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1. CMB 2013 (vol 57 pp. 526)

Heil, Wolfgang; Wang, Dongxu
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)

Bernardes, Nilson C.
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)

Iwase, Norio; Mimura, Mamoru; Oda, Nobuyuki; Yoon, Yeon Soo
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)

Chebolu, Sunil K.; Christensen, J. Daniel; Mináč, Ján
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)

Bernardes, Nilson C.
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)

Steinberg, Benjamin
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)

Bingham, N. H.; Ostaszewski, A. J.
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)

MacDonald, John; Scull, Laura
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)

Bullejos, M.; Cegarra, A. M.
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)

Kahl, Thomas
LS-catégorie algébrique 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

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