mathematics

Lusternik–Schnirelmann category

Lusternik–Schnirelmann category is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lusternik–Schnirelmann category rather than just read about it. In short: In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X {\displaystyle X} is the homotopy invariant defined to be the smallest integer number k {\displaystyle k} such that there is an open covering { U i } 1 ≤ i ≤ k {\displaystyle \{U_{i}\}_{1\leq i\leq k}} of X {\displaystyle X} with the property that each inclusion map U i ↪ X {\displaystyle…

Key takeaways

Lusternik–Schnirelmann category belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lusternik–Schnirelmann category to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lusternik–Schnirelmann category from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X {\displaystyle X} is the homotopy invariant defined to be the smallest integer number k {\displaystyle k} such that there is an open covering { U i } 1 ≤ i ≤ k {\displaystyle \{U_{i}\}_{1\leq i\leq k}} of X {\displaystyle X} with the property that each inclusion map U i ↪ X {\displaystyle U_{i}\hookrightarrow X} is nullhomotopic. For example, if X {\displaystyle X} is an n {\displaystyle n} -sphere, this takes the value 2 {\displaystyle 2} . Lusternik–Schnirelmann theorem implies that the LS-category of ⁠ R P n {\displaystyle \mathbb {RP} ^{n}} ⁠ is at least ⁠ n + 1 {\displaystyle n+1} ⁠ (and it is not difficult to show that ⁠ n + 1 {\displaystyle n+1} ⁠ sets suffice, so the category is exactly ⁠ n + 1 {\displaystyle n+1} ⁠). 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 {\displaystyle X} a manifold, the lower bound for the number of critical points that a real-valued function on X {\displaystyle 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.).

See also Ganea conjecture Systolic category

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

Worked examples

Example 1 — a first encounter with Lusternik–Schnirelmann category

Start with the simplest possible case. Write down what Lusternik–Schnirelmann category claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lusternik–Schnirelmann category before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lusternik–Schnirelmann category ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lusternik–Schnirelmann category

In research: Lusternik–Schnirelmann category appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lusternik–Schnirelmann category in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lusternik–Schnirelmann category is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic topology, Morse theory, so understanding it makes those chapters shorter.
In everyday life: Look for Lusternik–Schnirelmann category outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lusternik–Schnirelmann category in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lusternik–Schnirelmann category means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lusternik–Schnirelmann category out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lusternik–Schnirelmann category in simple terms?

Why does Lusternik–Schnirelmann category matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lusternik–Schnirelmann category?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lusternik–Schnirelmann category.