We construct a boundary of a finite-rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary, we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes.

This provides a characterization of virtually geometric multiwords: they are the multiwords that are built from geometric pieces. In particular, a multiword is virtually geometric if and only if the relative boundary is planar.

Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well known that only the if part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey and Murray, recent results of Stolz and Teichner on loop spaces, and some of our own results about string geometry and Brylinski–McLaughlin transgression.

We develop an alternative to the May–Thomason construction used to compare operad-based infinite loop machines to those of Segal, which rely on weak products. Our construction has the advantage that it can be carried out in $\mathcal{C}\mathit{at}$, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a $\Sigma$–free $\mathcal{C}\mathit{at}$ operad $\mathcal{O}$ is functorially weakly equivalent to a $\mathcal{C}\mathit{at}$ algebra over $\mathcal{O}$. When combined with the results of a previous paper, this allows us to conclude that, up to weak equivalences, the category of $\mathcal{O}$–categories is equivalent to the category of $B\mathcal{O}$–spaces, where $B:\mathcal{C}\mathit{at}\to \mathcal{T}\mathit{op}$ is the classifying space functor. In particular, $n$–fold loop spaces (and more generally $E_n$ spaces) are functorially weakly equivalent to classifying spaces of $n$–fold monoidal categories. Another application is a change of operads construction within $\mathcal{C}\mathit{at}$.

We provide a complete understanding of the rational homology of the space of long links of $m$ strands in ${ℝ}^d$ when $d\ge 4$. First, we construct explicitly a cosimplicial chain complex, $L_{\ast }^{\bullet }$, whose totalization is quasi-isomorphic to the singular chain complex of the space of long links. Next we show, using the fact that the Bousfield–Kan spectral sequence associated to $L_{\ast }^{\bullet }$ collapses at the $E^2$ page, that the homology Bousfield–Kan spectral sequence associated to the Munson–Volić cosimplicial model for the space of long links collapses at the $E^2$ page rationally, solving a conjecture of B Munson and I Volić. Our method enables us also to determine the rational homology of high-dimensional analogues of spaces of long links. Our last result states that the radius of convergence of the Poincaré series for the space of long links (modulo immersions) tends to zero as $m$ goes to infinity.

We give a necessary and sufficient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all $4$–dimensional orientable almost-flat manifolds with cyclic holonomy that do not admit a Spin structure.

We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in ${ℝ}^3$. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with nonreversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to $14$ crossings that have Legendrian representatives that are Lagrangian slice.

Two homotopy decompositions of suspensions of spaces involving polyhedral products are given. The first decomposition is motivated by the decomposition of suspensions of polyhedral products by Bahri, Bendersky, Cohen and Gitler, and is a generalization of a retractile argument of James. The second decomposition is on the union of an arrangement of subspaces called diagonal subspaces, and generalizes a result of Labassi.

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which includes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov–Rozansky polynomials in the case of nonnegative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov–Shende–Rasmussen conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at $a=0$, $q=1$ are conjectured to provide the Betti numbers of the Jacobian factors (compactified Jacobians) of the corresponding singularities.

The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor’s link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants obtained from iterated $p$–covers for homology cylinders. Using them, we show that the abelianization of the intersection of our filtration is of infinite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.

Any knot in a solid torus, called a pattern, induces a function, called a satellite operator, on concordance classes of knots in $S^3$ via the satellite construction. We introduce a generalization of patterns that form a group (unlike traditional patterns), modulo a generalization of concordance. Generalized patterns induce functions, called generalized satellite operators, on concordance classes of knots in homology spheres; using this we recover the recent result of Cochran and the authors that patterns with strong winding number $\pm 1$ induce injective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth $4$–dimensional Poincaré conjecture. We also obtain a characterization of patterns inducing surjective satellite operators, as well as a sufficient condition for a generalized pattern to have an inverse. As a consequence, we are able to construct infinitely many nontrivial patterns $P$ such that there is a pattern $\overline{P}$ for which $\overline{P}\left(P\left(K\right)\right)$ is concordant to $K$ (topologically as well as smoothly in a potentially exotic $S^3\times \left[0,1\right]$) for all knots $K$; we show that these patterns are distinct from all connected-sum patterns, even up to concordance, and that they induce bijective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth $4$–dimensional Poincaré conjecture.

We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of connected manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.

We give estimates for the torsion in the Postnikov sections ${\tau }_{\left[1,n\right]}{S}^0$ of the sphere spectrum, and we show that the $p$–localization is annihilated by $p^{n∕\left(2p-2\right)+O\left(1\right)}$. This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map ${\pi }_{\ast }\left(X\right)\to H_{\ast }\left(X;\mathbb{Z}\right)$ for a connective spectrum $X$. Such bounds were first considered by Arlettaz, although our estimates are tighter, and we prove that they are the best possible up to a constant factor. As applications, we sharpen existing bounds on the orders of $k$–invariants in a connective spectrum, sharpen bounds on the unstable Hurewicz map of an infinite loop space, and prove an exponent theorem for the equivariant stable stems.

Let $G\left(L_n\right)$ be the link group of a Brunnian $n$–link $L_n$ and $R_i$ be the normal closure of the $i^{th}$ meridian in $G\left(L_n\right)$ for $1\le i\le n$. In this article, we show that the intersecting subgroup $R_1\cap R_2\cap \cdots \cap R_m$ coincides with the iterated symmetric commutator subgroup ${\prod }_{\sigma \in {\Sigma }_m}\left[\left[R_{\sigma \left(1\right)},R_{\sigma \left(2\right)}\right],\dots ,R_{\sigma \left(m\right)}\right]$ for $2\le m\le n$ using the techniques of homotopy theory. Moreover, we give a presentation for the intersecting subgroup $R_1\cap R_2\cap \cdots \cap R_n$.

This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc). These are classes of knotted objects which are wider, but weaker than their “usual” counterparts.

The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke and was shown to be isomorphic to the McCool group of “basis-conjugating” automorphisms of a free group $F_n$: the smallest subgroup of $Aut\left(F_n\right)$ that contains both braids and permutations. Brendle and Hatcher, in work that traces back to Goldsmith, have shown this group to be a group of movies of flying rings in ${ℝ}^3$. Satoh studied several classes of w-knotted objects (under the name “weakly-virtual”) and has shown them to be closely related to certain classes of knotted surfaces in ${ℝ}^4$. So w-knotted objects are algebraically and topologically interesting.

Here we study finite-type invariants of w-braids and w-knots. Following Berceanu and Papadima, we construct homomorphic universal finite-type invariants of w-braids. The universal finite-type invariant of w-knots is essentially the Alexander polynomial.

Much as the spaces $\mathcal{A}$ of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces ${\mathcal{A}}^w$ of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Later in this paper series we re-interpret the work of Alekseev and Torossian on Drinfel’d associators and the Kashiwara–Vergne problem as a study of w-knotted trivalent graphs.

In the study of knot group epimorphisms, the existence of an epimorphism between two given knot groups is mostly (if not always) shown by giving an epimorphism which preserves meridians. A natural question arises: is there an epimorphism preserving meridians whenever a knot group is a homomorphic image of another? We answer in the negative by presenting infinitely many pairs of prime knot groups $\left(G,G^{\prime }\right)$ such that $G^{\prime }$ is a homomorphic image of $G$ but no epimorphism of $G$ onto $G^{\prime }$ preserves meridians.

We generalize two classical homotopy theory results, the Blakers–Massey theorem and Quillen’s Theorem B, to $G$–equivariant cubical diagrams of spaces, for a discrete group $G$. We show that the equivariant Freudenthal suspension theorem for permutation representations is a direct consequence of the equivariant Blakers–Massey theorem. We also apply this theorem to generalize to $G$–manifolds a result about cubes of configuration spaces from embedding calculus. Our proof of the equivariant Theorem B involves a generalization of the classical Theorem B to higher-dimensional cubes, as well as a categorical model for finite homotopy limits of classifying spaces of categories.

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity.

Our approach builds on the graded variant of $E_{\infty }$ spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded $E_{\infty }$ spaces and use it to exhibit our spectrum of units functor as a right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures.