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1. CJM Online first

Mihara, Tomoki
Cohomological Approach to Class Field Theory in Arithmetic Topology
We establish class field theory for $3$-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

Keywords:arithmetic topology, class field theory, branched covering, knots and prime numbers
Categories:11Z05, 18F15, 55N20, 57P05

2. CJM Online first

Kuribayashi, Katsuhiko; Menichi, Luc
The BV algebra in String Topology of classifying spaces
For almost any compact connected Lie group $G$ and any field $\mathbb{F}_p$, we compute the Batalin-Vilkovisky algebra $H^{*+\operatorname{dim }G}(LBG;\mathbb{F}_p)$ on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra is isomorphic to the Hochschild cohomology $HH^*(H_*(G),H_*(G))$. Over $\mathbb{F}_2$, such isomorphism of Batalin-Vilkovisky algebras does not hold when $G=SO(3)$ or $G=G_2$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

Keywords:string topology, Batalin-Vilkovisky algebra, classifying space
Categories:55P50, 55R35, 81T40

3. CJM 2017 (vol 70 pp. 265)

Cordova Bulens, Hector; Lambrechts, Pascal; Stanley, Don
Rational models of the complement of a subpolyhedron in a manifold with boundary
Let $W$ be a compact simply connected triangulated manifold with boundary and $K\subset W$ be a subpolyhedron. We construct an algebraic model of the rational homotopy type of $W\backslash K$ out of a model of the map of pairs $(K,K \cap \partial W)\hookrightarrow (W,\partial W)$ under some high codimension hypothesis. We deduce the rational homotopy invariance of the configuration space of two points in a compact manifold with boundary under 2-connectedness hypotheses. Also, we exhibit nice explicit models of these configuration spaces for a large class of compact manifolds.

Keywords:Lefschetz duality, Sullivan model, configuration space
Categories:55P62, 55R80

4. CJM 2017 (vol 70 pp. 191)

Kitchloo, Nitu; Lorman, Vitaly; Wilson, W. Stephen
The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C} \mathbb{P}^n$
The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}\mathbb{P}^n$ are computed along with the Atiyah-Hirzebruch spectral sequence for $ER(2)^*(\mathbb{C}\mathbb{P}^\infty)$. This, along with other papers in this series, gives us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces.

Keywords:complex projective space, cohomology theory, Eilenberg-MacLane space, Atiyah-Hirzebruch spectral sequence
Categories:55N20, 55N91, 55P20, 55T25

5. CJM 2015 (vol 68 pp. 463)

Sadykov, Rustam
The Weak b-principle: Mumford Conjecture
In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension $2$ approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension $2$ to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.

Keywords:generalized cohomology theories, fold singularities, h-principle, infinite loop spaces
Categories:55N20, 53C23

6. CJM 2014 (vol 67 pp. 1024)

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
Representation Stability of Power Sets and Square Free Polynomials
The symmetric group $\mathcal{S}_n$ acts on the power set $\mathcal{P}(n)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra
Categories:20C30, 13A50, 20F36, 55R80

7. CJM 2014 (vol 67 pp. 639)

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 cycles
Categories:14C17, 14F43, 55N22

8. 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 degree
Categories:05E45, 55U10, 05C07, 20K10

9. 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 category
Categories:55P35, 55P62, , , , 17B70

10. 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 group
Categories:53D19, 37J10, 55Q05

11. 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 integrals
Categories:14F35, 11F12, 55D35, 58A10

12. 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 fibres
Categories:18G50, 14F35, 55B30

13. 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 set
Categories:32S22, 55N25, 52C35

14. 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

15. 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

16. 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-map
Categories:55P45, 55Q05, 57T20

17. 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 fractions
Categories:57S15, 55N91, 19L47, 18D05, 18D35

18. CJM 2009 (vol 62 pp. 473)

Yun, Zhiwei
Goresky—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

19. 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

20. 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

21. 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

22. 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-$H$ spaces

23. 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, formality
Categories:55P91, 55P62, 55R35, 55S45

24. 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

25. 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
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