In mathematics,

\mathcal

-equivalence, or contact equivalence, is an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

between map germs. It was introduced by John Mather in his seminal work in

Singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

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

\scriptstyle\mathcal

-equivalent if ''ƒ''⁻¹(0) and ''g''⁻¹(0) are

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...

Definition

Two map germs

f,g:X \to (Y,0)

are

\scriptstyle\mathcal

-equivalent if there is a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...

\Psi: X \times Y \to X\times Y

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

\Psi(x,0) = (\varphi(x), 0)

, and :

\Psi(x,f(x)) = (\varphi(x), g(\varphi(x)))

. In other words, Ψ 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 solution of equations, and finds many applications in

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

and

bifurcation theory Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations ...

, for example. It is easy to see that this equivalence relation is ''weaker'' than

A-equivalence In mathematics, \mathcal-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let M and N be two manifolds, and let f, g : (M,x) \to (N,y) be two smooth map germs. We say that f and g are \mathc ...

, in that any pair of

\scriptstyle\mathcal

-equivalent map germs are necessarily

\scriptstyle\mathcal

-equivalent.

K_V-equivalence

This modification of

\scriptstyle\mathcal

-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\times\

but

X\times V

(that is,

y\in V \Rightarrow \psi(x,y)\in V

). In particular, Ψ maps ''f''⁻¹(V) to ''g''⁻¹(V).

References

* J. Martinet, ''Singularities of Smooth Functions and Maps'', Volume 58 of LMS Lecture Note Series. Cambridge University Press, 1982. * J. Damon, ''The Unfolding and Determinacy Theorems for Subgroups of

\scriptstyle

and

\scriptstyle\mathcal{K}

''. Memoirs Amer. Math. Soc. 50, no. 306 (1984). Functions and mappings Singularity theory Equivalence (mathematics)

Definition

KV-equivalence

See also

References

K_V-equivalence