In mathematics, -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 1970s 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 -equivalent if ƒ−1(0) and g−1(0) are diffeomorphic.
Two map germs are -equivalent if there is a diffeomorphism
of the form Ψ(x,y) = (φ(x),ψ(x,y)), satisfying,
- , and
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.
It is easy to see that this equivalence relation is weaker than A-equivalence, in that any pair of -equivalent map germs are necessarily -equivalent.
This modification of -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 but (that is, ). In particular, Ψ maps f−1(V) to g−1(V).
- 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 and . Memoirs Amer. Math. Soc. 50, no. 306 (1984).