Germ (mathematics)

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In mathematics, a germ of (continuous, differentiable or analytic) functions is an equivalence class of (continuous, differentiable or analytic) functions from a topological space to another (often from the real line to itself), grouped together on the basis of their equality on the neighborhood of a fixed reference point in their domain of definition. In the same way, a germ of sets is an equivalence class of subsets of a given topological space, grouped together on the basis of their equality on the neighborhood of a fixed reference point belonging to all of them.

The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is the "heart" of a function, as it is for a grain.

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[edit] Formal definition

[edit] Basic definition

Two functions f and g between the same topological space X and a set Y are said to be equivalent near a point x in their domain, if there is some open neighborhood U of x in X on which they agree, i.e.

f(x') = g(x') \quad \forall x'\in U\iff f|_U=g|_U \iff f \sim_x g

This is an equivalence relation on the space YX = Hom(X,Y) of maps between X and Y. For the proof, it is sufficient to note that equality is used in its definition: then reflexivity and symmetry are immediate consequences. For transitivity, given functions f,g,h such that f = g on U and g = h on V, then f = g = h on U ∩ V.

The equivalence classes have the following form

[f]_x=\left\{g \in Y^X\mid f \sim_x g\right\}\,

Then the space of germs of functions at x ∈ X, is the quotient set

\Gamma_x=Y^X/\!\!\sim_x\;=\left\{[f]_x \mid f\in Y^X\right\}\,

As it can be easily seen, the germ at x is the stalk of the sheaf of functions at x.

The basic definition of germ does not require a topology on the codomain: a topology is only necessary to define neighborhoods of points in the domain. Similarly, it does not require any continuity or smoothness condition on the functions. With even more generality, functions can only be defined on a neighborhood U of the given point \scriptstyle x \in X and need not be restrictions of globally defined functions, so only a presheaf is needed to define germs. Since the stalks of a presheaf agree with the stalks of its sheafification (indeed, a construction of sheafification uses the sheaf of stalks of the given presheaf), the wording stalks of a sheaf is commonly used.

[edit] Germs of various classes of functions

If X and Y have additional structure, it is possible to define subsets of YX, or more generally sub-presheaves of a given presheaf \scriptstyle\mathcal{F} and corresponding germs: some notable examples follow.

C^0(X,Y) \subset \mbox{Hom}(X,Y)\,
of continuous functions defines germs of continuous functions.
C^k(X,Y) \subset \mbox{Hom}(X,Y)\,
of k-times continuously differentiable functions, the subset
C^\infty(X,Y)=\bigcap_k C^k(X,Y)\subset \mbox{Hom}(X,Y)\,
of smooth functions and the subset
C^\omega(X,Y)\subset \mbox{Hom}(X,Y)
of analytic functions can be defined (ω here is the ordinal for infinity; this is an abuse of notation, by analogy with Ck and C∞), and then spaces of germs of (finitely) differentiable, smooth, analytic functions can be constructed.
  • If X,Y have an algebraic structure, then regular (and rational) functions between them can be defined, and germs of regular functions (and likewise rational) can be defined.

[edit] Germs of various classes of sets

Two subsets V,W of a topological space X are said to be equivalent near a point x belonging to them if there is some open neighborhood U of x in X such that

U \cap V = U \cap W \iff V \cong_x\! W

This is equivalent to say that the germs of the characteristic functions of the two subsets are equal, i.e.

V \cong_x\! W \iff \chi_V \sim_x \chi_W

In more abstract terms, the contravariant functor which maps a set to its power set is representable by the set Y = {0,1}, i.e. is naturally isomorphic to the hom functor Hom( − ,Y). Consequently, for a given set X, it is possible to alternatively analyze or its subsets or functions belonging to Hom(X,Y) where Y = {0,1}, since those objects are isomorphic.

Notably, if x is in the interior of both V and W, then they are equivalent near x.

This is an equivalence relation on the power set of the topological space X, to which V and W both belong: the equivalence classes have the following form

V_x=\left\{W \in \mathcal{P}(X)|\, V \cong_x\! W\right\}\,

Then the space of germs of sets at x in X is the quotient set

X_x=X/\!\!\cong_x\;=\left\{V_x|V\in \mathcal{P}(X)\right\}\,\,

Various spaces of germs of sets can be defined in the same way as it can be done for germs of functions. However, the space of germs of a variety is the most frequently encountered in standard mathematical research: the subset of the power set of the topological space X used in its construction is the class of analytic varieties.

[edit] Notation

The stalk of a sheaf \scriptstyle\mathcal{F} on a topological space X at a point x of X is commonly denoted by \scriptstyle\mathcal{F}_x . As a consequence germs, being stalks of sheaves of various kind of functions, borrow this scheme of notation:

  • \mathcal{C}_x^0\, is the space of germs of continuous functions at x.
  • \mathcal{C}_x^k\, for each natural number k is the space of germs of k-times-differentiable functions at x.
  • \mathcal{C}_x^\infty\, is the space of germs of infinitely differentiable ("smooth") functions at x.
  • \mathcal{C}_x^\omega\, is the space of germs of analytic functions at x.
  • \mathcal{O}_x\, is the space of germs of holomorphic functions (in complex geometry), or space of germs of regular functions (in algebraic geometry) at x.

For germs of sets and varieties, the notation is not so well established: some notations found in literature include:

  • \mathfrak{V}_x\, is the space of germs of analytic varieties at x.

When the point x is fixed and known (e.g. when X is a topological vector space and x = 0), it can be dropped in each of the above symbols: also, when dimX = n, a subscript before the symbol can be added. As example

  • {_n\mathcal{C}^0},{_n\mathcal{C}^k},{_n\mathcal{C}^\infty},{_n\mathcal{C}^\omega},{_n\mathcal{O}},{_n\mathfrak{V}}\, are the spaces of germs shown above when X is a n-dimensional vector space and x = 0.

[edit] Applications

The key word in the applications of germs is locality: all local properties of a function at a point can be studied analyzing its germ. They are a generalization of Taylor series, and indeed the Taylor series of a germ (of a differentiable function) is defined: you only need local information to compute derivatives.

Germs are useful in determining the properties of dynamical systems near chosen points of their phase space: they are one of the main tools in singularity theory and catastrophe theory.

When the topological spaces considered are Riemann surfaces or more generally analytic varieties, germs of holomorphic functions on them can be viewed as power series, and thus the set of germs can be considered to be the analytic continuation of an analytic function.

[edit] See also

[edit] References

  • Nicolas Bourbaki (1989). General Topology. Chapters 1-4, paperback ed., Springer-Verlag. ISBN 3-540-64241-2. , chapter I, paragraph 6, subparagraph 10 "Germs at a point".
  • Raghavan Narsimhan (1973). Analysis on Real and Complex Manifolds, 2nd ed., North-Holland Elsevier. ISBN 0-7204-2501-8. , chapter 2, paragraph 2.1, "Basic Definitions".
  • Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall. , chapter 2 "Local Rings of Holomorphic Functions", especially paragraph A "The Elementary Properties of the Local Rings" and paragraph E "Germs of Varieties".
  • Giuseppe Tallini (1973). Varietà differenziabili e coomologia di De Rham (Differentiable manifolds and De Rham cohomology). Edizioni Cremonese. ISBN 88-7083413-1. , paragraph 31, "Germi di funzioni differenziabili in un punto P di Vn (Germs of differentiable functions at a point P of Vn)" (in Italian).

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