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We extend some results of Hatcher and Quinn (1974) beyond the metastable range. We give a bordism-theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the assumption that can be deformed off of any proper subcollection in a homotopy coherent way. In a certain range of dimensions, is a complete obstruction to finding the desired deformation. We apply this machinery to embedding problems and to the study of linking phenomena.
The Rips filtration over a finite metric space and its corresponding persistent homology are prominent methods in topological data analysis to summarise the “shape” of data. Crucial to their use is the stability result that says if and are finite metric spaces then the (bottleneck) distance between the persistence diagrams constructed via the Rips filtration is bounded by (where is the Gromov–Hausdorff distance). A generalisation of the Rips filtration to any symmetric function was defined by Chazal, de Silva and Oudot (Geom. Dedicata 173 (2014) 193–214), where they showed it was stable with respect to the correspondence distortion distance. Allowing asymmetry, we consider four different persistence modules, definable for pairs where is any real valued function. These generalise the persistent homology of the symmetric Rips filtration in different ways. The first method is through symmetrisation. For each we can construct a symmetric function . We can then apply the standard theory for symmetric functions and get stability as a corollary. The second method is to construct a filtration of ordered tuple complexes where if for all . Both our first two methods have the same persistent homology as the standard Rips filtration when applied to a metric space, or more generally to a symmetric function. We then consider two constructions using an associated filtration of directed graphs or preorders. For each we can define a directed graph where directed edges are included in whenever (note this is when for a quasimetric). From this we construct a preorder where if there is a path from to in . We build persistence modules using the strongly connected components of the graphs , which are also the equivalence classes of the associated preorders. We also consider persistence modules using a generalisation of poset topology to preorders.
The Gromov–Hausdorff distance, when expressed via correspondence distortions, can be naturally extended as a correspondence distortion distance to set–function pairs . We prove that all these new constructions enjoy the same stability as persistence modules built via the original persistent homology for symmetric functions.
In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. We introduce the concept of a –algebraicdrawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on –algebras. We show that the construction is functorial and, in fact, it is the left adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable –categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant –theory. Finally, a method to analyze graph algebras in terms of trees is sketched.
We start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if is a finite collection of hyperbolic automorphisms of a CAT(0) square complex , then either there exists a pair of words of length at most in which freely generate a free semigroup, or all elements of stabilize a flat (of dimension or in ). As a corollary, we obtain a lower bound for the growth constant, , which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.
We prove that if a right-angled Artin group is abstractly commensurable to a group splitting nontrivially as an amalgam or HNN extension over , then must itself split nontrivially over for some . Consequently, if two right-angled Artin groups and are commensurable and has no separating –cliques for any , then neither does , so “smallest size of separating clique” is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for the braid group is not abstractly commensurable to any group that splits nontrivially over a “free group–free” subgroup, and the same holds for for the loop braid group . Our approach makes heavy use of the Bieri–Neumann–Strebel invariant.
Let be a smooth manifold and be a simplicial complex of codimension at least . Functor calculus methods lead to a homotopical formula of in terms of spaces where is a finite subset of . This is a generalization of the author’s previous work with Michael Weiss (Contemp. Math. 682, Amer. Math. Soc., Providence, RI (2017) 237–259), where the subset is assumed to be a smooth submanifold of and uses his generalization of manifold calculus adapted for simplicial complexes.
For oriented manifolds of dimension at least that are simply connected at infinity, it is known that end summing is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. We examine how and when uniqueness fails. Examples are given, in the categories top, pl and diff, of nonuniqueness that cannot be detected in a weaker category (including the homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends, and generalized to allow slides and cancellation of (possibly infinite) collections of – and –handles at infinity. Various applications are presented, including an analysis of how the monoid of smooth manifolds homeomorphic to acts on the smoothings of any noncompact –manifold.
In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne’s seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a –arrangement.
We study the –property for a certain class of subarrangements of Weyl arrangements, the so-called arrangements of ideal type . These stem from ideals in the set of positive roots of a reduced root system. We show that the –property holds for all arrangements if the underlying Weyl group is classical and that it extends to most of the if the underlying Weyl group is of exceptional type. Conjecturally this holds for all . In general, the are neither simplicial nor is their complexification of fiber type.
We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and nonorientable). We also give improved bounds for the topological complexity of unordered configuration spaces on all aspherical closed surfaces, reducing it to three possible values. The main methods used in the proofs were developed in 2015 by Grant, Lupton and Oprea to give bounds for the topological complexity of aspherical spaces. As such this paper is also part of the current effort to study the topological complexity of aspherical spaces and it presents many further examples where these methods strongly improve upon the lower bounds given by zero-divisor cup-length.
We prove that all atoroidal automorphisms of act on the space of projectivized geodesic currents with generalized north–south dynamics. As an application, we produce new examples of nonvirtually cyclic, free and purely atoroidal subgroups of such that the corresponding free group extension is hyperbolic. Moreover, these subgroups are not necessarily convex cocompact.
Our goal is to introduce a smaller, but equivalent version of the deformation –groupoid associated to a homotopy Lie algebra. In the case of differential graded Lie algebras, we represent it by a universal cosimplicial object.
We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady’s classifying space for braid groups and the Stein–Farley complexes used for various relatives of Thompson’s groups. It recovers the fact that Garside groups have finite classifying spaces.
We describe the categories and Garside structures underlying certain Thompson groups. The indirect product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our methods we introduce the group braided and show that it is of type .
Let be a pointed CW–complex. The generalized conjecture on spherical classes states that the Hurewicz homomorphismvanishes on classesof of Adams filtrationgreater than . Let denote the Lannes–Zarati homomorphism for the unstable –module . When , this homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the Lannes–Zarati homomorphism, , vanishes in any positive stem for and for any unstable –module .
We prove that, for an unstable –module of finite type, the Lannes–Zarati homomorphism, , vanishes on decomposable elements of the form in positive stems, where and with either , and , or , and . Consequently, we obtain a theorem proved by Hung and Peterson in 1998. We also prove that the fifth Lannes–Zarati homomorphism for vanishes on decomposable elements in positive stems.
We provide an extensive study of the homotopy theory of types of algebras with units, for instance unital associative algebras or unital commutative algebras. To this purpose, we endow the Koszul dual category of curved coalgebras, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of unital algebras. To prove such a result, we use recent methods based on presentable categories. This allows us to describe the homotopy properties of unital algebras in a simpler and richer way. Moreover, we endow the various model categories with several enrichments which induce suitable models for the mapping spaces and describe the formal deformations of morphisms of algebras.
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