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We investigate rational homology cobordisms of –manifolds with nonzero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology ’s bound rational homology ’s. We give a simple procedure to construct rational homology cobordisms between plumbed –manifolds. We introduce a family of plumbed –manifolds with . By adapting an obstruction based on Donaldson’s diagonalization theorem we characterize all manifolds in our family that bound rational homology ’s. For all these manifolds a rational homology cobordism to can be constructed via our procedure. Our family is large enough to include all Seifert fibered spaces over the –sphere with vanishing Euler invariant. In a subsequent paper we describe applications to arborescent link concordance.
Lie –groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie –groupoids called “Lie –groups” by integrating finite type Lie –algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie –algebras introduced in the companion paper (1371–1429), we present a homotopy theory for Lie –groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie –groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie –groupoids form an “incomplete category of fibrant objects” in which the weak equivalences correspond to “stalkwise” weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler’s results also hold in this more general context. As an application, we show that Henriques’ integration functor is an exact functor with respect to a class of distinguished fibrations, which we call “quasisplit fibrations”. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends –quasi-isomorphisms to weak equivalences and quasisplit fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasisplit fibrations.
We study Euler classes in groups of homeomorphisms of Seifert-fibered –manifolds. In contrast to the familiar Euler class for as a discrete group, we show that these Euler classes for as a discrete group are unbounded classes. In fact, we give examples of flat topological –bundles over a genus surface whose Euler class takes arbitrary values.
Previously (Adv. Math. 360 (2020) art. id. 106895), we introduced a class of –local finite spectra and showed that all spectra admit a –self-map of periodicity . The aim here is to compute the –local homotopy groups of all spectra using a homotopy fixed point spectral sequence, and we give an almost complete answer. The incompleteness lies in the fact that we are unable to eliminate one family of –differentials and a few potential hidden –extensions, though we conjecture that all these differentials and hidden extensions are trivial.
We show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang–Baxter equation (an involutive –matrix), which determines an extremal character on . These characters are classified by Thoma parameters, and Thoma parameters resulting from polynomial exponential functors are of a special kind. Moreover, we show that each –matrix with Thoma parameters of this kind yield a corresponding polynomial exponential functor.
In the second part of the paper we use these functors to construct a higher twist over for a localisation of –theory that generalises the one classified by the basic gerbe. We compute the indecomposable part of the rational characteristic classes of these twists in terms of the Thoma parameters of their –matrices.
We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of cube complexes. Examples of median spaces with compact intervals include all finite-rank median spaces and all proper median spaces of infinite rank. Our methods also apply to general median algebras, where we recover the zero-completions of Bandelt and Meletiou (1993). Along the way, we prove various properties of halfspaces in finite-rank median spaces and a duality result for locally convex median spaces.
Lie –algebras are the analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite-type Lie –algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie –groups, via a smooth analog of Sullivan’s realization functor. We provide an explicit proof that the category of finite-type Lie –algebras and (weak) –morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Roughly speaking, this CFO structure can be thought of as the transfer of the classical projective CFO structure on nonnegatively graded chain complexes via the tangent functor. In particular, the weak equivalences are precisely the –quasi-isomorphisms. Along the way, we give explicit constructions for pullbacks and factorizations of –morphisms between finite-type Lie –algebras. We also analyze Postnikov towers and Maurer–Cartan/deformation functors associated to such Lie –algebras. The main application of this work is our joint paper with C Zhu (1127–1219), which characterizes the compatibility of Henriques’ integration functor with the homotopy theory of Lie –algebras and that of Lie –groups.
We prove the recognition principle for relative –loop pairs of spaces for . If , this states that a pair of spaces homotopy equivalent to CW–complexes is homotopy equivalent to for a functorially determined relative space if and only if is a grouplike –space, where is any cofibrant resolution of the Swiss-cheese relative operad . The relative recognition principle for relative –loop pairs of spaces states that a pair of spaces homotopy equivalent to CW–complexes is homotopy equivalent to for a functorially determined relative spectrum if and only if is a grouplike –algebra, where is a contractible cofibrant relative operad or equivalently a cofibrant resolution of the terminal relative operad of continuous homomorphisms of commutative monoids. These principles are proved as equivalences of homotopy categories.
We prove that homotopy invariance and cancellation properties are satisfied by any category of correspondences that is defined, via Calmès and Fasel’s construction, by an underlying cohomology theory. In particular, this includes any category of correspondences arising from the cohomology theory defined by an –algebra.
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely –complicial sets, which are a proposed model for –categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models.
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