Moving homology classes to infinity

Moving Homology Classes to Infinity

Let q : X̃ → X be a regular covering over a finite polyhedron with free abelian group of covering translations. Each nonzero cohomology class ξ ∈ H(X;R) with q∗ξ = 0 determines a notion of “infinity” of the noncompact space X̃. In this paper we characterize homology classes z in X̃ which can be realized in arbitrary small neighborhoods of infinity in X̃. This problem was motivated by applications i...

Moving homology classes in finite covers of graphs

Let Y → X be a finite normal cover of a wedge of n ≥ 3 circles. We prove that for any v 6= 0 ∈ H1(Y ; Q) there exists a lift F̃ to Y of a homotopy equivalence F : X → X so that the set of iterates {F̃ (v) : d ∈ Z} ⊆ H1(Y ; Q) is infinite. The main achievement of this paper is the use of representation theory to prove the existence of a purely topological object that seems to be inaccessible via t...

Quantifying Homology Classes

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements’ size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(βn log n...

The compactificability classes: The behavior at infinity

We study the behavior of certain spaces and their compactificability classes at infinity. Among other results we show that every noncompact, locally compact, second countable Hausdorff space X such that each neighborhood of infinity (in the Alexandroff compactification) is uncountable, has (X) = (R). We also prove some criteria for (non-) comparability of the studied classes of mutual compactif...

Chow Homology and Chern Classes