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Starting categorically, we give simple and precise models for classifying spaces of equivariant principal bundles. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic –theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.
We show that if –surgery on a nontrivial knot yields the branched double cover of an alternating knot, then . This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched covers of alternating knots must be contained in an interval of width two and this full range can be realised only if the knot is a cable knot. The work of Greene and Gibbons shows that if bounds a sharp –manifold , then the intersection form of takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot , the slope and the Betti number .
Singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod , thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, some torus knots, and Montesinos knots, as well as for several families of two-component links.
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B Shipley based on J Cohen’s approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley’s and show that they are self-dual. Our main result is that the Adams differential in any Adams spectral sequence can be expressed as an –fold Toda bracket and as an order cohomology operation. We also show how the result simplifies under a sparseness assumption, discuss several examples and give an elementary proof of a result of Heller, which implies that the –fold Toda brackets in principle determine the higher Toda brackets.
We consider a finitely generated virtually abelian group acting properly and without inversions on a cube complex . We prove that stabilizes a finite-dimensional subcomplex that is isometrically embedded in the combinatorial metric. Moreover, we show that is a product of finitely many quasilines. The result represents a higher-dimensional generalization of Haglund’s axis theorem.
We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that, given a Hilbert modular variety of real dimension defined over a number field , the sequence of principal congruence coverings eventually satisfies
Picard –categories are symmetric monoidal –categories with invertible –, – and –cells. The classifying space of a Picard –category is an infinite loop space, the zeroth space of the –theory spectrum . This spectrum has stable homotopy groups concentrated in levels , and . We describe part of the Postnikov data of in terms of categorical structure. We use this to show that there is no strict skeletal Picard –category whose –theory realizes the –truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard –category from a Picard –category , and show that it commutes with –theory, in that is stably equivalent to .
Associated to every oriented link in the 3–sphere is its fundamental quandle and, for each , there is a certain quotient of the fundamental quandle called the –quandle of the link. We prove a conjecture of Przytycki which asserts that the –quandle of an oriented link in the 3–sphere is finite if and only if the fundamental group of the –fold cyclic branched cover of the 3–sphere, branched over , is finite. We do this by extending into the setting of –quandles, Joyce’s result that the fundamental quandle of a knot is isomorphic to a quandle whose elements are the cosets of the peripheral subgroup of the knot group. In addition to proving the conjecture, this relationship allows us to use the well-known Todd–Coxeter process to both enumerate the elements and find a multiplication table of a finite –quandle of a link. We conclude the paper by using Dunbar’s classification of spherical 3–orbifolds to determine all links in the 3–sphere with a finite –quandle for some .
Let be an infinite commutative ring with identity and an integer. We prove that for each integer , the –Betti number vanishes when is the general linear group , the special linear group or the group generated by elementary matrices. When is an infinite principal ideal domain, similar results are obtained when is the symplectic group , the elementary symplectic group , the split orthogonal group or the elementary orthogonal group . Furthermore, we prove that is not acylindrically hyperbolic if . We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of –rigid rings.
Given a rank-2 hermitian bundle over a –manifold that is nontrivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the –divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the –manifold. This formula counts flat connections on the induced adjoint bundle with Klein-four holonomy.
The author recently proved the existence of an infinite order cork: a compact, contractible submanifold of a 4–manifold and an infinite order diffeomorphism of such that cutting out and regluing it by distinct powers of yields pairwise nondiffeomorphic manifolds. The present paper exhibits the first handle diagrams of this phenomenon, by translating the earlier proof into this language (for each of the infinitely many corks arising in the first paper). The cork twists in these papers are twists on incompressible tori. We give conditions guaranteeing that such twists do not change the diffeomorphism type of a 4–manifold, partially answering a question from the original paper. We also show that the “–moves” recently introduced by Akbulut are essentially equivalent to torus twists.
We describe a family of hyperbolic knots whose character variety contain exactly two distinct components of characters of irreducible representations. The intersection points between the components carry rich topological information. In particular, these points are nonintegral and detect a Seifert surface.
We prove the existence of an exact triangle for the –monopole Floer homology groups of three-manifolds related by specific Dehn surgeries on a given knot. Unlike the counterpart in usual monopole Floer homology, only two of the three maps are those induced by the corresponding elementary cobordism. We use this triangle to describe the Manolescu correction terms of the manifolds obtained by –surgery on alternating knots with Arf invariant .
We show that the quantum covering group associated to has an associated colored quantum knot invariant à la Reshetikhin–Turaev, which specializes to a quantum knot invariant for , and to the usual quantum knot invariant for . In particular, this furnishes an “odd” variant of quantum invariants, even for knots labeled by spin representations. We then show that these knot invariants are essentially the same, up to a change of variables and a constant factor depending on the knot and weight.
We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid –algebra and thereby its topological –theory spectrum.
Khovanov and Rozansky’s categorification of the homfly-pt polynomial is invariant under braidlike isotopies for any general link diagram and Markov moves for braid closures. To define homfly-pt homology, they required a link to be presented as a braid closure, because they did not prove invariance under the other oriented Reidemeister moves. In this text we prove that the Reidemeister IIb move fails in homfly-pt homology by using virtual crossing filtrations of the author and Rozansky. The decategorification of homfly-pt homology for general link diagrams gives a deformed version of the homfly-pt polynomial, , which can be used to detect nonbraidlike isotopies. Finally, we will use to prove that homfly-pt homology is not an invariant of virtual links, even when virtual links are presented as virtual braid closures.
We give a complete characterization of the topological slice status of odd –strand pretzel knots, proving that an odd –strand pretzel knot is topologically slice if and only if it either is ribbon or has trivial Alexander polynomial. We also show that topologically slice even –strand pretzel knots, except perhaps for members of Lecuona’s exceptional family, must be ribbon. These results follow from computations of the Casson–Gordon –manifold signature invariants associated to the double branched covers of these knots.
We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold are obstructions to positive scalar curvature on an ambient manifold that contains as a submanifold. In the main result of this note, we show that the Rosenberg index of is an obstruction to positive scalar curvature on if is a fiber bundle of spin manifolds with aspherical and of finite asymptotic dimension. The proof is based on a new variant of the multipartitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension-one submanifolds. We also discuss some elementary obstructions using the -genus of certain submanifolds.
Greenlees established an equivalence of categories between the homotopy category of rational –spectra and the derived category of a certain abelian category. In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. Methods used in this paper provide the first step towards obtaining an algebraic model for the toral part of rational –spectra, for any compact Lie group .
Using factorization homology, we realize the rational homology of the unordered configuration spaces of an arbitrary manifold , possibly with boundary, as the homology of a Lie algebra constructed from the compactly supported cohomology of . By locating the homology of each configuration space within the Chevalley–Eilenberg complex of this Lie algebra, we extend theorems of Bödigheimer, Cohen and Taylor and of Félix and Thomas, and give a new, combinatorial proof of the homological stability results of Church and Randal-Williams. Our method lends itself to explicit calculations, examples of which we include.
We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal -categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.
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