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26. 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

27. 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, exponentCategory:55Q52

28. CMB 2001 (vol 44 pp. 459)

Kahl, Thomas
 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 attachmentsCategories:55M30, 18G55

29. 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 spaceCategories:55M10, 55S36, 54C20, 54F45

30. 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 conjectureCategories:55M10, 54F45, 55U20

31. 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 conjecturesCategories:57N15, 53C20, 55R65, 57N37

32. 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

33. 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

34. 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 operatorsCategories:55N20, 55N22, 55T15, 11F11, 11F25

35. 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

36. 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

37. 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

38. 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

39. CMB 1997 (vol 40 pp. 341)

Lee, Hyang-Sook
 The stable and unstable types of classifying spaces The main purpose of this paper is to study groups $G_1$, $G_2$ such that $H^\ast(BG_1,{\bf Z}/p)$ is isomorphic to $H^\ast(BG_2,{\bf Z}/p)$ in ${\cal U}$, the category of unstable modules over the Steenrod algebra ${\cal A}$, but not isomorphic as graded algebras over ${\bf Z}/p$. Categories:55R35, 20J06

40. CMB 1997 (vol 40 pp. 193)

Kucerovsky, Dan
 Finite rank operators and functional calculus on Hilbert modules over abelian $C^{\ast}$-algebras We consider the problem: If $K$ is a compact normal operator on a Hilbert module $E$, and $f\in C_0(\Sp K)$ is a function which is zero in a neighbourhood of the origin, is $f(K)$ of finite rank? We show that this is the case if the underlying $C^{\ast}$-algebra is abelian, and that the range of $f(K)$ is contained in a finitely generated projective submodule of $E$. Categories:55R50, 47A60, 47B38

41. CMB 1997 (vol 40 pp. 108)

Schaer, J.
 Continuous Self-maps of the Circle Given a continuous map $\delta$ from the circle $S$ to itself we want to find all self-maps $\sigma\colon S\to S$ for which $\delta\circ\sigma = \delta$. If the degree $r$ of $\delta$ is not zero, the transformations $\sigma$ form a subgroup of the cyclic group $C_r$. If $r=0$, all such invertible transformations form a group isomorphic either to a cyclic group $C_n$ or to a dihedral group $D_n$ depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: {\it Let $\Delta\colon\bbd R\to\bbd R$ be continuous, nowhere constant, and $\lim_{x\to -\infty}\Delta(x)=-\infty$, $\lim_{x\to+\infty}\Delta (x)=+\infty$; then the only continuous map $\Sigma\colon\bbd R\to\bbd R$ such that $\Delta\circ\Sigma=\Delta$ is the identity $\Sigma=\id_{\bbd R}$. Categories:53A04, 55M25, 55M35
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