Journal of Topology and Analysis

Let denote the graph whose set of vertices is {0,…,m - 1}^d, where two distinct vertices are adjacent if and only if they are either equal or adjacent in the m-cycle C_m in each coordinate. Let denote the graph on the same set of vertices in which two vertices are adjacent if and only if they are adjacent in one coordinate in C_m and equal in all others. Both graphs can be viewed as graphs of the d-dimensional torus. We prove that one can delete vertices of G₁ so that no topologically nontrivial cycles remain. This improves an O(d^{log₂ (3/2)}m^{d - 1}) estimate of Bollobás, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an fraction of the edges of G_∞ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous d-dimensional torus of surface area that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.

We prove that every length on a simple group over a locally compact field, is either bounded or proper.

Closed oriented 4-manifolds with the same geometrically two-dimensional fundamental group (satisfying certain properties) are classified up to s-cobordism by their w₂-type, equivariant intersection form and the Kirby–Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4-manifolds with solvable Baumslag–Solitar fundamental groups, including a precise realization result.

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first ℓ²-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher ℓ²-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans and Martin.

The main result is that a Banach space X is not super-reflexive if and only if the diamond graphs D_n Lipschitz embed into X with distortions independent of n. One of the consequences of that and previously known results is that dimension reduction a la Johnson–Lindenstrauss fails in any non-super-reflexive space with nontrivial type.

We also introduce the concept of Lipschitz (p,r)-summing map and prove that every Lipschitz mapping is Lipschitz (p,r)-summing for every 1 ≤ r < p.