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 ${\rm cat}\, Z\le k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results on the $C_{k}$-space are generalized to the $C_{k}^{f}$-space for a map $f\colon A \to X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of $C_{k}$-spaces, and non-$C_{k}$-spaces.

The fiber $W_{n}$ of the double suspension $S^{2n-1}\rightarrow\Omega^{2} S^{2n+1}$ is known to have a classifying space $BW_{n}$. An important conjecture linking the $EHP$ sequence to the homotopy theory of Moore spaces is that $BW_{n}\simeq\Omega T^{2np+1}(p)$, where $T^{2np+1}(p)$ is Anick's space. This is known if $n=1$. We prove the $n=p$ case and establish some related properties.

The usual constructions of classifying spaces for monoidal categories produce CW-complexes with many cells that, moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.

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.

We consider graded connected Gorenstein algebras with respect to the evaluation map $\ev_G = \Ext_G(k,\varepsilon )=:: \Ext_G(k,G) \longrightarrow \Ext_G(k,k)$. We prove that if $\ev_G \neq 0$, then the global dimension of $G$ is finite.