Singular homology

Summary

In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of the standard n-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation – mapping each n-dimensional simplex to its (n−1)-dimensional boundary – induces the singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups

Official source

https://en.wikipedia.org/wiki/Singular_homology

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (5)

Related publications

Related people (1)

Related people

Related units

Related concepts (30)

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined f

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology gro

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to

Related concepts

Related courses (4)

Related courses

Related lectures (18)

Related lectures

Related publications (5)

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

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

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