scholarly journals Diagonal approximation and the cohomology ring of torus fiber bundles

2015 ◽  
Vol 25 (03) ◽  
pp. 493-530 ◽  
Author(s):  
Sérgio Tadao Martins
Keyword(s):  
Cohomology Ring ◽  
Free Resolution ◽  
Fiber Bundles ◽  
Cohomology Groups ◽  
Torus Bundle ◽  
Diagonal Approximation ◽  
Cup Products

For a torus bundle (S1 × S1) → M → S1, we construct a finite free resolution of ℤ over ℤ[π1(M)] and compute the cohomology groups H*(π1(M); ℤ) and H*(π1(M);ℤp) for a prime p. We also construct a partial diagonal approximation for the resolution, which allows us to compute the cup products in H*(π1(M); ℤ) and H*(π1(M); ℤp).

scholarly journals The cohomology ring of the sapphires that admit the Sol geometry

2018 ◽  
Vol 28 (03) ◽  
pp. 365-380 ◽  
Author(s):  
Daciberg Lima Gonçalves ◽  
Sérgio Tadao Martins
Keyword(s):  
Fundamental Group ◽  
Cohomology Ring ◽  
Free Resolution ◽  
Multiplicative Structure ◽  
Torus Bundle ◽  
Cohomology Rings ◽  
Cup Product ◽  
Diagonal Approximation

Let [Formula: see text] be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of [Formula: see text] over [Formula: see text] and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings [Formula: see text] for [Formula: see text] and [Formula: see text] for an odd prime [Formula: see text], and indicate how to compute the groups [Formula: see text] and the multiplicative structure given by the cup product for any system of coefficients [Formula: see text].

scholarly journals ORBIFOLD CUP PRODUCTS AND RING STRUCTURES ON HOCHSCHILD COHOMOLOGIES

2011 ◽  
Vol 13 (01) ◽  
pp. 123-182 ◽  
Author(s):  
M. J. PFLAUM ◽  
H. B. POSTHUMA ◽  
X. TANG ◽  
H.-H. TSENG
Keyword(s):  
Cohomology Ring ◽  
Ring Structure ◽  
Hochschild Cohomology Ring ◽  
Convolution Algebras ◽  
First Case ◽  
De Rham ◽  
Ring Structures ◽  
Orbifold Cohomology ◽  
Cup Products ◽  
Equivariant Version

In this paper, we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case, the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an S1-equivariant version of the Chen–Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology.

Cup-products for the polyhedral product functor

2012 ◽  
Vol 153 (3) ◽  
pp. 457-469 ◽  
Author(s):  
A. BAHRI ◽  
M. BENDERSKY ◽  
F. R. COHEN ◽  
S. GITLER
Keyword(s):  
Simplicial Complex ◽  
Cohomology Ring ◽  
Decomposition Theorem ◽  
Ring Structure ◽  
Polyhedral Product ◽  
Cup Product ◽  
Rational Cohomology ◽  
Geometric Decomposition ◽  
Abstract Simplicial Complex ◽  
Cup Products

AbstractDavis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

scholarly journals Determination of the Homology and the Cohomology of a Few Groups of Isometries of the Hyperbolic Plane

2017 ◽  
Vol 36 ◽  
pp. 65-77
Author(s):  
Nasima Akhter ◽  
Subrata Majumdar
Keyword(s):  
Hyperbolic Plane ◽  
Free Resolution ◽  
Cohomology Groups ◽  
Graph Of Groups ◽  
Finitely Presented Groups ◽  
Full Resolution ◽  
Discontinuous Groups ◽  
Groups Of Isometries ◽  
Finitely Presented

In this paper we determine the homology and the cohomology groups of two properly discontinuous groups of isometries of the hyperbolic plane having non-compact orbit spaces and the fundamental group of a graph of groups with a finite vertex groups and no trivial edges by extending Lyndon’s partial free resolution for finitely presented groups. For the first two groups, we obtain partial extensions and the corresponding homology. We also compute the corresponding cohomology groups for one of these groups. Finally we obtain homology and cohomology in all dimensions for the last of the above mentioned groups by constructing a full resolution for this group.GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 65-77

scholarly journals The Homology and Cohomology Groups of H3

1970 ◽  
Vol 29 ◽  
pp. 139-146
Author(s):  
Subrata Majumdar ◽  
Quazi Selina Sultana
Keyword(s):  
Heisenberg Group ◽  
Key Words ◽  
Group Ring ◽  
Three Dimensional ◽  
Free Resolution ◽  
Integral Group Ring ◽  
Integral Group ◽  
Integral Homology ◽  
Cohomology Groups ◽  
Ams Classification

A free resolution of Z for the integral group ring of the three-dimensional Heisenberg group H3 has been constructed by extending Lyndon’s partial resolution. The integral homology and cohomology have been calculated from there.  AMS Classification: 18G, 20J. Key words: Fox derivatives; free resolution. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 139-146  DOI: http://dx.doi.org/10.3329/ganit.v29i0.8523 

Dupuytren's Contracture and Microvessels

1981 ◽  
Vol 39 ◽  
pp. 470-471
Author(s):  
C. W. Klscher ◽  
D. Speer
Keyword(s):  
Hypertrophic Scar ◽  
Active Form ◽  
Scar Tissue ◽  
Vascular Supply ◽  
Fiber Bundles ◽  
Dupuytren’S Contracture ◽  
Palmar Fascia ◽  
Immune Disease ◽  
Dupuytren's Contracture ◽  
Longitudinal Fiber

Dupuytren's Contracture is a nodular proliferation of the longitudinal fiber bundles of palmar fascia with its attendant contraction. The factors attributed to its etiology have included trauma, diabetes, alcoholism, arthritis, and auto-immune disease. The tissue has been observed by electron microscopy and found to contain myofibroblasts.Dupuytren's Contracture constitutes a scar, and as such, excessive collagen can be observed, along with an active form of fibroblast.Previous studies of the hypertrophic scar have led us to propose that integral in the initiation and sustenance of scar tissue is a profusion of microvascular regeneration, much of which becomes and remains occluded producing a hypoxia which stimulates fibroblast synthesis. Thus, when considering a study of Dupuytren's Contracture, we predicted finding occluded microvessels at or near the fascial scarring focus.Three cases of Dupuytren's Contracture yielded similar specimens, which were fixed in Karnovskys fluid for 2 to 20 days. Upon removal of the contracture bands care was taken to include the contiguous fatty and areolar tissue which contain the vascular supply and to identify the junctional area between old and new fascia.

Ultrastructural Features of the Oral Apparatus of Stentor

1976 ◽  
Vol 34 ◽  
pp. 142-143
Author(s):  
Elizabeth F. Howell
Keyword(s):  
Plasma Membrane ◽  
Osmium Tetroxide ◽  
Buccal Cavity ◽  
Fiber Bundles ◽  
Microtubular Root ◽  
Stentor Coeruleus ◽  
Root Fiber

The ultrastructure of the normal oral apparatus of Stentor has not been extensively studied. I report here on the ultrastructure of the buccal cavity of Stentor coeruleus.Stentor coeruleus was fixed in either a buffered mixture of osmium tetroxide and glutaraldehyde, or in buffered glutaraldehyde alone. Cells were then dehydrated and embedded in a mixture of Epon and Araldite.An extensive adoral zone of membranelles surrounds the anterior of the cell, and each membranelle consists of 2 parallel rows of cilia. These extend down into the buccal cavity. Two microtubular root fibers, or nemadesmata (Figs. 2 and 5), extend deeply into the cytoplasm from the base of each ciliary kinetosome. Mitochondria are usually closely associated with the root fiber bundles, and small vesicles are present between the nemadesmata of adjacent kinetosomes (Fig. 5). In the cytopharyngeal, non-ciliated areas of the buccal cavity, microtubular ribbons which extend into the cytoplasm are aligned perpendicular to the plasma membrane of the buccal cavity (Figs. 1 and 2).

Microstructures of SiC and Si3N4 with fibrous inclusions

1986 ◽  
Vol 44 ◽  
pp. 492-493
Author(s):  
N. J. Tighe ◽  
J. Sun ◽  
R.-M. Hu
Keyword(s):  
Vapor Deposition ◽  
Chemical Vapor ◽  
Crack Deflection ◽  
Fiber Bundles ◽  
High Temperature Strength ◽  
Graphite Fiber ◽  
Triple Junctions ◽  
Transmission Electron ◽  
Compact Graphite ◽  
Deposition Techniques

Particles of BN,and C are added in amounts of 1 to 40% to SiC and Si3N4 ceramics in order to improve their mechanical properties. The ceramics are then processed by sintering, hot-pressing and chemical vapor deposition techniques to produce dense products. Crack deflection at the particles can increase toughness. However the high temperature strength and toughness are determined byphase interactions in the environmental conditions used for testing. Examination of the ceramics by transmission electron microscopy has shown that the carbon and boron nitride particles have a fibrous texture. In the sintered aSiC ceramic the carbon appears as graphite fiber bundles in the triple junctions and as compact graphite particles within some grains. Examples of these inclusions are shown in Fig. 1A and B.

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