We define equivariant projective unitary stable bundles as the appropriate twists when defining K‐theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah–Segal.
the structural and ultrastructural features of gonads from endemic Mexican fish have received scarce attention. This study describes the histological and ultrastructural characteristics of the oocyte in Chirostoma humboldtianum. The ovary is asynchronic, and as such, most phases of oocyte development are found in the same ovary. The complete process of oogenesis was divided in five stages: oogonium and folliculogenesis, primary growth, cortical alveoli and lipid inclusions, vitellogenesis and maturation. The presence of big filaments, which appear at the end of primary growth, induces some common follicular adaptation. During primary growth, abundant ribosomes, rough endoplasmic reticulum, and mitochondria are grouped in the cytoplasm. At the end of this stage, the Z1 layer of the chorion is developed, while microvilli start to be evident as well. In the cortical alveoli and lipid droplets phase, intense PAS positive vesicles, some of them containing nucleoid material, are observed in the peripheral cytoplasm and the lipid droplets take a more central position. In vitellogenesis, the proteic yolk accumulates in a centripetal way while the chorion is completely formed. In maturation, the germinal vesicle migrates to the animal pole, meiosis is restored, and there is nuclear breakdown. The oocyte increases its size and holds some oil droplets and a big fluid mass of yolk. On the outside, filaments surround the oocyte completely. Rev. Biol. Trop. 56 (4): 1825-1835. Epub 2008 December 12.
We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion. 1 2 NOÉ BÁRCENAS, JESÚS ESPINOZA, BERNARDO URIBE, AND MARIO VELÁSQUEZFor this purpose we develop a twisted version of Bredon cohomology, cohomology which turns out to determine the E 2 -page of Segal's spectral sequence once it is applied to an equivariantly contractible cover.The construction of the spectral sequence extends and generalizes previous work of C. Dwyer [12], who only treated the twistings which are classified by cohomology classes of finite order which lie in the image of the canonical map H 3 (BG, Z) → H 3 (X× G EG; Z); these twistings take the name of discrete torsion twistings.The main result of this note, which is Theorem 4.7, relies on the construction and the properties of the universal stable equivariant projective unitary bundle carried out in [6]. Since this work can be seen as a continuation of what has been done in [6], we will use the notation, the definitions and the results of that paper. We will not reproduce any proof that already appears in [6], instead we will give appropriate references whenever a definition or a result of [6] is used.We emphasize that the topological issues that may appear when working with the Projective Unitary Group have all been resolved in [22, Section 15] when it is endowed with the norm topology. We therefore assume in this work that we are working with the norm topology when discussing topological properties of operator spaces.This note is organized as follows. In Section 1 a version of Bredon cohomology associated to an equivariant cover of a space is constructed. In Section 2 the basics of Transformation Groups and Parametrized Homotopy Theory needed for the construction are quickly reviewed. This is used to construct a version of Bredon cohomology with local coefficients. In Section 3 the construction of twisted equivariant K-theory for proper and discrete actions given in [6] is reviewed. In Section 4 , the Bredon cohomology with local coefficients in twisted representations is shown to be isomorphic to the second page of a spectral sequence converging to twisted equivariant K-theory. Some phenomena concerning the third differential of this spectral sequence is also analyzed. In Section 5 some simple examples are given including the case of discrete torsion which was developed by Dwyer in [12].Acknowledgements.
In this paper, we present an algorithm to compute the filtered generalizedČech complex for a finite collection of disks in the plane, which don't necessarily have the same radius.The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments.We present two applications of our main results. We give an algorithm for computing the 2-dimensional filtered generalizedČech complex of a finite collection of d-dimensional disks in R d . In addition, we show how the algorithm yields the minimal enclosing ball for a finite set of points in the plane.
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