Lyusternik–Schnirelmann category
Encyclopedia
In mathematics
, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category, or simply, category) of a topological space
is the topological invariant defined to be the smallest cardinality of an index set
such that there is an open covering 
of 
with the property that for each 
, the inclusion map
is nullhomotopic. For example, if 
is the circle
, this takes the value two.
Recently a different normalization of the invariant has been adopted, which is one less than the original definition by Lusternik and Schnirelmann. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).
In general it is not so easy to compute this invariant, which was initially introduced by Lazar Lyusternik
and Lev Schnirelmann
in connection with variational problems
. It has a close connection with algebraic topology
, in particular cup-length. In the modern normalization, the cup-length is a lower bound for LS category.
It was, as originally defined for the case of X a manifold
, the lower bound for the number of critical point
s that a real-valued function on X could possess (this should be compared with the result in Morse theory
that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category, or simply, category) of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
is the topological invariant defined to be the smallest cardinality of an index setIndex set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
such that there is an open covering of with the property that for each , the inclusion mapInclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
is nullhomotopic. For example, if is the circleCircle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, this takes the value two.
Recently a different normalization of the invariant has been adopted, which is one less than the original definition by Lusternik and Schnirelmann. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).
In general it is not so easy to compute this invariant, which was initially introduced by Lazar Lyusternik
Lazar Lyusternik
Lazar Aronovich Lyusternik was a Soviet mathematician....
and Lev Schnirelmann
Lev Schnirelmann
Lev Genrikhovich Schnirelmann , also Shnirelman, Shnirel'man was a Soviet mathematician who sought to prove Goldbach's conjecture...
in connection with variational problems
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. It has a close connection with algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, in particular cup-length. In the modern normalization, the cup-length is a lower bound for LS category.
It was, as originally defined for the case of X a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, the lower bound for the number of critical point
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
s that a real-valued function on X could possess (this should be compared with the result in Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).