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Search: All articles in the CMB digital archive with keyword homotopy

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1. CMB 2011 (vol 56 pp. 366)

Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.
 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 invarianceCategories:34B15, 34B18, 34C25, 58E05

2. CMB 2010 (vol 54 pp. 12)

Bingham, N. H.; Ostaszewski, A. J.
 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 homotopyCategory:26A03

3. 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

4. CMB 2008 (vol 51 pp. 508)

Cavicchioli, Alberto; Spaggiari, Fulvia
 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 mapCategories:57N65, 57R67, 57Q10

5. 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

6. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
 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, homotopyCategories:20J06, 20K01, 16W22, 18G35

7. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
 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, homotopyCategories:19K33, 46L06, 46L80, 20F99

8. 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

9. CMB 2006 (vol 49 pp. 628)

Zeron, E. S.
 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, homotopyCategories:32E30, 32Q55

10. CMB 2006 (vol 49 pp. 237)

Gauthier, P. M.; Zeron, E. S.
 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-homotopicCategories:32E30, 32C18

11. 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 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 typeCategories:18D10, 18G30, 55P15, 55P35, 55U40