scholarly journals A chain morphism for Adams operations on rational algebraic K-theory

2009 ◽  
Vol 5 (2) ◽  
pp. 349-402 ◽  
Author(s):  
Elisenda Feliu
Keyword(s):  
Noetherian Scheme ◽  
Chain Complexes ◽  
K Theory ◽  
A Chain ◽  
Adams Operations

AbstractFor any regular noetherian scheme X and every k ≥ 1, we define a chain morphism ψk between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by ℚ. It is shown that the morphisms ψk induce in homology the Adams operations defined by Gillet and Soulé or the ones defined by Grayson.

scholarly journals AN INVARIANT OF LEGENDRIAN AND TRANSVERSE LINKS FROM OPEN BOOK DECOMPOSITIONS OF CONTACT 3-MANIFOLDS

2020 ◽  
pp. 1-33
Author(s):  
ALBERTO CAVALLO
Keyword(s):  
Chain Complex ◽  
Open Book ◽  
Rational Homology ◽  
Open Book Decompositions ◽  
Chain Complexes ◽  
Legendrian Links ◽  
Sum Formula ◽  
A Chain ◽  
Special Cycle ◽  
Rational Homology Sphere

Abstract We introduce a generalization of the Lisca–Ozsváth–Stipsicz–Szabó Legendrian invariant ${\mathfrak L}$ to links in every rational homology sphere, using the collapsed version of link Floer homology. We represent a Legendrian link L in a contact 3-manifold ${(M,\xi)}$ with a diagram D, given by an open book decomposition of ${(M,\xi)}$ adapted to L, and we construct a chain complex ${cCFL^-(D)}$ with a special cycle in it denoted by ${\mathfrak L(D)}$ . Then, given two diagrams ${D_1}$ and ${D_2}$ which represent Legendrian isotopic links, we prove that there is a map between the corresponding chain complexes that induces an isomorphism in homology and sends ${\mathfrak L(D_1)}$ into ${\mathfrak L(D_2)}$ . Moreover, a connected sum formula is also proved and we use it to give some applications about non-loose Legendrian links; that are links such that the restriction of ${\xi}$ on their complement is tight.

Adams operations on the virtual K-theory of ℙ(1,n)

2016 ◽  
Vol 16 (08) ◽  
pp. 1750149
Author(s):  
Takashi Kimura ◽  
Ross Sweet
Keyword(s):  
Cotangent Bundle ◽  
Crepant Resolution ◽  
Chern Classes ◽  
Localization Theorem ◽  
Virtual Line ◽  
Power Operations ◽  
Power Operation ◽  
K Theory ◽  
Line Elements ◽  
Adams Operations

We analyze the structure of the virtual (orbifold) [Formula: see text]-theory ring of the complex orbifold [Formula: see text] and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin–Graham [D. Edidin and W. Graham, Nonabelian localization in equivariant [Formula: see text]-theory and Riemann–Roch for quotients, Adv. Math. 198(2) (2005) 547–582]. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual [Formula: see text]-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual [Formula: see text]-theory ring of [Formula: see text] to the ordinary [Formula: see text]-theory ring of a crepant resolution of the cotangent bundle of [Formula: see text] which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of [Formula: see text] which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin–Jarvis–Kimura [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial [Formula: see text]-theory, Ann. K-Theory (2016)], who proved the latter for [Formula: see text].

scholarly journals Assembly and annotation of the mitochondrial minicircle genome of a differentiation-competent strain of Trypanosoma brucei

2019 ◽  
Vol 47 (21) ◽  
pp. 11304-11325 ◽  
Author(s):  
Sinclair Cooper ◽  
Elizabeth S Wadsworth ◽  
Torsten Ochsenreiter ◽  
Alasdair Ivens ◽  
Nicholas J Savill ◽  
...  
Keyword(s):  
Trypanosoma Brucei ◽  
Small Rna ◽  
Mitochondrial Genomes ◽  
Transcriptome Data ◽  
Respiratory Chain Complexes ◽  
Data Set ◽  
Guide Rnas ◽  
Copy Numbers ◽  
Chain Complexes ◽  
A Chain

Abstract Kinetoplastids are protists defined by one of the most complex mitochondrial genomes in nature, the kinetoplast. In the sleeping sickness parasite Trypanosoma brucei, the kinetoplast is a chain mail-like network of two types of interlocked DNA molecules: a few dozen ∼23-kb maxicircles (homologs of the mitochondrial genome of other eukaryotes) and thousands of ∼1-kb minicircles. Maxicircles encode components of respiratory chain complexes and the mitoribosome. Several maxicircle-encoded mRNAs undergo extensive post-transcriptional RNA editing via addition and deletion of uridines. The process is mediated by hundreds of species of minicircle-encoded guide RNAs (gRNAs), but the precise number of minicircle classes and gRNA genes was unknown. Here we present the first essentially complete assembly and annotation of the kinetoplast genome of T. brucei. We have identified 391 minicircles, encoding not only ∼930 predicted ‘canonical’ gRNA genes that cover nearly all known editing events (accessible via the web at http://hank.bio.ed.ac.uk), but also ∼370 ‘non-canonical’ gRNA genes of unknown function. Small RNA transcriptome data confirmed expression of the majority of both categories of gRNAs. Finally, we have used our data set to refine definitions for minicircle structure and to explore dynamics of minicircle copy numbers.

scholarly journals TORSION OF QUASI-ISOMORPHISMS

2009 ◽  
Vol 18 (09) ◽  
pp. 1227-1258
Author(s):  
JAE-WOOK CHUNG ◽  
XIAO-SONG LIN
Keyword(s):  
Mild Condition ◽  
Chain Complex ◽  
Reidemeister Torsion ◽  
Homotopy Invariant ◽  
Chain Complexes ◽  
Chain Homotopy ◽  
Chain Map ◽  
A Chain

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.

Projective Approximations

1984 ◽  
Vol 36 (1) ◽  
pp. 178-192 ◽  
Author(s):  
K. Varadarajan
Keyword(s):  
Associative Ring ◽  
Chain Complex ◽  
Chain Complexes ◽  
Chain Map ◽  
Chain Maps ◽  
A Chain

Let R be an associative ring with 1 ≠ 0. Throughout we will be considering unitary left R-modules. Given a chain complex C over R, a free approximation of C is defined to be a free chain complex F over R together with an epimorphism τ:F → C of chain complexes with the property that H(τ):H(F) ≃ H(C). In Chapter 5, Section 2 of [3] it is proved that any chain complex C over Z has a free approximation τ:F → C. Moreover given a free approximation τ:F → C of C and any chain map f:F’ → C with F’ a free chain complex over Z, there exists a chain map φ:F’→ F with T O φ = f . Any two chain maps φ, ψ of F’ in F with T O φ = T O ψ are chain homotopic.

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