Recursos de colección

Project Euclid (Hosted at Cornell University Library) (192.674 recursos)

Homology, Homotopy and Applications

1. Universal enveloping crossed module of a Lie crossed module

We construct a pair of adjoint functors between the categories of crossed modules of Lie and associative algebras, which extends the classical one between the categories of Lie and associative algebras. This result is used to establish an equivalence of categories of modules over a Lie crossed module and its universal enveloping crossed module.

2. $L_{\infty}$-algebras of local observables from higher prequantum bundles

Fiorenza, Domenico; Rogers, Christopher L.; Schreiber, Urs
To any manifold equipped with a higher degree closed form, one can associate an $L_\infty$-algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this $L_\infty$-algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produce analogues of the (higher) Lie algebras of sections of the Atiyah Lie algebroid and of the Courant Lie 2-algebroid. We also exhibit the $L_\infty$-cocycle that realizes the $L_\infty$-algebra of local observables as a Kirillov-Kostant-Souriau-type $L_\infty$-extension of the Hamiltonian vector fields. When restricted along a...

3. Crossed modules of racks

Crans, Alissa S.; Wagemann, Friedrich
We generalize the notion of a crossed module of groups to that of a crossed module of racks. We investigate the relation to categorified racks, namely strict 2-racks, and trunk-like objects in the category of racks, generalizing the relation between crossed modules of groups and strict 2-groups. Then we explore topological applications. We show that by applying the rackspace functor, a crossed module of racks gives rise to a covering. Our main result shows how the fundamental racks associated to links upstairs and downstairs in a covering fit together to form a crossed module of racks.

4. Weak Lefschetz for Chow groups: Infinitesimal lifting

Patel, D.; Ravindra, G. V.
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero, and let $Y \subset X$ be a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states that the natural restriction map $\mathrm{CH}^p (X)_{\mathbb{Q}} \to \mathrm{CH}^p (Y)_{\mathbb{Q}}$ is an isomorphism for all $p \lt \dim (Y) / 2$. In this note, we revisit a strategy introduced by Grothendieck to attack this problem by using the Bloch-Quillen formula to factor this morphism through a continuous $\mathrm{K}$-cohomology group on the formal completion of $X$ along $Y$. This splits the conjecture into two smaller conjectures: one consisting of...

5. Derived categories of absolutely flat rings

Stevenson, Greg
Let $S$ be a commutative ring with topologically noetherian spectrum, and let $R$ be the absolutely flat approximation of $S$. We prove that subsets of the spectrum of $R$ parametrise the localising subcategories of $\mathsf{D}(R)$. Moreover, we prove the telescope conjecture holds for $\mathsf{D}(R)$. We also consider unbounded derived categories of absolutely flat rings that are not semi-artinian and exhibit a localising subcategory that is not a Bousfield class and a cohomological Bousfield class that is not a Bousfield class.

6. Homological descent for motivic homology theories

Geisser, Thomas
The purpose of this paper is to give homological descent theorems for motivic homology theories (for example, Suslin homology) and motivic Borel-Moore homology theories (for example, higher Chow groups) for certain hypercoverings.

7. Higher Morse moduli spaces and $n$-categories

Hohloch, Sonja
We generalize Cohen & Jones & Segal's flow category, whose objects are the critical points of a Morse function and whose morphisms are the Morse moduli spaces between the critical points to an $n$-category. The $n$-category construction involves repeatedly doing Morse theory on Morse moduli spaces for which we have to construct a class of suitable Morse functions. It turns out to be an 'almost strict' $n$-category, i.e. it is a strict $n$-category 'up to canonical isomorphisms'.

8. Cohomology of algebras over weak Hopf algebras

Álvarez, J. N. Alonso; Vilaboa, J. M. Fernández; Rodríguez, R. González
In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra $H$. We show that the second cohomology group classifies completely weak crossed products, having a common preunit, of $H$ with a commutative left $H$-module algebra $A$.

9. A purely homotopy-theoretci proof of the Blakers-Massey theorem for $n$-cubes

Munson, Brian A.
Goodwillie’s proof of the Blakers-Massey Theorem for n- cubes relies on a lemma whose proof invokes transversality. The rest of his proof follows from general facts about cubes of spaces and connectivities of maps. We present a purely homotopytheoretic proof of this lemma. The methods are elementary, using a generalization and modification of an argument originally due to Puppe used to prove the Blakers-Massey Theorem for squares.

10. Global orthogonal spectra

Bohmann, Anna Marie
For any compact Lie group $G$, there are several well-established definitions of a $G$-equivariant spectrum. In this paper, we develop the definition of a global orthogonal spectrum. Loosely speaking, this is a coherent choice of orthogonal $G$-spectrum for each compact Lie group $G$. We use the framework of enriched indexed categories to make this precise. We also consider equivariant $K$-theory and $\operatorname{Spin}^c$-cobordism from this perspective, and we show that the Atiyah-Bott-Shapiro orientation extends to the global context.

11. Graphs associated with simplicial complexes

Grigor'yan, A.; Muranov, Yu. V.; Yau, Shing-Tung
The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph $G_B$ of the partially ordered...

12. Mayer-Vietoris sequences in stable derivators

Groth, Moritz; Ponto, Kate; Shulman, Michael
We show that stable derivators, like stable model cate- gories, admit Mayer-Vietoris sequences arising from cocartesian squares. Along the way we characterize homotopy exact squares and give a detection result for colimiting diagrams in derivators. As an application, we show that a derivator is stable if and only if its suspension functor is an equivalence.

13. Describing high-order statistical dependence using "concurrence topology," with application to functional MRI brain data

Ellis, Steven P.; Klein, Arno
In multivariate data analysis dependence beyond pair-wise can be important. With many variables, however, the number of simple summaries of even third-order dependence can be unmanageably large. ¶ “Concurrence topology” is an apparently new method for describing high-order dependence among up to dozens of dichotomous (i.e., binary) variables (e.g., seventh-order dependence in 32 variables). This method generally produces summaries of dependence of manageable size. (But computing time can be lengthy.) For time series, this method can be applied in both the time and Fourier domains. ¶ Write each observation as a vector of 0’s and 1’s. A “concurrence” is a group of variables all labeled “1” in the same observation. The collection...

14. On connective $K$-theory of elementary abelian 2-groups and local duality

Powell, Geoffrey M. L.
The connective $ku$-(co)homology of elementary abelian $2$-groups is determined as a functor of the elementary abelian $2$-group, using the action of the Milnor operations $Q_0, Q_1$ on mod $2$ group cohomology, the Atiyah-Segal theorem for $KU$-cohomology, together with an analysis of the functorial structure of the integral group ring; the functorial structure then reduces calculations to the rank 1 case. ¶ These results are used to analyse the local cohomology spectral sequence calculating $ku$-homology, via a functorial version of local duality for Koszul complexes, giving a conceptual explanation of results of Bruner and Greenlees.

15. Exact sequences of commutative monoids and semimodules

Basic homological lemmas well known for modules over rings and, more generally, in the context of abelian categories, have been extended to many other concrete and abstract-categorical contexts by various authors. We propose a new such extension, specifically for commutative monoids and semimodules; these two contexts are equivalent since the forgetful functors from varieties of semimodules to the variety of commutative monoids preserve all limits and colimits.

Fialowski, Alice; Mukherjee, Goutam; Naolekar, Anita
We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base—the so-called versal deformation—which induces all other deformations of the given algebra.

17. Kei modules and unoriented link invariants

Grier, Michael; Nelson, Sam
We define invariants of unoriented knots and links by enhancing the integral kei counting invariant $\Phi_X^{\mathbb{Z}}(K)$ for a finite kei $X$ using representations of the kei algebra, $\mathbb{Z}_K[X]$, a quotient of the quandle algebra $\mathbb{Z}[X]$ defined by Andruskiewitsch and Graña. We give an example that demonstrates that the enhanced invariant is stronger than the unenhanced kei counting invariant. As an application, we use a quandle module over the Takasaki kei on $\mathbb{Z}_3$ which is not a $\mathbb{Z}_K[X]$-module to detect the non-invertibility of a virtual knot.

18. Complexification and homotopy

Kucharz, Wojciech; Maciejewski, Łukasz
Let $Y$ be a real algebraic variety. We are interested in determining the supremum, $\beta(Y)$, of all nonnegative integers $n$ with the following property: For every $n$-dimensional compact connected nonsingular real algebraic variety $X$, every continuous map from $X$ into $Y$ is homotopic to a regular map. We give an upper bound for $\beta(Y)$, based on a construction involving complexification of real algebraic varieties. In some cases, we obtain the exact value of $\beta(Y)$.

19. Postnikov towers with fibers generalized Eilenberg-Mac Lane spaces

Iriye, Kouyemon; Kishimoto, Daisuke
A generalized Postnikov tower (GPT) is defined as a tower of principal fibrations with the classifying maps into generalized Eilenberg–Mac Lane spaces. We study fundamental properties of GPT’s such as their existence, localization and length. We further consider the distribution of torsion in a GPT of a finite complex, motivated by the result of McGibbon and Neisendorfer. We also give an algebraic description of the length of a GPT of a rational space.

20. The homology graph of a precubical set

Kahl, Thomas
Precubical sets are used to model concurrent systems. We introduce the homology graph of a precubical set, which is a directed graph whose nodes are the homology classes of the precubical set. We show that the homology graph is invariant under weak morphisms that are homeomorphisms.

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