Some small aspherical spaces

Abstract

Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

Relating the Freiheitssatz to the asymptotic behavior of a group

“…This extends a result of Dyer and Vasquez in [11] for torsion-free one-relator groups (see also Lyndon's Identity Theorem in [21] for a more algebraic version).…”

On weak epimorphisms in homotopy theory

“…But the ^-dimensional obstruction to g = 0 lies in the (constant coefficient) cohomology group H k (K (H, 1); <π k Z\ k > 2. As H is acyclic and geometrically finite ( [DV;§4]), these obstructions vanish, g = 0, and hence g = 0.…”

On 2-dimensionalCW-complexes with a single 2-cell

“…If the single relator R is not a proper power, it is known that the cellular model C{&) is aspherical (see [10], [1], or [4]), hence it is determined up to homotopy type by its fundamental group. If the single relator R is a proper power, C(&) is not aspherical, nevertheless we are able to prove the following: THEOREM …”

“…Then 2 is torsion-free ( [11], Theorem 4.2, p. 266). This implies that π is torsion-free as well so that q = 1; thus X and Y are aspherical (see [10], [1], or [4]). Since by hypothesis 7t λ {X) -Ξ f& π = TC^Y), they have the same homotopy type and in fact there is a homotopy equivalence between X and Y inducing any isomorphism a: Ξ -> π.…”