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

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26. 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 map
Categories:57N65, 57R67, 57Q10

27. 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, homotopy
Categories:20J06, 20K01, 16W22, 18G35

28. CMB 2007 (vol 50 pp. 567)

Joshi, Kirti
Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence
In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsion
Categories:14F30, 14J30

29. CMB 2007 (vol 50 pp. 598)

Lorestani, Keivan Borna; Sahandi, Parviz; Yassemi, Siamak
Artinian Local Cohomology Modules
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\H^i_\fa(M)$ is finitely generated for all $i
Keywords:local cohomology module, Artinian module, reflexive module
Categories:13D45, 13E10, 13C05

30. CMB 2007 (vol 50 pp. 56)

Gourdeau, F.; Pourabbas, A.; White, M. C.
Simplicial Cohomology of Some Semigroup Algebras
In this paper, we investigate the higher simplicial cohomology groups of the convolution algebra $\ell^1(S)$ for various semigroups $S$. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than $a$ for some integer $a$. Our results are of two types: in some cases, we show that some cohomology groups are $0$, while in some other cases, we show that some cohomology groups are Banach spaces.

Keywords:simplicial cohomology, semigroup algebra

31. 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, homotopy
Categories:32E30, 32Q55

32. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$
In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$.

Keywords:$\overline\partial$-problem, cohomology groups, line bundles
Categories:32F20, 14F05, 32C16

33. CMB 2005 (vol 48 pp. 414)

Kaveh, Kiumars
Vector Fields and the Cohomology Ring of Toric Varieties
Let $X$ be a smooth complex projective variety with a holomorphic vector field with isolated zero set $Z$. From the results of Carrell and Lieberman there exists a filtration $F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of $\c$-valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X, \c)$ as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a $1$-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of $X$.

Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra
Categories:14M25, 52B20

34. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
Logarithms and the Topology of the Complement of a Hypersurface
This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$.

Keywords:Logarithm, homology groups and periods
Categories:32Q55, 14F45

35. CMB 2003 (vol 46 pp. 617)

Pak, Hong Kyung
On Harmonic Theory in Flows
Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and $H^*$-harmonic spaces associated to a H\"ormander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

Keywords:contact structure, geodesible flow, isometric flow, basic cohomology
Categories:53C20, 57R30

36. CMB 2003 (vol 46 pp. 268)

Puls, Michael J.
Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups
Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional
Categories:43A15, 20F65, 20F18

37. CMB 2000 (vol 43 pp. 3)

Adin, Ron; Blanc, David
Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedron
Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

38. CMB 1999 (vol 42 pp. 129)

Baker, Andrew
Hecke Operations and the Adams $E_2$-Term Based on Elliptic Cohomology
Hecke operators are used to investigate part of the $\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\Ext^1$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.

Keywords:Adams spectral sequence, elliptic cohomology, Hecke operators
Categories:55N20, 55N22, 55T15, 11F11, 11F25

39. CMB 1997 (vol 40 pp. 54)

Kechagias, Nondas E.
A note on $U_n\times U_m$ modular invariants
We consider the rings of invariants $R^G$, where $R$ is the symmetric algebra of a tensor product between two vector spaces over the field $F_p$ and $G=U_n\times U_m$. A polynomial algebra is constructed and these invariants provide Chern classes for the modular cohomology of $U_{n+m}$.

Keywords:Invariant theory, cohomology of the unipotent group
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