1,720,999 research outputs found
Coarse embeddings into products of trees
No full textWe give a short and elementary proof of the fact that every metric space of finite asymptotic dimension can be coarsely embedded into a finite product of trees
The asymptotic dimension of quotients by finite groups
Full text linkLet X be a proper metric space and let F be a finite group acting on X by isometries. We show that the asymptotic dimension of F \X is the same as the asymptotic dimension of X
On the K-theory of linear groups
Full text linkWe prove that for a finitely generated linear group over a field of positive characteristic the family of quotients by finite subgroups has finite asymptotic dimension. We use this to show that the K-theoretic assembly map for the family of finite subgroups is split injective for every finitely generated linear group G over a commutative ring with unit under the assumption that G admits a finite-dimensional model for the classifying space for the family of finite subgroups. Furthermore, we prove that this is the case if and only if an upper bound on the rank of the solvable subgroups of G exists
Topological 4-manifolds with 4-dimensional fundamental group
No full textLet be a group satisfying the Farrell-Jones conjecture and assume that is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group whose canonical map to has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby-Siebenmann invariant. If is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby-Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel's conjecture, and simply connected manifolds where rigidity is a consequence of Freedman's classification results
-theory of groups with finite decomposition complexity
Full text linkIt is proved that the assembly map in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups Γ with finite quotient finite decomposition complexity (a strengthening of finite decomposition complexity introduced by Guentner, Tessera and Yu) that admit a finite-dimensional model for EΓ and have an upper bound on the order of their finite subgroups. In particular, this applies to finitely generated linear groups over fields with characteristic zero with a finite-dimensional model for EΓ
Shortening binary complexes and commutativity of -theory with infinite products
Full text linkWe show that in Grayson's model of higher algebraic -theory using binary acyclic complexes, the complexes of length two suffice to generate the whole group. Moreover, we prove that the comparison map from Nenashev's model for to Grayson's model for is an isomorphism. It follows that algebraic -theory of exact categories commutes with infinite products
On the Farrell-Jones conjecture for localising invariants
No full textWe show the Farrell-Jones conjecture with coefficients in left-exact -categories for finitely -amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over -ring spectra
Counterexamples in 4-manifold topology
Full text linkWe illustrate the rich landscape of 4-manifold topology through the lens of
counterexamples. We consider several of the most commonly studied equivalence
relations on 4-manifolds and how they are related to one another. We explain
implications e.g. that -cobordant manifolds are stably homeomorphic, and we
provide examples illustrating the failure of other potential implications. The
information is conveniently organised in a flowchart and a table.Comment: 37 pages, 4 figures, 1 table; in v2, we have made several changes in
response to a referee report, including writing a more detailed introduction,
adding more details about the surgery exact sequence, uniformising the
structure of the subsections describing counterexamples, and adding
Proposition 5.6. This is the version published in EMS Survey
Homotopy classification of 4-manifolds with finite abelian 2-generator fundamental groups
Full text linkWe show that for an oriented 4-dimensional Poincaré complex with finite fundamental group, whose 2-Sylow subgroup is abelian with at most 2 generators, the homotopy type is determined by its quadratic 2-type.17 pages. Minor changes following a referee report. To appear in Mathematical Proceedings of the Cambridge Philosophical Societ
Long and thin covers for flow spaces
Full text linkLong and thin covers of flow spaces are important ingredients in the proof of the Farrell-Jones conjecture for certain classes of groups, like hyperbolic and CAT(0)-groups. In this paper we provide an alternative construction of such covers which holds in a more general setting and simplifies some of the arguments
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