mathematics

Homotopy extension property

Homotopy extension property 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 Homotopy extension property rather than just read about it. In short: In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.

Key takeaways

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

Reference excerpt

In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.

Definition Let X {\displaystyle X\,\!} be a topological space, and let A ⊂ X {\displaystyle A\subset X} . We say that the pair ( X , A ) {\displaystyle (X,A)\,\!} has the homotopy extension property if, given a homotopy f ∙ : A → Y I {\displaystyle f_{\bullet }\colon A\rightarrow Y^{I}} and a map f ~ 0 : X → Y {\displaystyle {\tilde {f}}_{0}\colon X\rightarrow Y} such that f ~ 0 ∘ ι = f ~ 0 | A = f 0 = π 0 ∘ f ∙ , {\displaystyle {\tilde {f}}_{0}\circ \iota =\left.{\tilde {f}}_{0}\right|_{A}=f_{0}=\pi _{0}\circ f_{\bullet },} then there exists an extension of f ∙ {\displaystyle f_{\bullet }} to a homotopy f ~ ∙ : X → Y I {\displaystyle {\tilde {f}}_{\bullet }\colon X\rightarrow Y^{I}} such that f ~ ∙ ∘ ι = f ~ ∙ | A = f ∙ {\displaystyle {\tilde {f}}_{\bullet }\circ \iota =\left.{\tilde {f}}_{\bullet }\right|_{A}=f_{\bullet }} . That is, the pair ( X , A ) {\displaystyle (X,A)\,\!} has the homotopy extension property if any map G : ( ( X × { 0 } ) ∪ ( A × I ) ) → Y {\displaystyle G\colon ((X\times \{0\})\cup (A\times I))\rightarrow Y} can be extended to a map G ′ : X × I → Y {\displaystyle G'\colon X\times I\rightarrow Y} (i.e. G {\displaystyle G\,\!} and G ′ {\displaystyle G'\,\!} agree on their common domain). If the pair has this property only for a certain codomain Y {\displaystyle Y\,\!} , we say that ( X , A ) {\displaystyle (X,A)\,\!} has the homotopy extension property with respect to Y {\displaystyle Y\,\!} .

Visualisation The homotopy extension property is depicted in the following diagram

If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map f ~ ∙ {\displaystyle {\tilde {f}}_{\bullet }} which makes the diagram commute. By currying, note that homotopies expressed as maps f ~ ∙ : X → Y I {\displaystyle {\tilde {f}}_{\bullet }\colon X\to Y^{I}} are in natural bijection with expressions as maps f ~ ∙ : X × I → Y {\displaystyle {\tilde {f}}_{\bullet }\colon X\times I\to Y} . Note that this diagram is dual to (opposite to) that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Homotopy extension property

Start with the simplest possible case. Write down what Homotopy extension property 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 Homotopy extension property 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 Homotopy extension property 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 Homotopy extension property

In research: Homotopy extension property 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 Homotopy extension property 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: Homotopy extension property is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic topology, Homotopy theory, so understanding it makes those chapters shorter.
In everyday life: Look for Homotopy extension property 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 Homotopy extension property in 20 minutes

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

Frequently asked questions

What is Homotopy extension property in simple terms?

Why does Homotopy extension property 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 Homotopy extension property?

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 Homotopy extension property.