We characterize the closed, oriented, Seifert fibered $3$–manifolds which are oriented boundaries of Stein manifolds. We also show that for this class of $3$–manifolds the existence of Stein fillings is equivalent to the existence of symplectic fillings.

A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.

Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $\left(G,H\right)$ acts on $\left(X,Y\right)$, we define the $k$–volume distortion function of $H$ in $G$ to measure the large-scale difference between the volumes of such fillings. We show that these functions are quasi-isometry invariants, and thus independent of the choice of spaces, and provide several bounds in terms of other group properties, such as Dehn functions. We also compute the volume distortion in a number of examples, including characterizing the $k$–volume distortion of ${\mathbb{Z}}^k$ in ${\mathbb{Z}}^k{⋊}_M\mathbb{Z}$, where $M$ is a diagonalizable matrix. We use this to prove a conjecture of Gersten.

A graph is $2$–apex if it is planar after the deletion of at most two vertices. Such graphs are not intrinsically knotted, IK. We investigate the converse, does not IK imply $2$–apex? We determine the simplest possible counterexample, a graph on nine vertices and 21 edges that is neither IK nor $2$–apex. In the process, we show that every graph of 20 or fewer edges is $2$–apex. This provides a new proof that an IK graph must have at least 21 edges. We also classify IK graphs on nine vertices and 21 edges and find no new examples of minor minimal IK graphs in this set.

By explicit machine computation we obtain the mod–$2$ cohomology ring of the third Conway group ${\mathit{Co}}_3$. It is Cohen–Macaulay, has dimension 4, and is detected on the maximal elementary abelian $2$–subgroups.

For each $g\ge 2$, we prove existence of a computable constant $ϵ\left(g\right)>0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic $3$–manifold $M$ and $\gamma$ is a simple geodesic of length less than $ϵ\left(g\right)$ in $M$, then $\gamma$ is isotopic into $S$.

We proved in a previous article that the bar complex of an $E_{\infty }$–algebra inherits a natural $E_{\infty }$–algebra structure. As a consequence, a well-defined iterated bar construction ${\mathtt{B}}^n\left(A\right)$ can be associated to any algebra over an $E_{\infty }$–operad. In the case of a commutative algebra $A$, our iterated bar construction reduces to the standard iterated bar complex of $A$.

The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of $E_{\infty }$–algebras. We use this effective definition to prove that the $n$–fold bar construction admits an extension to categories of algebras over $E_n$–operads.

Then we prove that the $n$–fold bar complex determines the homology theory associated to the category of algebras over an $E_n$–operad. In the case $n=\infty$, we obtain an isomorphism between the homology of an infinite bar construction and the usual $\Gamma$–homology with trivial coefficients.

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.

Let $V{\cup }_{S}W$ be a Heegaard splitting for a closed orientable $3$–manifold $M$. The inclusion-induced homomorphisms ${\pi }_1\left(S\right)\to {\pi }_1\left(V\right)$ and ${\pi }_1\left(S\right)\to {\pi }_1\left(W\right)$ are both surjective. The paper is principally concerned with the kernels $K=Ker\left({\pi }_1\left(S\right)\to {\pi }_1\left(V\right)\right)$, $L=Ker\left({\pi }_1\left(S\right)\to {\pi }_1\left(W\right)\right)$, their intersection $K\cap L$ and the quotient $\left(K\cap L\right)∕\left[K,L\right]$. The module $\left(K\cap L\right)∕\left[K,L\right]$ is of special interest because it is isomorphic to the second homotopy module ${\pi }_2\left(M\right)$. There are two main results.

(1) We present an exact sequence of $\mathbb{Z}\left({\pi }_1\left(M\right)\right)$–modules of the form

$$\left(K\cap L\right)∕\left[K,L\right]\hookrightarrow R\left\{x_1,\dots ,x_g\right\}∕J\stackrel{T^{\varphi }}{\to }R\left\{y_1,\dots ,y_g\right\}\stackrel{\theta }{\to }R\stackrel{ϵ}{\twoheadrightarrow }\mathbb{Z},$$

where $R=\mathbb{Z}\left({\pi }_1\left(M\right)\right)$, $J$ is a cyclic $R$–submodule of $R\left\{x_1,\dots ,x_g\right\}$, $T^{\varphi }$ and $\theta$ are explicitly described morphisms of $R$–modules and $T^{\varphi }$ involves Fox derivatives related to the gluing data of the Heegaard splitting $M=V{\cup }_{S}W$.

(2) Let $\mathcal{K}$ be the intersection kernel for a Heegaard splitting of a connected sum, and ${\mathcal{K}}_1$, ${\mathcal{K}}_2$ the intersection kernels of the two summands. We show that there is a surjection $\mathcal{K}\to {\mathcal{K}}_1\ast {\mathcal{K}}_2$ onto the free product with kernel being normally generated by a single geometrically described element.

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 $S^3$ 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 Ozsváth–Szabó 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 Matić from [J. Differential Geom. 83 (2009) 289–311].

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.

We describe an iterable construction of $\mathit{THH}$ for an $E_n$ ring spectrum. The reduced version is an iterable bar construction and its $n$th iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented $E_n$ algebra.

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.

We calculate the relative versions of embedded contact homology, contact homology and cylindrical contact homology of the sutured solid torus $\left(S^1\times D^2,\Gamma \right)$, where $\Gamma$ consists of $2n$ parallel longitudinal sutures.

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.

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 ${\mathbb{Z}}_{\left(2\right)}$–coefficients, where ${\mathbb{Z}}_{\left(2\right)}$ is the localized ring at 2.

The action–Maslov homomorphism $I:{\pi }_1\left(Ham\left(X,\omega \right)\right)\to ℝ$ 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 $\mathcal{D}$ (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 ${ℂ}^4$.

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 $S^3$ 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 $K$.

An earlier work of the author’s showed that it was possible to adapt the Alekseev–Meinrenken Chern–Weil proof of the Duflo isomorphism to obtain a completely combinatorial proof of the wheeling isomorphism. That work depended on a certain combinatorial identity, which said that a particular composition of elementary combinatorial operations arising from the proof was precisely the wheeling operation. The identity can be summarized as follows: The wheeling operation is just a graded averaging map in a space enlarging the space of Jacobi diagrams. The purpose of this paper is to present a detailed and self-contained proof of this identity. The proof broadly follows similar calculations in the Alekseev–Meinrenken theory, though the details here are somewhat different, as the algebraic manipulations in the original are replaced with arguments concerning the enumerative combinatorics of formal power series of graphs with graded legs.

Given a $C_{\infty }$ coalgebra $C_{\ast }$, a strict dg Hopf algebra $H_{\ast }$ and a twisting cochain $\tau :C_{\ast }\to H_{\ast }$ such that $Im\left(\tau \right)\subset Prim\left(H_{\ast }\right)$, we describe a procedure for obtaining an $A_{\infty }$ coalgebra on $C_{\ast }\otimes H_{\ast }$. This is an extension of Brown’s work on twisted tensor products. We apply this procedure to obtain an $A_{\infty }$ coalgebra model of the chains on the free loop space $LM$ based on the $C_{\infty }$ coalgebra structure of $H_{\ast }\left(M\right)$ induced by the diagonal map $M\to M\times M$ and the Hopf algebra model of the based loop space given by $T\left(H_{\ast }\left(M\right)\left[-1\right]\right)$. When $C_{\ast }$ has cyclic $C_{\infty }$ coalgebra structure, we describe an $A_{\infty }$ algebra on $C_{\ast }\otimes H_{\ast }$. This is used to give an explicit (nonminimal) $A_{\infty }$ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal $G$–bundles.

We define the meridional destabilizing number of a knot. This together with Heegaard genus (or tunnel number) gives a binary complexity of knots. We study its behavior under connected sum of tunnel number one knots.