It is shown that if $\{H_n\}_{n \in \omega}$ is a sequence of groups without involutions, with $1 \lt |H_n| \leq 2^{\aleph_0}$, then the topologist’s product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in \omega}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in \omega} A_n/\bigoplus_{n \in \omega} A_n$ is dependent on the sequence.

We show that under mild conditions, the connected sum $M\# N$ of simply connected, closed, orientable n-dimensional Poincaré Duality complexes M and N is hyperbolic and has no homotopy exponent at all but finitely many primes, verifying a weak version of Moore’s conjecture. This is derived from an elementary framework involving $CW$-complexes satisfying certain conditions.

For a path-connected metric space $(X,d)$, the $n$-th homotopy group $\pi _n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $\pi _n(X)$ the structure of a topological group, and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $\pi _n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on $\pi _n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps.

Inspired by a remarkable work of Félix, Halperin, and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using K-theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg–MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces, and unitary groups.

Let $\mathscr {A}$ be a topological Azumaya algebra of degree $mn$ over a CW complex X. We give conditions for the positive integers m and n, and the space X so that $\mathscr {A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees m and n. Then we prove that if $m<n$ and the dimension of X is higher than $2m+1$, $\mathscr {A}$ may not have such decomposition.

Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$-operations at a point $x\in X$. Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

We construct a spectral sequence converging to the $E_{2}$-term of the Bousfield–Kan spectral sequence (BKSS) for a wide variety of homology theories. Using this, the $E_{2}$-term of the BKSS based on $K(1)$-theory for the odd spheres is computed and the unstable $K(1)$-completion is computed.

Exponent information is proven about the Lie groups $SU(3),\,SU(4),\,Sp(2)$ , and ${{G}_{2}}$ by showing some power of the $H$ -space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are $8,\,32,\,64$ , and ${{2}^{8}}$ respectively. This null homotopy is best possible for $SU(3)$ given the number of loops, off by at most one power of 2 for $SU(4)$ and $Sp(2)$ , and off by at most two powers of 2 for ${{G}_{2}}$ .

Using spaces introduced by Anick, we construct a decomposition into indecomposable factors of the double loop spaces of odd primary Moore spaces when the powers of the primes are greater than the first power. If $n$ is greater than 1, this implies that the odd primary part of all the homotopy groups of the $2n\,+\,1$ dimensional sphere lifts to a mod ${{p}^{r}}$ Moore space.