Expand all Collapse all | Results 1 - 10 of 10 |
1. CMB Online first
On $3$-manifolds with Torus- or Klein-bottle Category Two A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$
is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors
homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any
base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a
subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this
latter property are called $\mathcal{G}_K$-contractible. We obtain a
list of the closed $3$-manifolds that can be covered by two open
$\mathcal{G}_K$-contractible subsets. This is applied to obtain a list
of the possible closed prime $3$-manifolds that can be covered by two
open $K$-contractible subsets.
Keywords:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible sets Categories:57N10, 55M30, 57M27, 57N16 |
2. CMB 2012 (vol 57 pp. 240)
Addendum to ``Limit Sets of Typical Homeomorphisms'' Given an integer $n \geq 3$,
a metrizable compact topological $n$-manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$-almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
3. CMB 2011 (vol 55 pp. 523)
The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps 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.
Keywords:Gottlieb sets for maps, L-S category, T-spaces Categories:55P45, 55P35 |
4. CMB 2011 (vol 55 pp. 48)
Freyd's Generating Hypothesis for Groups with Periodic Cohomology Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$
divides
the order of $G$.
Freyd's generating hypothesis for the stable module category of
$G$ is the statement that a map between finite-dimensional
$kG$-modules in the thick subcategory generated by $k$ factors through a
projective if the induced map on Tate cohomology is trivial. We show that if
$G$
has periodic cohomology, then the generating hypothesis holds if and only if
the Sylow
$p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions
that are equivalent to the GH
for groups with periodic cohomology.
Keywords:Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology Categories:20C20, 20J06, 55P42 |
5. CMB 2011 (vol 55 pp. 225)
Limit Sets of Typical Homeomorphisms Given an integer $n \geq 3$, a metrizable compact
topological $n$-manifold $X$ with boundary, and a finite positive Borel
measure $\mu$ on $X$, we prove that for the typical homeomorphism
$f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
non-sensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
6. CMB 2011 (vol 55 pp. 188)
Yet Another Solution to the Burnside Problem for Matrix Semigroups We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.
Keywords:Burnside problem, kernel category Categories:20M30, 20M32 |
7. CMB 2010 (vol 54 pp. 12)
Homotopy and the Kestelman-Borwein-Ditor Theorem
The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by
translation in (measure/category) ``large'' sets has two generalizations.
Miller replaces the translated sequence by a ``sequence homotopic
to the identity''. The authors, in a previous paper, replace points by functions:
a uniform functional null sequence replaces the null sequence, and
translation receives a functional form. We give a unified approach to
results of this kind. In particular, we show that (i) Miller's homotopy
version follows from the functional version, and (ii) the pointwise instance
of the functional version follows from Miller's homotopy version.
Keywords:measure, category, measure-category duality, differentiable homotopy Category:26A03 |
8. CMB 2009 (vol 52 pp. 273)
Amalgamations of Categories We consider the pushout of embedding functors in $\Cat$, the
category of small categories.
We show that if the embedding functors satisfy a 3-for-2
property, then the induced functors to the pushout category are
also embeddings. The result follows from the connectedness of
certain associated slice categories. The condition is motivated
by a similar result for maps of semigroups. We show that our
theorem can be applied to groupoids and to inclusions of full
subcategories. We also give an example to show that the theorem
does not hold when the
property only holds for one of the inclusion functors, or when it
is weakened to a one-sided condition.
Keywords:category, pushout, amalgamation Categories:18A30, 18B40, 20L17 |
9. CMB 2004 (vol 47 pp. 321)
Classifying Spaces for Monoidal Categories Through Geometric Nerves 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.
Keywords:monoidal category, pseudo-simplicial category,, simplicial set, classifying space, homotopy type Categories:18D10, 18G30, 55P15, 55P35, 55U40 |
10. CMB 2001 (vol 44 pp. 459)
LS-catÃ©gorie algÃ©brique et attachement de cellules Nous montrons que la A-cat\'egorie d'un espace simplement connexe de
type fini est inf\'erieure ou \'egale \`a $n$ si et seulement si son
mod\`ele d'Adams-Hilton est un r\'etracte homotopique d'une alg\`ebre
diff\'erentielle \`a $n$ \'etages. Nous en d\'eduisons que
l'invariant $\Acat$ augmente au plus de 1 lors de l'attachement
d'une cellule \`a un espace.
We show that the A-category of a simply connected space of finite type
is less than or equal to $n$ if and only if its Adams-Hilton model is
a homotopy retract of an $n$-stage differential algebra. We deduce
from this that the invariant $\Acat$ increases by at most 1 when a
cell is attached to a space.
Keywords:LS-category, strong category, Adams-Hilton models, cell attachments Categories:55M30, 18G55 |