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:S 2n-1 → Ω2 S 2n+1 is an H-space.