Lusternik–Schnirelmann category: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 12:33, 9 August 2019

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space $X$ is the homotopy invariant defined to be the smallest integer number $k$ such that there is an open covering $\{U_{i}\}_{1\leq i\leq k}$ of $X$ with the property that each inclusion map $U_{i}\hookrightarrow X$ is nullhomotopic. For example, if $X$ 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, 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 $X$ a manifold, the lower bound for the number of critical points 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).

The invariant has been generalized in several different directions (group actions, foliations, simplicial complexes, etc.).

References

Ralph H. Fox, On the Lusternik-Schnirelmann category, Annals of Mathematics 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 MR 0339147
Ioan James, On category, in the sense of Lusternik-Schnirelmann, Topology 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 ISBN 0-8218-3404-5

@@ Line 1: / Line 1: @@
-In [[mathematics]], the '''Lyusternik–Schnirelmann category''' of a [[topological space]] ''X'' is the [[topological invariant]] defined as the smallest [[cardinality]] of an [[open covering]] of ''X'' by [[contractible]] subsets. For example, if ''X'' is the [[circle]], this takes the value two.
+In [[mathematics]], the '''Lyusternik–Schnirelmann category''' (or, '''Lusternik–Schnirelmann category''', '''LS-category''') of a [[topological space]] <math>X</math> is the [[homotopy|homotopy invariant]] defined to be the smallest integer  number <math>k</math> such that there is an [[open covering]] <math>\{U_i\}_{1\leq i\leq k}</math> of <math>X</math> with the property that each  [[inclusion map]] <math>U_i\hookrightarrow X</math> is [[nullhomotopic]]. For example, if <math>X</math> 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 so easy to compute this invariant, which was initially introduced by [[Lazar Lyusternik]] and [[Lev Schnirelmann]] in connection with [[Calculus of variations|variational problems]]. It has a close connection with [[algebraic topology]], in particular [[cup-length]]. It was, as originally defined for the case of ''X'' a [[manifold]], the lower bound for the number of [[critical point]]s a [[Morse function]] on ''X'' could possess (cf. [[Morse theory]]).
+In general it is not easy to compute this invariant, which was initially introduced by [[Lazar Lyusternik]] and [[Lev Schnirelmann]] in connection with [[Calculus of variations|variational problems]]. It has a close connection with [[algebraic topology]], in particular [[cohomology ring|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 <math>X</math> a [[manifold]], the lower bound for the number of [[critical point (mathematics)|critical point]]s that a real-valued function on <math>X</math> 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).
-==References==
-R.H. Fox, ''On the Lusternik-Schnirelmann category'', Annals of Math. (1941), 333-370.
+The invariant has been generalized in several different directions (group actions, [[foliation]]s, [[simplicial complexes]], etc.).
-I.M. James, ''On category, in the sense of Lusternik-Schnirelmann'', Topology 17 (1978), 331-348.
+==See also==
+* [[Ganea conjecture]]
+* [[Systolic category]]
+==References==
-F. Takens, ''The minimal number of critical points of a function on compact manifolds and the Lusternik-Schnirelmann category'', Inventiones math. 6, (1968), 197-244.
+* [[Ralph Fox|Ralph H. Fox]], [https://www.jstor.org/stable/1968905 ''On the Lusternik-Schnirelmann category''], [[Annals of Mathematics]] '''42''' (1941), 333–370.
+* [[Floris Takens]], ''[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002087367 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.&nbsp;13 &ndash; 22    {{MathSciNet| id=0339147}}
+* [[Ioan James]], [https://dx.doi.org/10.1016/0040-9383(78)90002-2 ''On category, in the sense of Lusternik-Schnirelmann''], [[Topology (journal)|Topology]] '''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  {{ISBN|0-8218-3404-5}}
+{{DEFAULTSORT:Lusternik-Schnirelmann category}}
 [[Category:Algebraic topology]]
 [[Category:Morse theory]]