Homologie singulière

Résumé

En topologie algébrique, l'homologie singulière est une construction qui permet d'associer à un espace topologique X une suite homologique de groupes abéliens libres ou de modules. Cette association est un invariant topologique non complet, c'est-à-dire que si deux espaces sont homéomorphes alors ils ont mêmes groupes d'homologie singulière en chaque degré mais que la réciproque est fausse. Origine : intégration des formes différentielles fermées Le théorème de Stokes appliqué à des formes fermées donne des intégrales nulles. Cependant, il se fonde sur une hypothèse cruciale de compacité. En présence de trous dans la variété sous-jacente, on peut construire des formes fermées avec des intégrales de bord non nulles. Par exemple, \omega(x,y)=-\frac y{x^2+y^2}\mathrm dx+\frac x{x^2+y^2}\mathrm dy 1-forme définie sur ℝ \ { (0, 0) }. Vérifions sa fermeture : \mathrm d\omega=\left(\frac{\partial\omega_y}{\partial x}-\frac{\partial\omega_x}{\partial y}\righ

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https://fr.wikipedia.org/wiki/Homologie_singuli%C3%A8re

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Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand its properties and learn how to compute it. There will be many examples and applications.

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in algebra and topology.

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Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-modules is non-semisimple. In this thesis, we develop new techniques for studying Krull-Schmidt decompositions of tensor products of G-modules.More specifically, we use minimal complexes of tilting modules to define a tensor ideal of singular G-modules, and we show that, up to singular direct summands, taking tensor products of G-modules respects the decomposition of Rep(G) into linkage classes. In analogy with the classical translation principle, this allows us to reduce questions about tensor products of G-modules in arbitrary l-regular linkage classes to questions about tensor products of G-modules in the principal block of G. We then identify a particular non-singular indecomposable direct summand of the tensor product of two simple G-modules in the principal block (with highest weights in two given l-alcoves), which we call the generic direct summand because it appears generically in Krull-Schmidt decompositions of tensor products of simple G-modules (with highest weights in the given l-alcoves). We initiate the study of generic direct summands, and we use them to prove a necessary condition for the complete reducibility of tensor products of simple G-modules, when G is a simple algebraic group of type A_n.

EPFL2022

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A Shadow Perspective on Equivariant Hochschild Homologies

Kathryn Hess Bellwald, Inbar Klang

Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category V, as well as for small V-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and C-n-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic K-theory to twisted topological Hochschild homology.

OXFORD UNIV PRESS2022

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Bifurcation points of a singular boundary-value problem on (0,1)

Charles Stuart

Following earlier work on some special cases [17,11] and on the analogous problem in higher dimensions [10,20], we make a more thorough investigation of the bifurcation points for a nonlinear boundary value problem of the form -{A(x)u' (x)}'.= f (lambda, x, u(x), u' (x)) for 0 < x < 1, integral(1)(0) A(x)u' (x)(2)dx < infinity and u(1) = 0, where, for all lambda is an element of R and x is an element of (0, 1), f (lambda, x, 0, 0) = 0 and partial derivative(3)f(lambda, x, 0, 0) = lambda - V (x) and partial derivative(4) f (lambda, x, 0, 0) = 0, so that the formal linearization about a trivial solution u equivalent to 0 is -{A(x)v'(x)}' + V(x)v(x) = lambda v(x) for 0 < x < 1. Even when f is a smooth function of all its variables, standard bifurcation theory does not apply to the problem and the results differ from the usual conclusions. This is because we deal with the case where the coefficient A has a critical degeneracy as x -> 0 in the sense that [GRAPHICS] It was observed in [17,11] that if the exponent 2 is replaced by a value less than 2 then classical bifurcation theory can be used to treat the problem. The paper [17] deals with the case f (lambda, x, s, t) = lambda sin s whereas [11] covers the more general form f (lambda, x, s, t) = lambda F(s). Here we admit a much broader class of nonlinearities and some new phenomena appear. In particular, we encounter situations where bifurcation does not occur at a simple eigenvalue of the linearization lying below the essential spectrum, (C) 2015 Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2016

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