K-equivalence

Key takeaways

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

Reference excerpt

In mathematics, K {\displaystyle {\mathcal {K}}} -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technical tool for studying stable maps. Since then it has proved important in its own right. Roughly speaking, two map germs ƒ, g are K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent if ƒ−1(0) and g−1(0) are diffeomorphic.

Definition Two map germs f , g : X → ( Y , 0 ) {\displaystyle f,g:X\to (Y,0)} are K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent if there is a diffeomorphism

Ψ : X × Y → X × Y {\displaystyle \Psi :X\times Y\to X\times Y}

of the form Ψ(x,y) = (φ(x),ψ(x,y)), satisfying,

Ψ ( x , 0 ) = ( φ ( x ) , 0 ) {\displaystyle \Psi (x,0)=(\varphi (x),0)} , and

Ψ ( x , f ( x ) ) = ( φ ( x ) , g ( φ ( x ) ) ) {\displaystyle \Psi (x,f(x))=(\varphi (x),g(\varphi (x)))} . In other words, &Psi. Maps the graph of f to the graph of g, as well as the graph of the zero map to itself. In particular, the diffeomorphism φ maps f−1(0) to g−1(0). The name contact is explained by the fact that this equivalence is measuring the contact between the graph of f and the graph of the zero map. Contact equivalence is the appropriate equivalence relation for studying the sets of solutions of equations, and finds many applications in dynamical systems and bifurcation theory, for example. It is easy to see that this equivalence relation is weaker than A-equivalence, in that any pair of A {\displaystyle \scriptstyle {\mathcal {A}}} -equivalent map germs are necessarily K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalent.

KV-equivalence This modification of K {\displaystyle \scriptstyle {\mathcal {K}}} -equivalence was introduced by James Damon in the 1980s. Here V is a subset (or subvariety) of Y, and the diffeomorphism Ψ above is required to preserve not X × { 0 } {\displaystyle X\times \{0\}} but X × V {\displaystyle X\times V} (that is, y ∈ V ⇒ ψ ( x , y ) ∈ V {\displaystyle y\in V\Rightarrow \psi (x,y)\in V} ). In particular, Ψ maps f−1(V) to g−1(V).

Worked examples

Example 1 — a first encounter with K-equivalence

Start with the simplest possible case. Write down what K-equivalence 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 K-equivalence 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 K-equivalence 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 K-equivalence

In research: K-equivalence 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 K-equivalence 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: K-equivalence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Equivalence (mathematics), Functions and mappings, Singularity theory, so understanding it makes those chapters shorter.
In everyday life: Look for K-equivalence 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 K-equivalence in 20 minutes

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

Frequently asked questions

What is K-equivalence in simple terms?

In mathematics, K {\displaystyle {\mathcal {K}}} -equivalence, or contact equivalence, is an equivalence relation between map germs. It was introduced by John Mather in his seminal work in Singularity theory in the 1960s as a technical tool for studying stable maps.

Why does K-equivalence 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 K-equivalence?

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 K-equivalence.