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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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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