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1. CMB 2017 (vol 61 pp. 272)

Franz, Matthias
 Symmetric Products of Equivariantly Formal Spaces Let $$X$$ be a CW complex with a continuous action of a topological group $$G$$. We show that if $$X$$ is equivariantly formal for singular cohomology with coefficients in some field $$\Bbbk$$, then so are all symmetric products of $$X$$ and in fact all its $$\Gamma$$-products. In particular, symmetric products of quasi-projective M-varieties are again M-varieties. This generalizes a result by Biswas and D'Mello about symmetric products of M-curves. We also discuss several related questions. Keywords:symmetric product, equivariant formality, maximal variety, Gamma productCategories:55N91, 55S15, 14P25

2. CMB 2017 (vol 60 pp. 478)

Carrell, Jim; Kaveh, Kiumars
 Springer's Weyl Group Representation via Localization Let $G$ denote a reductive algebraic group over $\mathbb{C}$ and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer variety $\mathcal{B}_x$ is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing the Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable property that the Weyl group $W$ of $G$ admits a representation on the cohomology of $\mathcal{B}_x$ even though $W$ rarely acts on $\mathcal{B}_x$ itself. Well-known constructions of this action due to Springer et al use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when $x$ is what we call parabolic-surjective. The idea is to use localization to construct an action of $W$ on the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic subtorus of $G$. This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type $A$ and, more generally, all nilpotents for which it is known that $W$ acts on $H_S^*(\mathcal{B}_x)$ for some torus $S$. Our result is deduced from a general theorem describing when a group action on the cohomology of the fixed point set of a torus action on a space lifts to the full cohomology algebra of the space. Keywords:Springer variety, Weyl group action, equivariant cohomologyCategories:14M15, 14F43, 55N91

3. CMB 2017 (vol 60 pp. 235)

Basu, Samik; Subhash, B
 Topology of Certain Quotient Spaces of Stiefel Manifolds We compute the cohomology of the right generalised projective Stiefel manifolds. Following this, we discuss some easy applications of the computations to the ranks of complementary bundles, and bounds on the span and immersibility. Keywords:projective Stiefel manifold, span, spectral sequenceCategories:55R20, 55R25, 57R20

4. CMB 2017 (vol 61 pp. 124)

Hemasundar, Gollakota V. V.; Simha, R. R.
 The Jordan Curve Theorem via Complex Analysis The aim of this article is to give a proof of the Jordan Curve Theorem via complex analysis. Keywords:Jordan Curve TheoremCategories:55P15, 30G12

5. CMB 2016 (vol 59 pp. 682)

Carlson, Jon F.; Chebolu, Sunil K.; Mináč, Ján
 Ghosts and Strong Ghosts in the Stable Category Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\gt 0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd's generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps which mimic ghosts in high degrees. Keywords:Tate cohomology, ghost maps, stable module category, almost split sequence, periodic cohomologyCategories:20C20, 20J06, 55P42

6. CMB 2016 (vol 59 pp. 483)

Crooks, Peter; Holden, Tyler
 Generalized Equivariant Cohomology and Stratifications For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group. Keywords:equivariant cohomology theory, stratification, affine GrassmannianCategories:55N91, 19L47

7. CMB 2015 (vol 58 pp. 575)

Martinez-Torres, David
 The Diffeomorphism Type of Canonical Integrations Of Poisson Tensors on Surfaces A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit canonical integration, $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic to the cotangent bundle $T^*\Sigma$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini. Keywords:Poisson tensor, Lie groupoid, cotangent bundleCategories:58H05, 55R10, 53D17

8. CMB 2014 (vol 58 pp. 80)

Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya
 The Equivariant Cohomology Rings of Peterson Varieties in All Lie Types Let $G$ be a complex semisimple linear algebraic group and let $Pet$ be the associated Peterson variety in the flag variety $G/B$. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring $H^*_S(Pet)$ of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal torus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type $A$. Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formulaCategories:55N91, 14N15

9. CMB 2013 (vol 57 pp. 526)

Heil, Wolfgang; Wang, Dongxu
 On $3$-manifolds with Torus or Klein Bottle Category Two A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this latter property are called $\mathcal{G}_K$-contractible. We obtain a list of the closed $3$-manifolds that can be covered by two open $\mathcal{G}_K$-contractible subsets. This is applied to obtain a list of the possible closed prime $3$-manifolds that can be covered by two open $K$-contractible subsets. Keywords:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible setsCategories:57N10, 55M30, 57M27, 57N16

10. CMB 2013 (vol 57 pp. 225)

 Small Flag Complexes with Torsion We classify flag complexes on at most $12$ vertices with torsion in the first homology group. The result is moderately computer-aided. As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly $13$ elements. Keywords:clique complex, order complex, homology, torsion, minimal modelCategories:55U10, 06A11, 55P40, 55-04, 05-04

11. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
 Assouad-Nagata Dimension of Wreath Products of Groups Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$. Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groupsCategories:54F45, 55M10, 54C65

12. CMB 2011 (vol 55 pp. 523)

Iwase, Norio; Mimura, Mamoru; Oda, Nobuyuki; Yoon, Yeon Soo
 The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps The concept of $C_{k}$-spaces is introduced, situated at an intermediate stage between $H$-spaces and $T$-spaces. The $C_{k}$-space corresponds to the $k$-th Milnor-Stasheff filtration on spaces. It is proved that a space $X$ is a $C_{k}$-space if and only if the Gottlieb set $G(Z,X)=[Z,X]$ for any space $Z$ with ${\rm cat}\, Z\le k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results on the $C_{k}$-space are generalized to the $C_{k}^{f}$-space for a map $f\colon A \to X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of $C_{k}$-spaces, and non-$C_{k}$-spaces. Keywords:Gottlieb sets for maps, L-S category, T-spacesCategories:55P45, 55P35

13. CMB 2011 (vol 55 pp. 319)

Jardine, J. F.
 The Verdier Hypercovering Theorem This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant. Keywords:simplicial presheaf, hypercover, cocycleCategories:14F35, 18G30, 55U35

14. CMB 2011 (vol 55 pp. 48)

Chebolu, Sunil K.; Christensen, J. Daniel; Mináč, Ján
 Freyd's Generating Hypothesis for Groups with Periodic Cohomology Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology, then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology. Keywords:Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomologyCategories:20C20, 20J06, 55P42

15. CMB 2010 (vol 53 pp. 730)

Theriault, Stephen D.
 A Case When the Fiber of the Double Suspension is the Double Loops on Anick's Space The fiber $W_{n}$ of the double suspension $S^{2n-1}\rightarrow\Omega^{2} S^{2n+1}$ is known to have a classifying space $BW_{n}$. An important conjecture linking the $EHP$ sequence to the homotopy theory of Moore spaces is that $BW_{n}\simeq\Omega T^{2np+1}(p)$, where $T^{2np+1}(p)$ is Anick's space. This is known if $n=1$. We prove the $n=p$ case and establish some related properties. Keywords:double suspension, Anick's spaceCategories:55P35, 55P10

16. CMB 2010 (vol 53 pp. 438)

Chigogidze, A.; Nagórko, A.
 Near-Homeomorphisms of Nöbeling Manifolds We characterize maps between $n$-dimensional NÃ¶beling manifolds that can be approximated by homeomorphisms. Keywords:n-dimensional Nöbeling manifold, Z-set unknotting, near-homeomorphismCategories:55M10, 54F45

17. CMB 2009 (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

18. CMB 2009 (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

19. CMB 2009 (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

20. CMB 2008 (vol 51 pp. 535)

Csorba, Péter
 On the Simple $\Z_2$-homotopy Types of Graph Complexes and Their Simple $\Z_2$-universality We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal. Keywords:graph complexes, simple $\Z_2$-homotopy, universalityCategories:57Q10, 05C10, 55P10

21. CMB 2008 (vol 51 pp. 310)

Witbooi, P. J.
 Relative Homotopy in Relational Structures The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs $(X,A)$ where $X$ is a poset and $A$ is a subposet of $X$. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif. Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalenceCategories:55Q05, 54A05;, 18B30

22. CMB 2007 (vol 50 pp. 365)

Godinho, Leonor
 Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations. Categories:53D20, 55N91, 57S15

23. CMB 2007 (vol 50 pp. 440)

Raghuram, A.
 A KÃ¼nneth Theorem for $p$-Adic Groups Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$. Categories:22E50, 18G15, 55U25

24. CMB 2007 (vol 50 pp. 206)

Golasiński, Marek; Gonçalves, Daciberg Lima
 Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well. Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space formCategories:55M35, 55P15, 20E22, 20F28, 57S17

25. CMB 2006 (vol 49 pp. 407)

Jardine, J. F.
 Intermediate Model Structures for Simplicial Presheaves This note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense. Categories:18G30, 18F20, 55U35
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