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1. CJM Online first

Gonzalez, Jose Luis; Karu, Kalle
 Projectivity in Algebraic Cobordism The algebraic cobordism group of a scheme is generated by cycles that are proper morphisms from smooth quasiprojective varieties. We prove that over a field of characteristic zero the quasiprojectivity assumption can be omitted to get the same theory. Keywords:algebraic cobordism, quasiprojectivity, cobordism cyclesCategories:14C17, 14F43, 55N22

2. CJM 2013 (vol 65 pp. 843)

Jonsson, Jakob
 3-torsion in the Homology of Complexes of Graphs of Bounded Degree For $\delta \ge 1$ and $n \ge 1$, consider the simplicial complex of graphs on $n$ vertices in which each vertex has degree at most $\delta$; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When $\delta = 1$, we obtain the matching complex, for which it is known that there is $3$-torsion in degree $d$ of the homology whenever $\frac{n-4}{3} \le d \le \frac{n-6}{2}$. This paper establishes similar bounds for $\delta \ge 2$. Specifically, there is $3$-torsion in degree $d$ whenever $\frac{(3\delta-1)n-8}{6} \le d \le \frac{\delta (n-1) - 4}{2}$. The procedure for detecting torsion is to construct an explicit cycle $z$ that is easily seen to have the property that $3z$ is a boundary. Defining a homomorphism that sends $z$ to a non-boundary element in the chain complex of a certain matching complex, we obtain that $z$ itself is a non-boundary. In particular, the homology class of $z$ has order $3$. Keywords:simplicial complex, simplicial homology, torsion group, vertex degreeCategories:05E45, 55U10, 05C07, 20K10

3. CJM 2012 (vol 65 pp. 82)

Félix, Yves; Halperin, Steve; Thomas, Jean-Claude
 The Ranks of the Homotopy Groups of a Finite Dimensional Complex Let $X$ be an $n$-dimensional, finite, simply connected CW complex and set $\alpha_X =\limsup_i \frac{\log\mbox{ rank}\, \pi_i(X)}{i}$. When $0\lt \alpha_X\lt \infty$, we give upper and lower bound for $\sum_{i=k+2}^{k+n} \textrm{rank}\, \pi_i(X)$ for $k$ sufficiently large. We show also for any $r$ that $\alpha_X$ can be estimated from the integers rk$\,\pi_i(X)$, $i\leq nr$ with an error bound depending explicitly on $r$. Keywords:homotopy groups, graded Lie algebra, exponential growth, LS categoryCategories:55P35, 55P62, , , , 17B70

4. CJM 2012 (vol 65 pp. 553)

Godinho, Leonor; Sousa-Dias, M. E.
 Addendum and Erratum to "The Fundamental Group of $S^1$-manifolds" This paper provides an addendum and erratum to L. Godinho and M. E. Sousa-Dias, "The Fundamental Group of $S^1$-manifolds". Canad. J. Math. 62(2010), no. 5, 1082--1098. Keywords:symplectic reduction; fundamental groupCategories:53D19, 37J10, 55Q05

5. CJM 2012 (vol 65 pp. 544)

Deitmar, Anton; Horozov, Ivan
 Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants. Keywords:higher order forms, iterated integralsCategories:14F35, 11F12, 55D35, 58A10

6. CJM 2011 (vol 63 pp. 1345)

Jardine, J. F.
 Pointed Torsors This paper gives a characterization of homotopy fibres of inverse image maps on groupoids of torsors that are induced by geometric morphisms, in terms of both pointed torsors and pointed cocycles, suitably defined. Cocycle techniques are used to give a complete description of such fibres, when the underlying geometric morphism is the canonical stalk on the classifying topos of a profinite group $G$. If the torsors in question are defined with respect to a constant group $H$, then the path components of the fibre can be identified with the set of continuous maps from the profinite group $G$ to the group $H$. More generally, when $H$ is not constant, this set of path components is the set of continuous maps from a pro-object in sheaves of groupoids to $H$, which pro-object can be viewed as a Grothendieck fundamental groupoid". Keywords:pointed torsors, pointed cocycles, homotopy fibresCategories:18G50, 14F35, 55B30

7. CJM 2011 (vol 63 pp. 1038)

Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
 Critical Points and Resonance of Hyperplane Arrangements If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement $\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship between the critical set of $\Phi_\lambda$, the variety defined by the vanishing of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$. For arrangements satisfying certain conditions, we show that if $\lambda$ is resonant in dimension $p$, then the critical set of $\Phi_\lambda$ has codimension at most $p$. These include all free arrangements and all rank $3$ arrangements. Keywords:hyperplane arrangement, master function, resonant weights, critical setCategories:32S22, 55N25, 52C35

8. CJM 2010 (vol 62 pp. 1387)

Pamuk, Mehmetcik
 Homotopy Self-Equivalences of 4-manifolds with Free Fundamental Group We calculate the group of homotopy classes of homotopy self-equivalences of $4$-manifolds with free fundamental group and obtain a classification of such $4$-manifolds up to $s$-cobordism. Categories:57N13, 55P10, 57R80

9. CJM 2010 (vol 62 pp. 1082)

Godinho, Leonor; Sousa-Dias, M. E.
 The Fundamental Group of $S^1$-manifolds We address the problem of computing the fundamental group of a symplectic $S^1$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian $S^1$-action. Several examples are presented to illustrate our main results. Categories:53D20, 37J10, 55Q05

10. CJM 2009 (vol 62 pp. 614)

Pronk, Dorette; Scull, Laura
 Translation Groupoids and Orbifold Cohomology We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams. Keywords:orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractionsCategories:57S15, 55N91, 19L47, 18D05, 18D35

11. CJM 2009 (vol 62 pp. 473)

Yun, Zhiwei
 GoreskyâMacPherson Calculus for the Affine Flag Varieties We use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated by degree 2. We use this result to show that the vertices of the moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid. Categories:14L30, 55N91

12. CJM 2009 (vol 62 pp. 284)

Grbić, Jelena; Theriault, Stephen
 Self-Maps of Low Rank Lie Groups at Odd Primes Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given. Keywords:Lie group, self-map, H-mapCategories:55P45, 55Q05, 57T20

13. CJM 2008 (vol 60 pp. 348)

Santos, F. Guillén; Navarro, V.; Pascual, P.; Roig, Agust{\'\i}
 Monoidal Functors, Acyclic Models and Chain Operads We prove that for a topological operad $P$ the operad of oriented cubical singular chains, $C^{\ord}_\ast(P)$, and the operad of simplicial singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{\ord}_\ast(P\nsemi\mathbb{Q})$ is formal if and only if $S_\ast(P\nsemi\mathbb{Q})$ is formal, thus linking together some formality results which are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by $R$-simplicial differential graded algebras. Categories:18G80, 55N10, 18D50

14. CJM 2007 (vol 59 pp. 1154)

Boardman, J. Michael; Wilson, W. Stephen
 $k(n)$-Torsion-Free $H$-Spaces and $P(n)$-Cohomology The $H$-space that represents Brown--Peterson cohomology $\BP^k (-)$ was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum $P(n)$, by constructing idempotent operations in $P(n)$-cohomology $P(n)^k ($--$)$ in the style of Boardman--Johnson--Wilson; this relies heavily on the Ravenel--Wilson determination of the relevant Hopf ring. The resulting $(i- 1)$-connected $H$-spaces $Y_i$ have free connective Morava $\K$-homology $k(n)_* (Y_i)$, and may be built from the spaces in the $\Omega$-spectrum for $k(n)$ using only $v_n$-torsion invariants. We also extend Quillen's theorem on complex cobordism to show that for any space $X$, the \linebeak$P(n)_*$-module $P(n)^* (X)$ is generated by elements of $P(n)^i (X)$ for $i \ge 0$. This result is essential for the work of Ravenel--Wilson--Yagita, which in many cases allows one to compute $\BP$-cohomology from Morava $\K$-theory. Categories:55N22, 55P45

15. CJM 2007 (vol 59 pp. 1008)

Kaczynski, Tomasz; Mrozek, Marian; Trahan, Anik
 Ideas from Zariski Topology in the Study of Cubical Homology Cubical sets and their homology have been used in dynamical systems as well as in digital imaging. We take a fresh look at this topic, following Zariski ideas from algebraic geometry. The cubical topology is defined to be a topology in $\R^d$ in which a set is closed if and only if it is cubical. This concept is a convenient frame for describing a variety of important features of cubical sets. Separation axioms which, in general, are not satisfied here, characterize exactly those pairs of points which we want to distinguish. The noetherian property guarantees the correctness of the algorithms. Moreover, maps between cubical sets which are continuous and closed with respect to the cubical topology are precisely those for whom the homology map can be defined and computed without grid subdivisions. A combinatorial version of the Vietoris-Begle theorem is derived. This theorem plays the central role in an algorithm computing homology of maps which are continuous with respect to the Euclidean topology. Categories:55-04, 52B05, 54C60, 68W05, 68W30, 68U10

16. CJM 2006 (vol 58 pp. 877)

Selick, P.; Theriault, S.; Wu, J.
 Functorial Decompositions of Looped Coassociative Co-$H$ Spaces Selick and Wu gave a functorial decomposition of $\Omega\Sigma X$ for path-connected, $p$-local \linebreak$\CW$\nbd-com\-plexes $X$ which obtained the smallest nontrivial functorial retract $A^{\min}(X)$ of $\Omega\Sigma X$. This paper uses methods developed by the second author in order to extend such functorial decompositions to the loops on coassociative co-$H$ spaces. Keywords:homotopy decomposition, coassociative co-$H$ spacesCategory:55P53

17. CJM 2004 (vol 56 pp. 1290)

Scull, Laura
 Equivariant Formality for Actions of Torus Groups This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use $S^1$-equivariant minimal models to give a number of examples of $S^1$-spaces illustrating the properties of the various definitions. Keywords:Equivariant homotopy, circle action, minimal model,, rationalization, formalityCategories:55P91, 55P62, 55R35, 55S45

18. CJM 2003 (vol 55 pp. 181)

Theriault, Stephen D.
 Homotopy Decompositions Involving the Loops of Coassociative Co-$H$ Spaces James gave an integral homotopy decomposition of $\Sigma\Omega\Sigma X$, Hilton-Milnor one for $\Omega (\Sigma X\vee\Sigma Y)$, and Cohen-Wu gave $p$-local decompositions of $\Omega\Sigma X$ if $X$ is a suspension. All are natural. Using idempotents and telescopes we show that the James and Hilton-Milnor decompositions have analogues when the suspensions are replaced by coassociative co-$H$ spaces, and the Cohen-Wu decomposition has an analogue when the (double) suspension is replaced by a coassociative, cocommutative co-$H$ space. Categories:55P35, 55P45

19. CJM 2002 (vol 54 pp. 608)

Stanley, Donald
 On the Lusternik-Schnirelmann Category of Maps We give conditions which determine if $\cat$ of a map go up when extending over a cofibre. We apply this to reprove a result of Roitberg giving an example of a CW complex $Z$ such that $\cat(Z)=2$ but every skeleton of $Z$ is of category $1$. We also find conditions when $\cat (f\times g) < \cat(f) + \cat(g)$. We apply our result to show that under suitable conditions for rational maps $f$, $\mcat(f) < \cat(f)$ is equivalent to $\cat(f) = \cat (f\times \id_{S^n})$. Many examples with $\mcat(f) < \cat(f)$ satisfying our conditions are constructed. We also answer a question of Iwase by constructing $p$-local spaces $X$ such that $\cat (X\times S^1) = \cat(X) = 2$. In fact for our spaces and every $Y \not\simeq *$, $\cat (X\times Y) \leq \cat(Y) +1 < \cat(Y) + \cat(X)$. We show that this same $X$ has the property $\cat(X) = \cat (X\times X) = \cl (X\times X) = 2$. Categories:55M30, 55P62

20. CJM 2002 (vol 54 pp. 396)

Lebel, André
 Framed Stratified Sets in Morse Theory In this paper, we present a smooth framework for some aspects of the geometry of CW complexes'', in the sense of Buoncristiano, Rourke and Sanderson \cite{[BRS]}. We then apply these ideas to Morse theory, in order to generalize results of Franks \cite{[F]} and Iriye-Kono \cite{[IK]}. More precisely, consider a Morse function $f$ on a closed manifold $M$. We investigate the relations between the attaching maps in a CW complex determined by $f$, and the moduli spaces of gradient flow lines of $f$, with respect to some Riemannian metric on~$M$. Categories:57R70, 57N80, 55N45

21. CJM 2001 (vol 53 pp. 212)

Puppe, V.
 Group Actions and Codes A $\mathbb{Z}_2$-action with maximal number of isolated fixed points'' ({\it i.e.}, with only isolated fixed points such that $\dim_k (\oplus_i H^i(M;k)) =|M^{\mathbb{Z}_2}|, k = \mathbb{F}_2)$ on a $3$-dimensional, closed manifold determines a binary self-dual code of length $=|M^{\mathbb{Z}_2}|$. In turn this code determines the cohomology algebra $H^*(M;k)$ and the equivariant cohomology $H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary self-dual codes one gets information about the cohomology type of $3$-manifolds which admit involutions with maximal number of isolated fixed points. In particular, most'' cohomology types of closed $3$-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, {\it e.g.}, one gets that most'' cohomology types (over $\mathbb{F}_2)$ of closed $3$-manifolds do not admit a non-trivial involution. Keywords:Involutions, $3$-manifolds, codesCategories:55M35, 57M60, 94B05, 05E20

22. CJM 1999 (vol 51 pp. 897)

Astey, L.; Gitler, S.; Micha, E.; Pastor, G.
 Cohomology of Complex Projective Stiefel Manifolds The cohomology algebra mod $p$ of the complex projective Stiefel manifolds is determined for all primes $p$. When $p=2$ we also determine the action of the Steenrod algebra and apply this to the problem of existence of trivial subbundles of multiples of the canonical line bundle over a lens space with $2$-torsion, obtaining optimal results in many cases. Categories:55N10, 55S10

23. CJM 1999 (vol 51 pp. 3)

Allday, C.; Puppe, V.
 On a Conjecture of Goresky, Kottwitz and MacPherson We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, \ie, to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra. Keywords:Koszul duality, Hirsch-Brown modelCategories:13D25, 18E30, 18G35, 55U15

24. CJM 1999 (vol 51 pp. 49)

Ndombol, Bitjong; El haouari, M.
 AlgÃ¨bres quasi-commutatives et carrÃ©s de Steenrod Soit $k$ un corps de caract\'eristique $p$ quelconque. Nous d\'efinissons la cat\'egorie des $k$-alg\ebres de cocha\^{\i}nes fortement quasi-commutatives et nous donnons une condition n\'ecessaire et suffisante pour que l'alg\ebre de cohomologie \a coefficients dans $\mathbb{Z}_2$ d'un objet de cette cat\'egorie soit un module instable sur l'alg\ebre de Steenrod \a coefficients dans $\mathbb{Z}_2$. A tout c.w.\ complexe simplement connexe de type fini $X$ on associe une $k$-alg\ebre de cocha\^{\i}nes fortement quasi-commutative; la structure de module sur l'alg\ebre de Steenrod d\'efinie sur l'alg\ebre de cohomologie de celle-ci co\"\i ncide avec celle de $H^*(X; \mathbb{Z}_2)$. We define the category of strongly quasi-commutative cochain $k$-algebras, where $k$ is a field of any characteristic $p$. We give a necessary and sufficient condition which enables the cohomology algebra with $\mathbb{Z}_2$-coefficients of an object in this category to be an unstable module on the $\mathbb{Z}_2$-Steenrod algebra. To each simply connected c.w.\ complex of finite type $X$ is associated a strongly quasi-commutative model and the module structure over the $\mathbb{Z}_2$-Steenrod algebra defined on the cohomology of this model is the usual structure on $H^*(X; \mathbb{Z}_2)$. Keywords:algÃ¨bres de cocha\^{\i}nes (fortement) quasi-commutatives, $T (V)$-modÃ¨le, carrÃ©s de Steenrod, quasi-isomorphismeCategories:55P62, 55S05

25. CJM 1998 (vol 50 pp. 1048)

Goerss, P. G.; Jardine, J. F.
 Localization theories for simplicial presheaves Most extant localization theories for spaces, spectra and diagrams of such can be derived from a simple list of axioms which are verified in broad generality. Several new theories are introduced, including localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra, and a theory of localization along a geometric topos morphism. The $f$-localization concept has an analog for simplicial presheaves, and specializes to the $\hbox{\Bbbvii A}^1$-local theory of Morel-Voevodsky. This theory answers a question of Soul\'e concerning integral homology localizations for diagrams of spaces. Categories:55P60, 19E08, 18F20
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