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Map Germ


Consider the local behavior of a map f:R^m->R^n by choosing a point x in R^m and an open neighborhood U subset R^m such that x in U. Now consider the set of all mappings f:U->R^n. It is possible to put these mappings into categories by introducing an equivalence relation. Given two mappings f_1:U_1->R^n and f_2:U_2->R^n, write f_1∼f_2 provided there exists a neighborhood U of x such that U subset= U_1 and U subset= U_2 and the restriction f_1|U coincides with f_2|U. These equivalence classes are called map germs and members are called representatives of the germ. It follows from this that f_1(x)=f_2(x), hence it is common to write f:(R^m,x)->(R^n,y) for the germ where f_i(x)=y.

Consider a map germ f:(R^m,0)->(R^n,0), which is an equivalence class of maps agreeing in a small neighborhood of the origin. The group of germs of diffeomorphisms phi:(R^m,0)->(R^m,0) is denoted R, whereas the psi:(R^n,0)->(R^n,0) is denoted L. These give coordinate changes in the source and target respectively.

Denote the space of all analytic map germs (R^m,0)->(R^n,0) by A(m,n). The group R×L=A acts on A(m,n) in a standard way. Let phi in R, psi in L, and f in A(m,n). Then (phi,psi)·f=psi degreesf degreesphi^(-1).

MapGerm

The orbit of f under this action is

 Orb(f)={(phi,psi)·f:phi in R and psi in L}.

If g in A(m,n) and g in Orb(f), then g=psi degreesf degreesphi^(-1) for some phi and psi. This is the same as g degreesphi=psi degreesf and means the above diagram commutes. In this case, f and g are said to be A-equivalent as map germs.


See also

Germ

This entry contributed by Declan Davis

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Cite this as:

Davis, Declan. "Map Germ." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MapGerm.html

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