We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

The method for computing the $p$-localization of the group $\left[ X,\,\text{U}\left( n \right) \right]$, by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is given and the homotopy nilpotency of $\text{Sp}\left( 3 \right)$ localized at $p\,\ge \,5$ is determined.

The concept of ${{C}_{k}}$-spaces is introduced, situated at an intermediate stage between $H$-spaces and $T$-spaces. The ${{C}_{k}}$-space corresponds to the $k$-th Milnor–Stasheff filtration on spaces. It is proved that a space $X$ is a ${{C}_{k}}$-space if and only if the Gottlieb set $G(Z,\,X)\,=\,[Z,\,X]$ for any space $Z$ with cat $Z\,\le \,k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\sum B,\,X)\,=\,[\sum B,\,X]$ for any space $B$. Some results on the ${{C}_{k}}$-space are generalized to the $C_{k}^{f}$-space for a map $f\,:\,A\,\to \,X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of ${{C}_{k}}$-spaces, and non-${{C}_{k}}$-spaces.

Let $G$ be a simple, compact, simply-connected Lie group localized at an odd prime $p$. We study the group of homotopy classes of self-maps $\left[ G,\,G \right]$ when the rank of $G$ is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H\left[ G,\,G \right]$. The low rank condition gives $G$ certain structural properties which make calculations accessible. Several examples and applications are given.

The $H$-space that represents Brown-Peterson cohomology $\text{B}{{\text{P}}^{k}}\left( - \right)$ was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum $P\left( n \right)$, by constructing idempotent operations in $P\left( n \right)$-cohomology $P{{(n)}^{k}}\left( - \right)$ in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting $\left( i\,-\,1 \right)$-connected $H$-spaces ${{Y}_{i}}$ have free connective Morava $K$-homology $k{{(n)}_{*}}({{Y}_{i}})$, and may be built from the spaces in the $\Omega$-spectrum for $k\left( n \right)$ using only ${{v}_{n}}$-torsion invariants.

We also extend Quillen's theorem on complex cobordism to show that for any space $X$, the $P{{\left( n \right)}_{*}}$-module $P{{(n)}^{*}}\,(X)$ is generated by elements of $P{{(n)}^{i}}(X)$ for $i\,\ge \,0$. This result is essential for the work of Ravenel-Wilson-Yagita, which in many cases allows one to compute BP-cohomology from Morava $K$-theory.

James gave an integral homotopy decomposition of $\sum \Omega \sum X$, Hilton-Milnor one for $\Omega (\sum X\,\vee \,\sum Y)$, and Cohen-Wu gave $p$-local decompositions of $\Omega \sum X$ if $X$ is a suspension. All are natural. Using idempotents and telescopes we show that the James and Hilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative $\text{co-}H$ spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative $\text{co-}H$ space.

It is shown that all the generalized Whitehead products vanish in X and all the components of XΣA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.

We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

In this paper we show that the homotopy-theoretic fibre of the double suspension map E2:S2n-1 → Ω2S2n+1 is an H-space.