26. CJM 1999 (vol 51 pp. 3)
 Allday, C.; Puppe, V.

On a Conjecture of Goresky, Kottwitz and MacPherson
We settle a conjecture of Goresky, Kottwitz and MacPherson related
to Koszul duality, \ie, to the correspondence between differential
graded modules over the exterior algebra and those over the
symmetric algebra.
Keywords:Koszul duality, HirschBrown model Categories:13D25, 18E30, 18G35, 55U15 

27. CJM 1999 (vol 51 pp. 49)
 Ndombol, Bitjong; El haouari, M.

AlgÃ¨bres quasicommutatives 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 quasicommutatives 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 quasicommutative; la
structure de module sur l'alg\`ebre de Steenrod d\'efinie sur
l'alg\`ebre de cohomologie de celleci co\"\i ncide avec celle de
$H^*(X; \mathbb{Z}_2)$.
We define the category of strongly quasicommutative 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 quasicommutative 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) quasicommutatives, $T (V)$modÃ¨le, carrÃ©s de Steenrod, quasiisomorphisme Categories:55P62, 55S05 

28. CJM 1998 (vol 50 pp. 1048)
 Goerss, P. G.; Jardine, J. F.

Localization theories for simplicial presheaves
Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$local theory of
MorelVoevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.
Categories:55P60, 19E08, 18F20 

29. CJM 1998 (vol 50 pp. 845)
 Scheerer, H.; Tanré, D.

LusternikSchnirelmann 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 LemaireSigrist and F\'elixHalperin.
Categories:55P50, 55P62 

30. CJM 1998 (vol 50 pp. 342)
 Giraldo, Antonio

Shape fibrations, multivalued maps and shape groups
The notion of shape fibration with the near lifting of near
multivalued paths property is studied. The relation of these
mapswhich agree with shape fibrations having totally disconnected
fiberswith Hurewicz fibrations with the unique path lifting
property is completely settled. Some results concerning homotopy and
shape groups are presented for shape fibrations with the near lifting
of near multivalued paths property. It is shown that for this class of
shape fibrations the existence of liftings of a fine multivalued map,
is equivalent to an algebraic problem relative to the homotopy, shape
or strong shape groups associated.
Keywords:Shape fibration, multivalued map, homotopy groups, shape, groups, strong shape groups Categories:54C56, 55P55, 55Q05, 55Q07, 55R05 

31. CJM 1997 (vol 49 pp. 1323)
 Sankaran, Parameswaran; Zvengrowski, Peter

Stable parallelizability of partially oriented flag manifolds II
In the first paper with the same title the authors
were able to determine all partially oriented flag
manifolds that are stably parallelizable or
parallelizable, apart from four infinite families
that were undecided. Here, using more delicate
techniques (mainly Ktheory), we settle these
previously undecided families and show that none of
the manifolds in them is stably parallelizable,
apart from one 30dimensional manifold which still
remains undecided.
Categories:57R25, 55N15, 53C30 

32. 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, SullivanHaefliger model. Categories:55P62, 55P15, 58D99. 
