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1. CJM 2012 (vol 65 pp. 82)
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 category Categories:55P35, 55P62, , , , 17B70 |
2. CJM 2011 (vol 63 pp. 1345)
Pointed Torsors This paper gives a characterization of homotopy fibres of inverse
image maps on groupoids of torsors that are induced by geometric
morphisms, in terms of both pointed torsors and pointed cocycles,
suitably defined. Cocycle techniques are used to give a complete
description of such fibres, when the underlying geometric morphism is
the canonical stalk on the classifying topos of a profinite group
$G$. If the torsors in question are defined with respect to a constant
group $H$, then the path components of the fibre can be identified with
the set of continuous maps from the profinite group $G$ to the group
$H$. More generally, when $H$ is not constant, this set of path components
is the set of continuous maps from a pro-object in sheaves of
groupoids to $H$, which pro-object can be viewed as a ``Grothendieck
fundamental groupoid".
Keywords:pointed torsors, pointed cocycles, homotopy fibres Categories:18G50, 14F35, 55B30 |
3. CJM 2011 (vol 63 pp. 1388)
Nonabelian $H^1$ and the Ãtale Van Kampen Theorem
Generalized Ã©tale homotopy pro-groups $\pi_1^{\operatorname{Ã©t}}(Ä{C}, x)$
associated with pointed, connected, small Grothendieck
sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois
theory and the theory of pointed torsors for discrete
groups is explained.
Applications include new rigorous proofs of some folklore results around $\pi_1^{\operatorname{Ã©t}}(Ã©t(X), x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new Ã©tale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows. Keywords:Ã©tale homotopy theory, simplicial sheaves Categories:18G30, 14F35 |
4. CJM 2009 (vol 62 pp. 614)
Translation Groupoids and Orbifold Cohomology We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.
Keywords:orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions Categories:57S15, 55N91, 19L47, 18D05, 18D35 |
5. CJM 2006 (vol 58 pp. 877)
Functorial Decompositions of Looped Coassociative Co-$H$ Spaces Selick and Wu gave a functorial decomposition of
$\Omega\Sigma X$ for path-connected, $p$-local \linebreak$\CW$\nbd-com\-plexes $X$
which obtained the smallest nontrivial functorial retract $A^{\min}(X)$
of $\Omega\Sigma X$. This paper uses methods developed by
the second author in order to extend such functorial
decompositions to the loops on coassociative co-$H$ spaces.
Keywords:homotopy decomposition, coassociative co-$H$ spaces Category:55P53 |
6. CJM 2004 (vol 56 pp. 1290)
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, formality Categories:55P91, 55P62, 55R35, 55S45 |
7. CJM 1998 (vol 50 pp. 342)
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
maps---which agree with shape fibrations having totally disconnected
fibers---with 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 |
8. CJM 1997 (vol 49 pp. 855)
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. |