1. CMB Online first
 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 Grassmannian Categories:55N91, 19L47 

2. CMB 2015 (vol 58 pp. 575)
 MartinezTorres, 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 4dimensional manifold with a (symplectic) Lie groupoid
structure.
In this short note we show that if $\pi$ is not an area
form on the 2sphere, 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 bundle Categories:58H05, 55R10, 53D17 

3. 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 FukukawaHaradaMasuda,
which was only for Lie type $A$.
Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula Categories:55N91, 14N15 

4. 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:LusternikSchnirelmann category, coverings of $3$manifolds by open $K$contractible sets Categories:57N10, 55M30, 57M27, 57N16 

5. CMB 2013 (vol 57 pp. 225)
 Adamaszek, Michał

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 computeraided.
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 model Categories:55U10, 06A11, 55P40, 5504, 0504 

6. CMB 2013 (vol 57 pp. 245)
 Brodskiy, N.; Dydak, J.; Lang, U.

AssouadNagata 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 AssouadNagata 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:AssouadNagata dimension, asymptotic dimension, wreath product, growth of groups Categories:54F45, 55M10, 54C65 

7. CMB 2011 (vol 55 pp. 523)
 Iwase, Norio; Mimura, Mamoru; Oda, Nobuyuki; Yoon, Yeon Soo

The MilnorStasheff 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 MilnorStasheff 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, LS category, Tspaces Categories:55P45, 55P35 

8. CMB 2011 (vol 55 pp. 319)
 Jardine, J. F.

The Verdier Hypercovering Theorem
This note gives a simple cocycletheoretic 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, cocycle Categories:14F35, 18G30, 55U35 

9. 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 finitedimensional
$kG$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 cohomology Categories:20C20, 20J06, 55P42 

10. 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
$S^{2n1}\rightarrow\Omega^{2} S^{2n+1}$
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 space Categories:55P35, 55P10 

11. CMB 2010 (vol 53 pp. 438)
12. CMB 2008 (vol 51 pp. 535)
13. 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 equivalence Categories:55Q05, 54A05;, 18B30 

14. 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 

15. 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 

16. CMB 2007 (vol 50 pp. 206)
 Golasiński, Marek; Gonçalves, Daciberg Lima

Spherical Space Forms: Homotopy Types and SelfEquivalences 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 $X(2dn1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn1)/\mu$ associated with
free and cellular $G$actions $\mu$ on $X(2dn1)$ are determined as
well.
Keywords:automorphism group, $CW$complex, free and cellular $G$action, group of self homotopy equivalences, LyndonHochschildSerre spectral sequence, special (linear) group, spherical space form Categories:55M35, 55P15, 20E22, 20F28, 57S17 

17. 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 

18. CMB 2006 (vol 49 pp. 41)
19. CMB 2005 (vol 48 pp. 614)
 Tuncali, H. Murat; Valov, Vesko

On FinitetoOne Maps
Let $f\colon X\to Y$ be a $\sigma$perfect $k$dimensional surjective
map of metrizable spaces such that $\dim Y\leq m$. It is shown that
for every positive integer $p$ with $ p\leq m+k+1$ there exists a
dense $G_{\delta}$subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$
with the source limitation topology such that each fiber of
$f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most
$\max\{k+mp+2,1\}$ points. This result
provides a proof the following conjectures of
S. Bogatyi, V. Fedorchuk and J. van Mill.
Let $f\colon X\to Y$ be a $k$dimensional map between compact
metric spaces with $\dim Y\leq m$. Then:
\begin{inparaenum}[\rm(1)]
\item there exists a map
$h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+2k}$ is 2toone provided $k\geq 1$;
\item there exists a
map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+k+1}$ is $(k+1)$toone.
\end{inparaenum}
Keywords:finitetoone maps, dimension, setvalued maps Categories:54F45, 55M10, 54C65 

20. CMB 2004 (vol 47 pp. 321)
 Bullejos, M.; Cegarra, A. M.

Classifying Spaces for Monoidal Categories Through Geometric Nerves
The usual constructions of classifying spaces for monoidal categories
produce CWcomplexes with
many cells that, moreover, do not have any proper geometric meaning.
However, geometric nerves of
monoidal categories are very handy simplicial sets whose simplices
have
a pleasing geometric
description: they are diagrams with the shape of the 2skeleton of
oriented standard simplices. The
purpose of this paper is to prove that geometric realizations of
geometric nerves are classifying
spaces for monoidal categories.
Keywords:monoidal category, pseudosimplicial category,, simplicial set, classifying space, homotopy type Categories:18D10, 18G30, 55P15, 55P35, 55U40 

21. 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 crosssection body $CK$ of a convex body
$K \subset \sR^d, \, d \ge 2$,
we prove theorems about maximal $k$sections of convex bodies,
$k \in \{1, \dots, d1\}$,
and, simultaneously, statements
about common maximal
$(d1)$ and $1$transversals of families
of convex bodies.
Categories:52A20, 55Mxx 

22. 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 connectedcover) 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 Category:55Q52 

23. CMB 2001 (vol 44 pp. 459)
 Kahl, Thomas

LScatÃ©gorie algÃ©brique et attachement de cellules
Nous montrons que la Acat\'egorie d'un espace simplement connexe de
type fini est inf\'erieure ou \'egale \`a $n$ si et seulement si son
mod\`ele d'AdamsHilton 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 Acategory of a simply connected space of finite type
is less than or equal to $n$ if and only if its AdamsHilton 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:LScategory, strong category, AdamsHilton models, cell attachments Categories:55M30, 18G55 

24. 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 finitedimensional 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 

25. CMB 2001 (vol 44 pp. 80)