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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 2013 (vol 57 pp. 225)

Adamaszek, Michał
Small Flag Complexes with Torsion
We classify flag complexes on at most $12$ vertices with torsion in the first homology group. The result is moderately computer-aided. As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly $13$ elements.

Keywords:clique complex, order complex, homology, torsion, minimal model
Categories:55U10, 06A11, 55P40, 55-04, 05-04

3. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
Assouad-Nagata Dimension of Wreath Products of Groups
Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.

Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groups
Categories:54F45, 55M10, 54C65

4. 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

5. CMB 2011 (vol 55 pp. 319)

Jardine, J. F.
The Verdier Hypercovering Theorem
This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant.

Keywords:simplicial presheaf, hypercover, cocycle
Categories:14F35, 18G30, 55U35

6. 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

7. CMB 2010 (vol 53 pp. 730)

Theriault, Stephen D.
A Case When the Fiber of the Double Suspension is the Double Loops on Anick's Space
The fiber $W_{n}$ of the double suspension $S^{2n-1}\rightarrow\Omega^{2} S^{2n+1}$ is known to have a classifying space $BW_{n}$. An important conjecture linking the $EHP$ sequence to the homotopy theory of Moore spaces is that $BW_{n}\simeq\Omega T^{2np+1}(p)$, where $T^{2np+1}(p)$ is Anick's space. This is known if $n=1$. We prove the $n=p$ case and establish some related properties.

Keywords:double suspension, Anick's space
Categories:55P35, 55P10

8. CMB 2010 (vol 53 pp. 438)

Chigogidze, A.; Nagórko, A.
Near-Homeomorphisms of Nöbeling Manifolds
We characterize maps between $n$-dimensional Nöbeling manifolds that can be approximated by homeomorphisms.

Keywords:n-dimensional Nöbeling manifold, Z-set unknotting, near-homeomorphism
Categories:55M10, 54F45

9. CMB 2008 (vol 51 pp. 535)

Csorba, Péter
On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality
We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal.

Keywords:graph complexes, simple $\Z_2$-homotopy, universality
Categories:57Q10, 05C10, 55P10

10. CMB 2008 (vol 51 pp. 310)

Witbooi, P. J.
Relative Homotopy in Relational Structures
The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs $(X,A)$ where $X$ is a poset and $A$ is a subposet of $X$. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif.

Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalence
Categories:55Q05, 54A05;, 18B30

11. CMB 2007 (vol 50 pp. 440)

Raghuram, A.
A Künneth Theorem for $p$-Adic Groups
Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$.

Categories:22E50, 18G15, 55U25

12. CMB 2007 (vol 50 pp. 365)

Godinho, Leonor
Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds
Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

Categories:53D20, 55N91, 57S15

13. CMB 2007 (vol 50 pp. 206)

Golasiński, Marek; Gonçalves, Daciberg Lima
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$
Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well.

Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space form
Categories:55M35, 55P15, 20E22, 20F28, 57S17

14. CMB 2006 (vol 49 pp. 407)

Jardine, J. F.
Intermediate Model Structures for Simplicial Presheaves
This note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense.

Categories:18G30, 18F20, 55U35

15. CMB 2006 (vol 49 pp. 41)

Doeraene, Jean-Paul; El Haouari, Mohammed
The Ganea and Whitehead Variants of the\\Lusternik--Schnirelmann Category
The Lusternik--Schnirelmann category has been described in different ways. Two major ones, the first by Ganea, the second by Whitehead, are presented here with a number of variants. The equivalence of these variants relies on the axioms of Quillen's model category, but also sometimes on an additional axiom, the so-called ``cube axiom''.

Category:55P30

16. CMB 2005 (vol 48 pp. 614)

Tuncali, H. Murat; Valov, Vesko
On Finite-to-One Maps
Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that for every positive integer $p$ with $ p\leq m+k+1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that each fiber of $f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most $\max\{k+m-p+2,1\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f\colon X\to Y$ be a $k$-dimensional map between compact metric spaces with $\dim Y\leq m$. Then: \begin{inparaenum}[\rm(1)] \item there exists a map $h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to Y\times\uin^{m+2k}$ is 2-to-one provided $k\geq 1$; \item there exists a map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to Y\times\uin^{m+k+1}$ is $(k+1)$-to-one. \end{inparaenum}

Keywords:finite-to-one maps, dimension, set-valued maps
Categories:54F45, 55M10, 54C65

17. 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

18. CMB 2004 (vol 47 pp. 246)

Makai, Endre; Martini, Horst
On Maximal $k$-Sections and Related Common Transversals of Convex Bodies
Generalizing results from [MM1] referring to the intersection body $IK$ and the cross-section body $CK$ of a convex body $K \subset \sR^d, \, d \ge 2$, we prove theorems about maximal $k$-sections of convex bodies, $k \in \{1, \dots, d-1\}$, and, simultaneously, statements about common maximal $(d-1)$- and $1$-transversals of families of convex bodies.

Categories:52A20, 55Mxx

19. CMB 2004 (vol 47 pp. 119)

Theriault, Stephen D.
$2$-Primary Exponent Bounds for Lie Groups of Low Rank
Exponent information is proven about the Lie groups $SU(3)$, $SU(4)$, $Sp(2)$, and $G_2$ by showing some power of the $H$-space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are $8$, $32$, $64$, and $2^8$ respectively. This null homotopy is best possible for $SU(3)$ given the number of loops, off by at most one power of~$2$ for $SU(4)$ and $Sp(2)$, and off by at most two powers of $2$ for $G_2$.

Keywords:Lie group, exponent
Category:55Q52

20. 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

21. CMB 2001 (vol 44 pp. 266)

Cencelj, M.; Dranishnikov, A. N.
Extension of Maps to Nilpotent Spaces
We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$.

Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space
Categories:55M10, 55S36, 54C20, 54F45

22. CMB 2001 (vol 44 pp. 80)

Levin, Michael
Constructing Compacta of Different Extensional Dimensions
Applying the Sullivan conjecture we construct compacta of certain cohomological and extensional dimensions.

Keywords:cohomological dimension, Eilenberg-MacLane complexes, Sullivan conjecture
Categories:55M10, 54F45, 55U20

23. CMB 2000 (vol 43 pp. 343)

Hughes, Bruce; Taylor, Larry; Williams, Bruce
Controlled Homeomorphisms Over Nonpositively Curved Manifolds
We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.

Keywords:controlled topology, controlled homeomorphism, nonpositive curvature, Novikov conjectures
Categories:57N15, 53C20, 55R65, 57N37

24. CMB 2000 (vol 43 pp. 226)

Neisendorfer, Joseph
James-Hopf Invariants, Anick's Spaces, and the Double Loops on Odd Primary Moore Spaces
Using spaces introduced by Anick, we construct a decomposition into indecomposable factors of the double loop spaces of odd primary Moore spaces when the powers of the primes are greater than the first power. If $n$ is greater than $1$, this implies that the odd primary part of all the homotopy groups of the $2n+1$ dimensional sphere lifts to a $\mod p^r$ Moore space.

Categories:55Q52, 55P35

25. CMB 2000 (vol 43 pp. 37)

Bousaidi, M. A.
Multiplicative Structure of the Ring $K \bigl( S(T^*\R P^{2n+1}) \bigr)$
We calculate the additive and multiplicative structure of the ring $K\bigl(S(T^*\R P^{2n+1})\bigr)$ using the eta invariant.

Categories:19L64, 19K56, 55C35
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