In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is the
homotopy invariant defined to be the smallest integer number
such that there is an
open covering
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
of
with the property that each
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
is
nullhomotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
. For example, if
is a sphere, this takes the value two.
Sometimes a different normalization of the invariant is adopted, which is one less than the definition above. Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).
In general it is not easy to compute this invariant, which was initially introduced by
Lazar Lyusternik and
Lev Schnirelmann
Lev Genrikhovich Schnirelmann (also Shnirelman, Shnirel'man; ; 2 January 1905 – 24 September 1938) was a Soviet mathematician who worked on number theory, topology and differential geometry.
Work
Schnirelmann sought to prove Goldbach's conjec ...
in connection with
variational problems. It has a close connection with
algebraic topology
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 homeomorphism, though usually most classify ...
, in particular
cup-length. In the modern normalization, the cup-length is a lower bound for the LS-category.
It was, as originally defined for the case of
a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, the lower bound for the number of
critical points that a real-valued function on
could possess (this should be compared with the result in
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).
The invariant has been generalized in several different directions (group actions,
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
s,
simplicial complexes, etc.).
See also
*
Ganea conjecture Ganea's conjecture is a claim in algebraic topology, now disproved. It states that
: \operatorname(X \times S^n)=\operatorname(X) +1
for all n>0, where \operatorname(X) is the Lusternik–Schnirelmann category of a topological space ''X'', and ' ...
*
Systolic category
References
*
Ralph H. Fox''On the Lusternik-Schnirelmann category'' Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as th ...
42 (1941), 333–370.
*
Floris Takens
Floris Takens (12 November 1940 – 20 June 2010) was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.
Together with David Ruelle, he predicted that fluid turbulence could develop through a strange attr ...
,
The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category',
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editor ...
6 (1968), 197–244.
*
Tudor Ganea
Tudor Ganea (October 17, 1922 –August 1971) was a Romanian-American mathematician, known for his work in algebraic topology, especially homotopy theory. Ganea left Communist Romania to settle in the United States in the early 1960s. He taug ...
, ''Some problems on numerical homotopy invariants'', Lecture Notes in Math. 249 (Springer, Berlin, 1971), pp. 13 – 22
*
Ioan James
Ioan Mackenzie James FRS (born 23 May 1928) is a British mathematician working in the field of topology, particularly in homotopy theory.
Biography
James was born in Croydon, Surrey, England, and was educated at St Paul's School, London and ...
''On category, in the sense of Lusternik-Schnirelmann'' Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
17 (1978), 331–348.
*
Mónica Clapp
Mónica Alicia Clapp Jiménez Labora is a mathematician at the Universidad Nacional Autónoma de México (UNAM) known for her work in nonlinear partial differential equations and algebraic topology.
Life and work
Clapp was born in Mexico City. S ...
and Dieter Puppe, ''Invariants of the Lusternik-Schnirelmann type and the topology of critical sets'',
Transactions of the American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 p ...
298 (1986), no. 2, 603–620.
* Octav Cornea, Gregory Lupton, John Oprea, Daniel Tanré, ''Lusternik-Schnirelmann category'', Mathematical Surveys and Monographs, 103.
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...
, Providence, RI, 2003
{{DEFAULTSORT:Lusternik-Schnirelmann category
Algebraic topology
Morse theory