Retraction (topology)

Key takeaways

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

Reference excerpt

In topology, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace. An absolute neighborhood retract (ANR) is a particularly well-behaved type of topological space. For example, every topological manifold is an ANR. Every ANR has the homotopy type of a very simple topological space, a CW complex.

Definitions

Retract Let X be a topological space and A a subspace of X. Then a continuous map

r : X → A {\displaystyle r\colon X\to A}

is a retraction if the restriction of r to A is the identity map on A; that is, r ( a ) = a {\textstyle r(a)=a} for all a in A. Equivalently, denoting by

ι : A ↪ X {\displaystyle \iota \colon A\hookrightarrow X}

the inclusion, a retraction is a continuous map r such that

r ∘ ι = id A , {\displaystyle r\circ \iota =\operatorname {id} _{A},}

that is, the composition of r with the inclusion is the identity of A. Note that, by definition, a retraction maps X onto A. A subspace A is called a retract of X if such a retraction exists. For instance, any non-empty space retracts to a point in the obvious way (any constant map yields a retraction). If X is Hausdorff, then A must be a closed subset of X. If r : X → A {\textstyle r:X\to A} is a retraction, then the composition ι ∘ r {\displaystyle \iota \circ r} is an idempotent continuous map from X to X. Conversely, given any idempotent continuous map s : X → X , {\textstyle s:X\to X,} we obtain a retraction onto the image of s by restricting the codomain.

Deformation retract and strong deformation retract A continuous map

F : X × [ 0 , 1 ] → X {\displaystyle F\colon X\times [0,1]\to X}

is a deformation retraction of a space X onto a subspace A if, for every x in X and a in A,

F ( x , 0 ) = x , F ( x , 1 ) ∈ A , and F ( a , 1 ) = a . {\displaystyle F(x,0)=x,\quad F(x,1)\in A,\quad {\mbox{and}}\quad F(a,1)=a.}

In other words, a deformation retraction is a homotopy between a retraction (strictly, between its composition with the inclusion) and the identity map on X. The subspace A is called a deformation retract of X. A deformation retraction is a special case of a homotopy equivalence. A retract need not be a deformation retract. For instance, having a single point as a deformation retract of a space X would imply that X is path connected (and in fact that X is contractible). Note: An equivalent definition of deformation retraction is the following. A continuous map r : X → A {\textstyle r:X\to A} is itself called a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on X. In this language, a deformation retraction still carries with it a homotopy between the identity map on X and itself, but we refer to the map r {\textstyle r} rather than the homotopy as a deformation retraction. If, in the definition of a deformation retraction, we add the requirement that

F ( a , t ) = a {\displaystyle F(a,t)=a}

for all t in [0, 1] and a in A, then F is called a strong deformation retraction. In other words, a strong deformation retraction leaves points in A fixed throughout the homotopy. (Some authors, such as Hatcher, take this as the definition of deformation retraction.) As an example, the n-sphere S n {\textstyle S^{n}} is a strong deformation retract of R n + 1 ∖ { 0 } ; {\textstyle \mathbb {R} ^{n+1}\backslash \{0\};} as strong deformation retraction one can choose the map

F ( x , t ) = ( 1 − t ) x + t x ‖ x ‖ . {\displaystyle F(x,t)=(1-t)x+t{x \over \|x\|}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Retraction (topology)

Start with the simplest possible case. Write down what Retraction (topology) 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 Retraction (topology) 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 Retraction (topology) 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 Retraction (topology)

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

How to study Retraction (topology) in 20 minutes

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

Frequently asked questions

What is Retraction (topology) in simple terms?

In topology, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space.

Why does Retraction (topology) 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 Retraction (topology)?

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 Retraction (topology).