Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When $0\lt \alpha_X\lt \infty$, we give upper and lower bound for $ \sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X) $ for $k$ sufficiently large. We show also for any $r$ that $\alpha_X$ can be estimated from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound depending explicitly on $r$.

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use $S^1$-equivariant minimal models to give a number of examples of $S^1$-spaces illustrating the properties of the various definitions.

We give conditions which determine if $\cat$ of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a CW complex $Z$ such that $\cat(Z)=2$ but every skeleton of $Z$ is of category $1$. We also find conditions when $\cat (f\times g) < \cat(f) + \cat(g)$. We apply our result to show that under suitable conditions for rational maps $f$, $\mcat(f) < \cat(f)$ is equivalent to $\cat(f) = \cat (f\times \id_{S^n})$. Many examples with $\mcat(f) < \cat(f)$ satisfying our conditions are constructed. We also answer a question of Iwase by constructing $p$-local spaces $X$ such that $\cat (X\times S^1) = \cat(X) = 2$. In fact for our spaces and every $Y \not\simeq *$, $\cat (X\times Y) \leq \cat(Y) +1 < \cat(Y) + \cat(X)$. We show that this same $X$ has the property $\cat(X) = \cat (X\times X) = \cl (X\times X) = 2$.

Soit $k$ un corps de caract\'eristique $p$ quelconque. Nous d\'efinissons la cat\'egorie des $k$-alg\`ebres de cocha\^{\i}nes fortement quasi-commutatives et nous donnons une condition n\'ecessaire et suffisante pour que l'alg\`ebre de cohomologie \`a coefficients dans $\mathbb{Z}_2$ d'un objet de cette cat\'egorie soit un module instable sur l'alg\`ebre de Steenrod \`a coefficients dans $\mathbb{Z}_2$. A tout c.w.\ complexe simplement connexe de type fini $X$ on associe une $k$-alg\`ebre de cocha\^{\i}nes fortement quasi-commutative; la structure de module sur l'alg\`ebre de Steenrod d\'efinie sur l'alg\`ebre de cohomologie de celle-ci co\"\i ncide avec celle de $H^*(X; \mathbb{Z}_2)$. We define the category of strongly quasi-commutative cochain $k$-algebras, where $k$ is a field of any characteristic $p$. We give a necessary and sufficient condition which enables the cohomology algebra with $\mathbb{Z}_2$-coefficients of an object in this category to be an unstable module on the $\mathbb{Z}_2$-Steenrod algebra. To each simply connected c.w.\ complex of finite type $X$ is associated a strongly quasi-commutative model and the module structure over the $\mathbb{Z}_2$-Steenrod algebra defined on the cohomology of this model is the usual structure on $H^*(X; \mathbb{Z}_2)$.

In this paper, we define the notion of $R_{\ast}$-$\LS$ category associated to an increasing system of subrings of $\Q$ and we relate it to the usual $\LS$-category. We also relate it to the invariant introduced by F\'elix and Lemaire in tame homotopy theory, in which case we give a description in terms of Lie algebras and of cocommutative coalgebras, extending results of Lemaire-Sigrist and F\'elix-Halperin.

Let $M(X,Y)$ denote the space of all continous functions between $X$ and $Y$ and $M_f(X,Y)$ the path component corresponding to a given map $f: X\rightarrow Y.$ When $X$ and $Y$ are classical flag manifolds, we prove the components of $M(X,Y)$ corresponding to ``simple'' maps $f$ are classified up to rational homotopy type by the dimension of the kernel of $f$ in degree two cohomology. In fact, these components are themselves all products of flag manifolds and odd spheres.