Given an $l$–component pointed oriented link $\left(L,p\right)$ in an oriented three-manifold $Y$, one can construct its link Floer chain complex $CFL\left(Y,L,p\right)$ over the polynomial ring ${\mathbb{F}}_2\left[U_1,\dots ,U_l\right]$. Moving the basepoint $p_i\in L_i$ once around the link component $L_i$ induces an automorphism of $CFL\left(Y,L,p\right)$. We study a (possibly different) automorphism of $CFL\left(Y,L,p\right)$ defined explicitly in terms of holomorphic disks; for links in $S^3$, we show that these two automorphisms are the same.

We show that Khovanov homology (and its ${\mathfrak{s}\mathfrak{l}}_3$ variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of $2$–representations of categorified quantum ${\mathfrak{s}\mathfrak{l}}_m$ via categorical skew Howe duality. Utilizing Cautis–Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones–Wenzl projectors and their ${\mathfrak{s}\mathfrak{l}}_3$ analogs purely from the higher representation theory of categorified quantum groups. In the ${\mathfrak{s}\mathfrak{l}}_2$ case, this work reveals the importance of a modified class of foams introduced by Christian Blanchet which in turn suggest a similar modified version of the ${\mathfrak{s}\mathfrak{l}}_3$ foam category introduced here.

The homotopy theory of small functors is a useful tool for studying various questions in homotopy theory. In this paper, we develop the homotopy theory of small functors from spectra to spectra, and study its interplay with Spanier–Whitehead duality and enriched representability in the dual category of spectra.

We note that the Spanier–Whitehead duality functor $D:Sp\to {Sp}^{op}$ factors through the category of small functors from spectra to spectra, and construct a new model structure on the category of small functors, which is Quillen equivalent to ${Sp}^{op}$. In this new framework for the Spanier–Whitehead duality, $Sp$ and ${Sp}^{op}$ are full subcategories of the category of small functors and dualization becomes just a fibrant replacement in our new model structure.

The Temperley–Lieb algebra is a fundamental component of $SU\left(2\right)$ topological quantum field theories. We construct chain complexes corresponding to minimal idempotents in the Temperley–Lieb algebra. Our results apply to the framework which determines Khovanov homology. Consequences of our work include semi-orthogonal decompositions of categorifications of Temperley–Lieb algebras and Postnikov decompositions of all Khovanov tangle invariants.

Pontryagin classes $p_i\left(M\right)$ are basic invariants of a smooth manifold $M$, and many topological problems can be reduced to computing these classes. For a locally symmetric manifold, Borel and Hirzebruch gave an algorithm to determine if $p_i\left(M\right)$ is nonzero. In addition they implemented their algorithm for a few well-known $M$ and for $i=1$, $2$. Nevertheless, there remained several $M$ for which their algorithm was not implemented. In this note we compute low-degree Pontryagin classes for every closed, locally symmetric manifold of noncompact type. As a result of this computation, we answer the question: Which closed locally symmetric $M$ have at least one nonzero Pontryagin class?

Let $f:{ℂ}^{n+1}\to ℂ$ be a polynomial that is transversal (or regular) at infinity. Let $\mathcal{U}={ℂ}^{n+1}\setminus f^{-1}\left(0\right)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give several estimates for the (infinite cyclic) Alexander polynomials of $\mathcal{U}$ induced by $f$, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of $\mathcal{U}$.

The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples. Surprisingly, we show that there are (topologically embedded) smooth suborbifolds which do not arise as the image of a smooth orbifold embedding. We are also able to characterize those suborbifolds that can arise as the images of orbifold embeddings. As an application, we show that a length-minimizing curve (a geodesic segment) in a Riemannian orbifold can always be realized as the image of an orbifold embedding.

In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states.

A twisted torus knot is obtained from a torus knot by adding a number of full twists to some adjacent strands of the torus knot. In this paper, we show that if a twisted torus knot is a torus knot, then the number of added full twists is generically at most two in absolute value. We also show that this bound is the best possible by classifying twisted torus knots for which the upper bound is attained.

We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock and is related to questions asked by Farb and Mosher and by Farb.

We define quasihomomorphisms from braid groups to the concordance group of knots and examine their properties and consequences of their existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. We also provide applications to the geometry of the infinite braid group $B_{\infty }$. In particular, we show that the commutator subgroup $\left[B_{\infty },B_{\infty }\right]$ admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich.

Given a chord-generic, horizontally displaceable Legendrian submanifold $\Lambda \subset P\times ℝ$ with the property that its characteristic algebra admits a finite-dimensional matrix representation, we prove an Arnold-type lower bound for the number of Reeb chords on $\Lambda$. This result is a generalization of the results of Ekholm, Etnyre, Sabloff and Sullivan, which hold for Legendrian submanifolds whose Chekanov–Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds $\Lambda$ of ${ℂ}^n\times ℝ$, $n\ge 1$, whose characteristic algebras admit finite-dimensional matrix representations but whose Chekanov–Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct a Legendrian submanifold $\Lambda \subset {ℂ}^n\times ℝ$ with the property that the characteristic algebra of $\Lambda$ does not satisfy the rank property. Finally, in the case when a Legendrian submanifold $\Lambda$ has a non-acyclic Chekanov–Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of $\Lambda$. These bounds are slightly better than the number of Reeb chords it is possible to achieve with a Legendrian submanifold whose Chekanov–Eliashberg algebra is acyclic.

We show how the existing proof of the Farrell–Jones conjecture for virtually poly-$\mathbb{Z}$–groups can be improved to rely only on the usual inheritance properties in combination with transfer reducibility as a sufficient criterion for the validity of the conjecture.

This paper introduces even triangulations of $n$–dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension $3$, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken $n$–manifolds, an interesting connection between very short hierarchies and even triangulations is observed.

We reduce Rudyak’s conjecture that a degree-one map between closed manifolds cannot raise the Lusternik–Schnirelmann category to the computation of the category of the product of two lens spaces $L_p^n\times L_q^n$ with relatively prime $p$ and $q$. We have computed $cat\left(L_p^n\times L_q^n\right)$ for values $p$, $q>n∕2$. It turns out that our computation supports the conjecture.

For spin manifolds $M$ we establish a criterion for the equality $catM=dimM-1$, which is a K–theoretic refinement of the Katz–Rudyak criterion for $catM=dimM$. We apply it to obtain the inequality $cat\left(L_p^n\times L_q^n\right)\le 2n-2$ for all odd $n$ and odd relatively prime $p$ and $q$.

We find infinitely many lattices in $SL\left(4,R\right)$, each of which contains thin subgroups commensurable with the figure-eight knot group.

For equivariant stable homotopy theory, equivariant KK–theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite étale extensions in algebraic geometry.

We give an easy proof that the Morava $K$–theories for $BO\left(q\right)$ and $MO\left(q\right)$ are in even degrees. Although this is a known result, it had followed from a difficult proof that ${BP}^{\ast }\left(BO\left(q\right)\right)$ was Landweber flat. Landweber flatness follows from the even Morava $K$–theory. We go further and compute an explicit description of $K{\left(n\right)}_{\ast }\left(BO\left(q\right)\right)$ and $K{\left(n\right)}_{\ast }\left(MO\left(q\right)\right)$ and reconcile it with the purely algebraic construct from Landweber flatness.

A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. In this paper, we describe all the fiberwise homogeneous fibrations of Euclidean and hyperbolic $3$–space by geodesics. Our main result is that, up to fiber-preserving isometries, there is precisely a one-parameter family of such fibrations of Euclidean $3$–space, and a two-parameter family in hyperbolic $3$–space.

We study the asymptotic geometry of Teichmüller space equipped with the Weil–Petersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as a similar characterization for right angled Artin groups and for mapping class groups. As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type, thereby answering questions of Behrstock, Druţu and Mosher, and Brock and Masur. In particular, we prove that Teichmüller space of the genus-two surface with one boundary component (or puncture) is the only Teichmüller space which is thick of order two.