A direct proof is presented of a form of Alvise Trevisan’s theorem, that every monomial ideal ring is represented by the cohomology of a topological space. Certain of these rings are shown to be realized by polyhedral products indexed by simplicial complexes.

The main objective of this paper is to determine generators of the topological filtrations on the higher $K$-theory of a noetherian commutative ring with unit A. We introduce the concept of Koszul cubes and give a comparison theorem between the $K$-theory of Koszul cubes with that of topological filtrations.

We define homological dimensions for $S$-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real $K$-theory $KO$ and its connective version $ko$ at the prime 2. We show that the global dimension of $KO$ is 2 or 3, and the global dimension of $ko$ is 4 or 5.

We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces $X$ whose higher topological complexity ${\sf TC}_n(X)$ is as low as possible, namely $n-1$.

A simple, simply-connected, compact Lie group $G$ is $p$-regular if it is homotopy equivalent to a product of spheres when localized at $p$. If $A$ is the corresponding wedge of spheres, then it is well known that there is a $p$-local retraction of $G$ off $\Omega\Sigma A$. We show that that complementary factor is very well behaved, and this allows us to deduce properties of $G$ from those of $\Omega\Sigma A$. We apply this to show that, localized at $p$, the $p$^th-power map on $G$ is an $H$-map. This is a significant step forward in Arkowitz-Curjel and McGibbon's programme for identifying which power maps between finite $H$-spaces are $H$-maps.