaces of q-tuples of points in a Euclidean space, without k-wise coincidences (configuration-like spaces). A transitive group action by permuting these points is considered, and some new upper bounds on the genus (in the sense of Krasnosel'skii--Schwarz and Clapp--Puppe) for this action are given. Some theorems of Cohen--Lusk type for coincidence points of continuous maps to Euclidean spaces are deduced.
(9) Properties of Bott manifolds and cohomological rigidity by Suyoung Choi and Dong Youp Suh
The cohomological rigidity problem for toric manifolds asks whether the integral cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these results are also generalized to the case with Z_(2)-coefficients, where Z_(2) is the localized ring at 2.
(10) Symplectic manifolds with vanishing action-Maslov homomorphism by Mark Branson
The action--Maslov homomorphism I: pi_1(Ham(X,omega)) --> R is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this homomorphism, and show that it is identically zero when the Seidel element has finite order and the homology satisfies property D (a generalization of having homology generated by divisor classes). We use these results to show that I=0 for products of projective spaces and the Grassmannian of 2 planes in C^4.
(11) Unexpected local minima in the width complexes for knots by Alexander Zupan
In [Pacific J. Math. 239 (2009) 135--156], Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the three-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the width complex for a knot can have local minima that are not global minima. In this paper, we find an embedding of the unknot 0_1 that is a local minimum but not a global minimum in the width complex for 0_1, resolving a question of Scharlemann. We use this embedding to exhibit for any knot K infinitely many distinct local minima that are not global minima of the width complex for K.
(12) Braid ordering and the geometry of closed braid by Tetsuya Ito
We study the relationships between the Dehornoy ordering of the braid groups and the topology and geometry of the closed braid complements. We show that the Dehornoy floor of braids, which is a nonnegative integer determined by the Dehornoy ordering, tells us the position of essential surfaces in the closed braid complements. Furthermore, we prove that if the Dehornoy floor of a braid is bigger than or equal to two, then the Nielsen-Thurston classification of braids and the geometric structure of the closed braid complements are in one-to-one correspondence.
(13) Erratum to the article A simply connected surface of general type with p_g=0 and K^2=3 by Heesang Park, Jongil Park and Dongsoo Shin
We give a different configuration constructed from a rational elliptic surface to correct an example from [Geom. Topol. 13 (2009) 743-767].
(14) Galois actions on homotopy groups of algebraic varieties by Jonathan P Pridham
We study the Galois actions on the l-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the l-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever l is not equal to the residue characteristic p of K. For quasiprojective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When l=p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic homotopy types. Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin-Mazur homotopy groups.
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