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There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realised as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary –handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies.
Here we focus on the spectral sequence due to Kronheimer and Mrowka from Khovanov homology to instanton knot Floer homology, and on that due to Ozsváth and Szabó to the Heegaard Floer homology of the branched double cover. For example, we use the –handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the –torus knot, determining these up to an ambiguity in a pair of degrees; to determine the Ozsváth–Szabó spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the Kronheimer–Mrowka and the Ozsváth–Szabó spectral sequences necessarily lower the delta grading for all pretzel knots.
We find bounds for the Hofer–Zehnder capacity of spherically monotone coadjoint orbits of compact Lie groups with respect to the Kostant–Kirillov–Souriau symplectic form in terms of the combinatorics of their Bruhat graphs. We show that our bounds are sharp for coadjoint orbits of the unitary group and equal in that case to the diameter of a weighted Cayley graph.
Wright (1992) showed that, if a –ended, simply connected, locally compact ANR with pro-monomorphic fundamental group at infinity (ie representable by an inverse sequence of monomorphisms) admits a –action by covering transformations, then that fundamental group at infinity can be represented by an inverse sequence of finitely generated free groups. Geoghegan and Guilbault (2012) strengthened that result, proving that also satisfies the crucial semistability condition (ie representable by an inverse sequence of epimorphisms).
Here we get a stronger theorem with weaker hypotheses. We drop the “pro-monomorphic hypothesis” and simply assume that the –action is generated by what we call a “coaxial” homeomorphism. In the pro-monomorphic case every –action by covering transformations is generated by a coaxial homeomorphism, but coaxials occur in far greater generality (often embedded in a cocompact action). When the generator is coaxial, we obtain the sharp conclusion: is proper –equivalent to the product of a locally finite tree with . Even in the pro-monomorphic case this is new: it says that, from the viewpoint of the fundamental group at infinity, the “end” of looks like the suspension of a totally disconnected compact set.
We describe two locally finite graphs naturally associated to each knot type , called Reidemeister graphs. We determine several local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly, we introduce another object, relating the Reidemeister and Gordian graphs, and determine some of its properties.
We reinterpret Kim’s nonabelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of étale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer–Manin obstruction, allowing us to determine when Kim’s maps recover the Brauer–Manin locus. A tower based on relative completions yields nontrivial reciprocity maps even for Shimura varieties; for the stacky modular curve, these take values in Galois cohomology of modular forms, and give obstructions to an adèlic elliptic curve with global Tate module underlying a global elliptic curve.
The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in . In recent years, the realization problem for C–, T–, O– and I–type spherical manifolds has been solved, leaving the D–type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized as for a pair of relatively prime integers and . We determine a list of prism manifolds that can possibly be realized by positive integral surgeries on knots in when . Based on the forthcoming work of Berge and Kang, we are confident that this list is complete. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces.
We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the –homology of , where is a ring spectrum, is a commutative –algebra and is a connective commutative –algebra. The input of the spectral sequence are the topological Hochschild homology groups of with coefficients in the –homology groups of . The mod and topological Hochschild homology of connective complex –theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We present an alternative, short computation using the generalized Brun spectral sequence.
For each rational homology –sphere which bounds simply connected definite –manifolds of both signs, we construct an infinite family of irreducible rational homology –spheres which are homology cobordant to but cannot bound any simply connected definite –manifold. As a corollary, for any coprime integers and , we obtain an infinite family of irreducible rational homology –spheres which are homology cobordant to the lens space but cannot be obtained by a knot surgery.
We study the mod- homotopy type of classifying spaces for commutativity, , at a prime . We show that the mod- homology of depends on the mod- homotopy type of when is a compact connected Lie group, in the sense that a mod- homology isomorphism for such groups induces a mod- homology isomorphism . In order to prove this result, we study a presentation of as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gómez. We also study the relationship between the mod- type of a Lie group and the locally finite group , where is a Chevalley group. We see that the naïve analogue for of the celebrated Friedlander–Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a action on .
We study the arithmeticity of the Couwenberg–Heckman–Looijenga lattices in , and show that they contain a nonarithmetic lattice in which is not commensurable to the nonarithmetic Deligne–Mostow lattice in .
We formulate and prove a version of the Segal conjecture for infinite groups. For finite groups it reduces to the original version. The condition that is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups that the zeroth stable cohomotopy of the classifying space is isomorphic to the –adic completion of the ring given by the zeroth equivariant stable cohomotopy of for the augmentation ideal.
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with –bridge links. This characterizes fibering for many families of state surfaces.
We apply mapping class group techniques and trisections to study intersection forms of smooth –manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology –sphere can be obtained from the standard Heegaard decomposition of by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of –manifolds. Specifically, if and admit handle decompositions without – or –handles and have isomorphic intersection forms, then a trisection of can be obtained from a trisection of by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology –spheres can be applied, via this result, to obstruct intersection forms of smooth –manifolds. As an application, we use the Casson invariant to recover Rohlin’s theorem on the signature of spin –manifolds.
We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the –action on Lee’s perturbation of Khovanov homology.
We study the –dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number , and show that a knot has immersed crosscap number if and only if it is a nontrivial –torus or –cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the –dimensional crosscap number.
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