In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is the
homotopy invariant defined to be the smallest integer number
such that there is an
open covering of
with the property that each
inclusion map is
nullhomotopic. 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 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, 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, 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 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,
foliations,
simplicial complexes, etc.).
See also
*
Ganea conjecture
*
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 t ...
42 (1941), 333–370.
*
Floris Takens,
The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category',
Inventiones Mathematicae 6 (1968), 197–244.
*
Tudor Ganea, ''Some problems on numerical homotopy invariants'', Lecture Notes in Math. 249 (Springer, Berlin, 1971), pp. 13 – 22
*
Ioan James''On category, in the sense of Lusternik-Schnirelmann'' Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
17 (1978), 331–348.
*
Mónica Clapp and Dieter Puppe, ''Invariants of the Lusternik-Schnirelmann type and the topology of critical sets'',
Transactions of the American Mathematical Society 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, Providence, RI, 2003
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Algebraic topology
Morse theory