location:  Publications → journals
Search results

Search: MSC category 55P62 ( Rational homotopy theory )

 Expand all        Collapse all Results 1 - 6 of 6

1. CJM 2012 (vol 65 pp. 82)

Félix, Yves; Halperin, Steve; Thomas, Jean-Claude
 The Ranks of the Homotopy Groups of a Finite Dimensional Complex 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 $r$. Keywords:homotopy groups, graded Lie algebra, exponential growth, LS categoryCategories:55P35, 55P62, , , , 17B70

2. CJM 2004 (vol 56 pp. 1290)

Scull, Laura
 Equivariant Formality for Actions of Torus Groups 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. Keywords:Equivariant homotopy, circle action, minimal model,, rationalization, formalityCategories:55P91, 55P62, 55R35, 55S45

3. CJM 2002 (vol 54 pp. 608)

Stanley, Donald
 On the Lusternik-Schnirelmann Category of Maps 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 $\cat(X) = \cat (X\times X) = \cl (X\times X) = 2$. Categories:55M30, 55P62

4. CJM 1999 (vol 51 pp. 49)

Ndombol, Bitjong; El haouari, M.
 AlgÃ¨bres quasi-commutatives et carrÃ©s de Steenrod 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 $H^*(X; \mathbb{Z}_2)$. Keywords:algÃ¨bres de cocha\^{\i}nes (fortement) quasi-commutatives, $T (V)$-modÃ¨le, carrÃ©s de Steenrod, quasi-isomorphismeCategories:55P62, 55S05

5. CJM 1998 (vol 50 pp. 845)

Scheerer, H.; Tanré, D.
 Lusternik-Schnirelmann category and algebraic $R$-local homotopy theory 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. Categories:55P50, 55P62

6. CJM 1997 (vol 49 pp. 855)

Smith, Samuel Bruce
 Rational Classification of simple function space components for flag manifolds. 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. Keywords:Rational homotopy theory, Sullivan-Haefliger model.Categories:55P62, 55P15, 58D99.