Expand all Collapse all | Results 1 - 11 of 11 |
1. CMB 2011 (vol 56 pp. 366)
Multiple Solutions for Nonlinear Periodic Problems We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p-1)$-superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$-Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).
Keywords:$C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance Categories:34B15, 34B18, 34C25, 58E05 |
2. CMB 2010 (vol 54 pp. 12)
Homotopy and the Kestelman-Borwein-Ditor Theorem
The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by
translation in (measure/category) ``large'' sets has two generalizations.
Miller replaces the translated sequence by a ``sequence homotopic
to the identity''. The authors, in a previous paper, replace points by functions:
a uniform functional null sequence replaces the null sequence, and
translation receives a functional form. We give a unified approach to
results of this kind. In particular, we show that (i) Miller's homotopy
version follows from the functional version, and (ii) the pointwise instance
of the functional version follows from Miller's homotopy version.
Keywords:measure, category, measure-category duality, differentiable homotopy Category:26A03 |
3. CMB 2008 (vol 51 pp. 535)
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, universality Categories:57Q10, 05C10, 55P10 |
4. CMB 2008 (vol 51 pp. 508)
A Result in Surgery Theory We study the topological $4$-dimensional surgery problem
for a closed connected orientable
topological $4$-manifold $X$ with vanishing
second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has
one end and $F(r)$ is the free group of rank $r\ge 1$.
Our result is related to a theorem of Krushkal and Lee, and
depends on the validity of the Novikov conjecture for
such fundamental groups.
Keywords:four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map Categories:57N65, 57R67, 57Q10 |
5. CMB 2008 (vol 51 pp. 310)
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 |
6. CMB 2008 (vol 51 pp. 81)
Homotopy Formulas for Cyclic Groups Acting on Rings The positive cohomology groups of a finite group acting on a ring
vanish when the ring has a norm one element. In this note we give
explicit homotopies on the level of cochains when the group is cyclic,
which allows us to express any cocycle of a cyclic group
as the coboundary of an explicit cochain.
The formulas in this note are closely related to the effective problems considered in previous joint work
with Eli Aljadeff.
Keywords:group cohomology, norm map, cyclic group, homotopy Categories:20J06, 20K01, 16W22, 18G35 |
7. CMB 2007 (vol 50 pp. 268)
On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups Using ideas of S. Wassermann on non-exact $C^*$-algebras and
property T groups, we show that one of his examples of non-invertible
$C^*$-extensions is not semi-invertible. To prove this, we
show that a certain element vanishes in the asymptotic tensor
product. We also show that a modification of the example gives
a $C^*$-extension which is not even invertible up to homotopy.
Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopy Categories:19K33, 46L06, 46L80, 20F99 |
8. CMB 2007 (vol 50 pp. 206)
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ |
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 form Categories:55M35, 55P15, 20E22, 20F28, 57S17 |
9. CMB 2006 (vol 49 pp. 628)
Approximation and the Topology of Rationally Convex Sets Considering a mapping $g$ holomorphic on a neighbourhood of a rationally
convex set $K\subset\cc^n$, and range into the complex projective space
$\cc\pp^m$, the main objective of this paper is to show that we can
uniformly approximate $g$ on $K$ by rational mappings defined from
$\cc^n$ into $\cc\pp^m$. We only need to ask that the second \v{C}ech
cohomology group $\check{H}^2(K,\zz)$ vanishes.
Keywords:Rationally convex, cohomology, homotopy Categories:32E30, 32Q55 |
10. CMB 2006 (vol 49 pp. 237)
Approximation by Rational Mappings, via Homotopy Theory Continuous mappings defined from compact subsets $K$ of complex
Euclidean space $\cc^n$ into complex projective space $\pp^m$ are
approximated by rational mappings. The fundamental tool employed
is homotopy theory.
Keywords:Rational approximation, homotopy type, null-homotopic Categories:32E30, 32C18 |
11. CMB 2004 (vol 47 pp. 321)
Classifying Spaces for Monoidal Categories Through Geometric Nerves The usual constructions of classifying spaces for monoidal categories
produce CW-complexes 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 2-skeleton 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, pseudo-simplicial category,, simplicial set, classifying space, homotopy type Categories:18D10, 18G30, 55P15, 55P35, 55U40 |