This paper studies compactifications of moduli spaces involving closed Riemann surfaces. The first main result identifies the homeomorphism types of these compactifications. The second main result introduces orbicell decompositions on these spaces using semistable ribbon graphs extending the earlier work of Looijenga.

We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus-two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus-two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus-two mutation interchanges the groups in $\delta$–gradings $k$ and $-k$.

We construct explicit multiplicative and additive structures as well as integration maps on differential extensions of rationally even cohomology theories in the Hopkins–Singer cocycle model. To this end, we consider also a pair-theory for which a long exact sequence is established.

Given a knot $K$ we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of $K$. This lower bound subsumes the lower bounds given by the Levine–Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for $25$ knots with up to $12$ crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise.

We give an operator algebraic model for the first group of the unit spectrum ${\mathit{gl}}_1\left(KU\right)$ of complex topological $K$–theory, ie $\left[X,{BGL}_1\left(KU\right)\right]$, by bundles of stabilized infinite Cuntz $C^{\ast }$–algebras ${\mathcal{O}}_{\infty }\otimes \mathbb{K}$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathcal{ℐ}$–monoid model for the units of $K$–theory by Sagave and Schlichtkrull and it was motivated by the goal of finding connections between the infinite loop space structure of the classifying space of the automorphism group of stabilized strongly self-absorbing $C^{\ast }$–algebras that arose in our generalization of the Dixmier–Douady theory and classical spectra from algebraic topology.

We develop a homology of vertex groups as a tool for studying orbifolds and orbifold cobordism and its torsion. To a pair $\left(G,H\right)$ of conjugacy classes of degree-$n$ and degree-$\left(n-1\right)$ finite subgroups of $SO\left(n\right)$ and $SO\left(n-1\right)$ we associate the parity with which $H$ occurs up to $O\left(n\right)$ conjugacy as a vertex group in the orbifold $S^{n-1}∕G$. This extends to a map $d_n:{\beta }_n\to {\beta }_{n-1}$ between the $Z_2$ vector spaces whose bases are all such conjugacy classes in $SO\left(n\right)$ and then $SO\left(n-1\right)$. Using orbifold graphs, we prove $d:\beta \to \beta$ is a differential and defines a homology, ${\mathcal{\mathscr{H}}}_{\ast }$. We develop a map $s:{\beta }_{\ast }^{-}\to {\beta }_{\ast +1}^{-}$ for a subcomplex of groups which admit orientation-reversing automorphisms. We then look at examples and algebraic properties of $d$ and $s$, including that $d$ is a derivation. We prove that the natural map $\psi$ between the set of diffeomorphism classes of closed, locally oriented $n$–orbifolds and ${\beta }_n$ maps into $ker{d}_n$ and that this map is onto $ker{d}_n$ for $n\le 4$. We relate $d$ to orbifold cobordism and surgery and show that $\psi$ quotients to a map between oriented orbifold cobordism and ${\mathcal{\mathscr{H}}}_{\ast }$.

We obtain an exact sequence of cyclic Legendrian homology for Legendrian links. We present some examples in $3$ dimensions and higher. In higher dimensions we count holomorphic curves via Morse flow trees developed by Ekholm.

This paper investigates the geometry of a symplectic $4$–manifold $\left(M,\omega \right)$ relative to a $J$–holomorphic normal crossing divisor $\mathcal{S}$. Extending work by Biran, we give conditions under which a homology class $A\in H_2\left(M;\mathbb{Z}\right)$ with nontrivial Gromov invariant has an embedded $J$–holomorphic representative for some $\mathcal{S}$–compatible $J$. This holds for example if the class $A$ can be represented by an embedded sphere, or if the components of $\mathcal{S}$ are spheres with self-intersection $-2$. We also show that inflation relative to $\mathcal{S}$ is always possible, a result that allows one to calculate the relative symplectic cone. It also has important applications to various embedding problems, for example of ellipsoids or Lagrangian submanifolds.

The purpose of this paper is to study finite-dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite-dimensional equivariant moduli problem. In addition, we define a coindex for a $G$–vector bundle that is determined by the $G$–action on the vector bundle and prove that if the coindex of an oriented equivariant moduli problem is bigger than $1$, then we obtain an invariant Euler cycle via equivariant perturbation. In particular, we get a localization formula for the stratified transversal intersection of $S^1$–moduli problems.

For a smooth algebraic curve $X$ over a field, applying $H_1$ to the Abel map $X\to PicX∕\partial X$ to the Picard scheme of $X$ modulo its boundary realizes the Poincaré duality isomorphism

$$H_1\left(X,\mathbb{Z}∕\ell \right)\to H^1\left(X∕\partial X,\mathbb{Z}∕\ell \left(1\right)\right)\cong H_c^1\left(X,\mathbb{Z}∕\ell \left(1\right)\right).$$

We show the analogous statement for the Abel map $X∕\partial X\to \overline{Pic}X∕\partial X$ to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism $H_1\left(X∕\partial X,\mathbb{Z}∕\ell \right)\to H^1\left(X,\mathbb{Z}∕\ell \left(1\right)\right)$. In particular, $H_1$ of this Abel map is an isomorphism.

In proving this result, we prove some results about $\overline{Pic}$ that are of independent interest. The singular curve $X∕\partial X$ has a unique singularity that is an ordinary fold point, and we describe the compactified Picard scheme of such a curve up to universal homeomorphism using a presentation scheme. We construct a Mayer–Vietoris sequence for certain pushouts of schemes, and an isomorphism of functors ${\pi }_1^{\ell }{Pic}^0\left(-\right)\cong H^1\left(-,{\mathbb{Z}}_{\ell }\left(1\right)\right).$

The Ptolemy coordinates for boundary-unipotent $SL\left(n,ℂ\right)$–representations of a $3$–manifold group were introduced by Garoufalidis, Thurston and Zickert [arXiv:1111.2828] inspired by the $\mathcal{A}$–coordinates on higher Teichmüller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) $PSL\left(2,ℂ\right)$-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.

We provide an alternative proof of a sufficient condition for the fundamental group of the $n^{th}$ cyclic branched cover of $S^3$ along a prime knot $K$ to be left-orderable, which is originally due to Boyer, Gordon and Watson. As an application of this sufficient condition, we show that for any $\left(p,q\right)$ two-bridge knot, with $p\equiv 3\text{mod }4$, there are only finitely many cyclic branched covers whose fundamental groups are not left-orderable. This answers a question posed by Da̧bkowski, Przytycki and Togha.

Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots and suppose

$$\cdots \subset {\mathcal{T}}_{n+1}\subset {\mathcal{T}}_n\subset \cdots \subset {\mathcal{T}}_2\subset {\mathcal{T}}_1\subset {\mathcal{T}}_0\subset \mathcal{T}$$

is the bipolar filtration of $\mathcal{T}$. We show that ${\mathcal{T}}_0∕{\mathcal{T}}_1$ has infinite rank, even modulo Alexander polynomial one knots. Recall that knots in ${\mathcal{T}}_0$ (a topologically slice $0$–bipolar knot) necessarily have zero $\tau$–, $s$– and $ϵ$–invariants. Our invariants are detected using certain $d$–invariants associated to the $2$–fold branched covers.

We define Tits rigidity for visual boundaries of $CAT\left(0\right)$ groups, and prove that the join of two Cantor sets and its suspension are Tits rigid.

In this paper we shall illustrate that each polytopal moment-angle complex can be understood as the intersection of the minima of corresponding Siegel leaves and the unit sphere, with respect to the maximum norm. Consequently, an alternative proof of a rigidity theorem of Bosio and Meersseman is obtained; as piecewise linear manifolds, polytopal real moment-angle complexes can be smoothed in a natural way.

Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and prove that it is infinite-dimensional in many cases. The main idea is to use different transfer maps to distinguish between classes in uniformly finite homology. Furthermore we show that there are infinitely many classes in degree zero that cannot be detected by means.

In this article we consider a space $B_{com}G$ assembled from commuting elements in a Lie group $G$ first defined by Adem, Cohen and Torres-Giese. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that $\mathbb{Z}\times B_{com}U$ is a loop space and define a notion of commutative K–theory for bundles over a finite complex $X$, which is isomorphic to $\left[X,\mathbb{Z}\times B_{com}U\right]$. We compute the rational cohomology of $B_{com}G$ for $G$ equal to any of the classical groups $SU\left(r\right)$, $U\left(q\right)$ and $Sp\left(k\right)$, and exhibit the rational cohomologies of $B_{com}U$, $B_{com}SU$ and $B_{com}Sp$ as explicit polynomial rings.

We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally enriched functor from a spectrally enriched version of permutative categories to the category of spectra that is built using an appropriate version of $K$–theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg–Mac Lane spectra for Mackey functors and for suspension spectra for finite $G$–sets.

Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3–manifold group into $SL\left(n,ℂ\right)$ using Ptolemy coordinates, which were inspired by $\mathcal{A}$–coordinates on higher Teichmüller space due to Fock and Goncharov. We parametrize representations into $PGL\left(n,ℂ\right)$ using shape coordinates, which are a $3$–dimensional analogue of Fock and Goncharov’s $\mathcal{X}$–coordinates. These coordinates satisfy equations generalizing Thurston’s gluing equations. These equations are of Neumann–Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.