Expand all Collapse all | Results 1 - 25 of 31 |
1. CJM Online first
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 |
2. CJM Online first
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 |
3. CJM Online first
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 |
4. CJM 2013 (vol 65 pp. 843)
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 |
5. CJM 2012 (vol 65 pp. 82)
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 |
6. CJM 2012 (vol 65 pp. 544)
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 |
7. CJM 2012 (vol 65 pp. 553)
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 |
8. CJM 2011 (vol 63 pp. 1345)
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 |
9. CJM 2011 (vol 63 pp. 1038)
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 |
10. CJM 2010 (vol 62 pp. 1387)
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 |
11. CJM 2010 (vol 62 pp. 1082)
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 |
12. CJM 2009 (vol 62 pp. 284)
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 |
13. CJM 2009 (vol 62 pp. 473)
GoreskyâMacPherson 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 |
14. CJM 2009 (vol 62 pp. 614)
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 |
15. CJM 2008 (vol 60 pp. 348)
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 |
16. CJM 2007 (vol 59 pp. 1154)
$k(n)$-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 |
17. CJM 2007 (vol 59 pp. 1008)
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 |
18. CJM 2006 (vol 58 pp. 877)
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 Category:55P53 |
19. CJM 2004 (vol 56 pp. 1290)
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 |
20. CJM 2003 (vol 55 pp. 181)
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 |
21. CJM 2002 (vol 54 pp. 608)
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 |
22. CJM 2002 (vol 54 pp. 396)
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 |
23. CJM 2001 (vol 53 pp. 212)
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, codes Categories:55M35, 57M60, 94B05, 05E20 |
24. CJM 1999 (vol 51 pp. 897)
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 |
25. CJM 1999 (vol 51 pp. 3)
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 model Categories:13D25, 18E30, 18G35, 55U15 |