mathematics

Regular homotopy

Regular homotopy 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 Regular homotopy rather than just read about it. In short: In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.

Key takeaways

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

Reference excerpt

In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions f , g : M → N {\displaystyle f,g:M\to N} are homotopic if they represent points in the same path-components of the mapping space C ( M , N ) {\displaystyle C(M,N)} , given the compact-open topology. The space of immersions is the subset of C ( M , N ) {\displaystyle C(M,N)} consisting of immersions, denoted by Imm ⁡ ( M , N ) {\displaystyle \operatorname {Imm} (M,N)} , given the C 1 {\displaystyle C^{1}} topology. Two immersions f , g : M → N {\displaystyle f,g:M\to N} are regularly homotopic if they represent points in the same path-component of Imm ⁡ ( M , N ) {\displaystyle \operatorname {Imm} (M,N)} .

Examples Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if their Gauss maps have the same degree/winding number.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Regular homotopy

Start with the simplest possible case. Write down what Regular homotopy 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 Regular homotopy 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 Regular homotopy 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 Regular homotopy

In research: Regular homotopy 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 Regular homotopy 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: Regular homotopy is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic topology, Differential topology, so understanding it makes those chapters shorter.
In everyday life: Look for Regular homotopy 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Regular homotopy in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Regular homotopy 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 Regular homotopy out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Regular homotopy in simple terms?

In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions.

Why does Regular homotopy 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 Regular homotopy?

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 Regular homotopy.