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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. CMB 1999 (vol 42 pp. 248)

Weber, Christian
The Classification of $\Pin_4$-Bundles over a $4$-Complex
In this paper we show that the Lie-group $\Pin_4$ is isomorphic to the semidirect product $(\SU_2\times \SU_2)\timesv \Z/2$ where $\Z/2$ operates by flipping the factors. Using this structure theorem we prove a classification theorem for $\Pin_4$-bundles over a finite $4$-complex $X$.

Categories:55N25, 55R10, 57S15

38. CMB 1999 (vol 42 pp. 129)

Baker, Andrew
Hecke Operations and the Adams $E_2$-Term Based on Elliptic Cohomology
Hecke operators are used to investigate part of the $\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\Ext^1$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.

Keywords:Adams spectral sequence, elliptic cohomology, Hecke operators
Categories:55N20, 55N22, 55T15, 11F11, 11F25

39. CMB 1999 (vol 42 pp. 52)

Edmonds, Allan L.
Embedding Coverings in Bundles
If $V\to X$ is a vector bundle of fiber dimension $k$ and $Y\to X$ is a finite sheeted covering map of degree $d$, the implications for the Euler class $e(V)$ in $H^k(X)$ of $V$ implied by the existence of an embedding $Y\to V$ lifting the covering map are explored. In particular it is proved that $dd'e(V)=0$ where $d'$ is a certain divisor of $d-1$, and often $d'=1$.

Categories:57M10, 55R25, 55S40, 57N35

40. CMB 1998 (vol 41 pp. 28)

Félix, Yves; Murillo, Aniceto
Gorenstein graded algebras and the evaluation map
We consider graded connected Gorenstein algebras with respect to the evaluation map $\ev_G = \Ext_G(k,\varepsilon )=:: \Ext_G(k,G) \longrightarrow \Ext_G(k,k)$. We prove that if $\ev_G \neq 0$, then the global dimension of $G$ is finite.

Categories:55P35, 13C11

41. CMB 1998 (vol 41 pp. 20)

Brunetti, Maurizio
A new cohomological criterion for the $p$-nilpotence of groups
Let $G$ be a finite group, $H$ a copy of its $p$-Sylow subgroup, and $\kn$ the $n$-th Morava $K$-theory at $p$. In this paper we prove that the existence of an isomorphism between $K(n)^\ast(BG)$ and $K(n)^\ast(BH)$ is a sufficient condition for $G$ to be $p$-nilpotent.

Categories:55N20, 55N22
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