Germ (mathematics)


Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the sets or maps in question will have specific properties, such as being analytic or smooth, but in general this is not needed (the maps or functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local have some sense.

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

Contents

Formal definition

Basic definition

Given a point x of a topological space X, and two maps f, g : X → Y (where Y is any set), then f and g define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal; meaning that f(x) = g(x) for all x in U. Similarly, if S and T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that

S\,\cap \,U = T \,\cap\, U.

It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written \scriptstyle f \sim_x g or \scriptstyle S \sim_x T.

Given a map f on X, then its germ at x is usually denoted [f ]x. Similarly, the germ at x of a set S is written [S]x. Thus,

[f]_x = \{g:X\to Y \mid g \sim_x f\}.

To denote a map germ at x in X which maps the point x in X to the point y in Y, one writes

f:(X,x) \to (Y,y),

such an f is then an entire equivalence class of maps, and it is usual to use the same letter f for any representative map.

Notice that two sets are germ-equivalent at x if and only if their characteristic functions are germ-equivalent at x:

S\sim_x T \; \Longleftrightarrow \; \mathbf{1}_S \sim_x \mathbf{1}_T.

More generally

Maps need not be defined on all of X, and in particular they don't need to have the same domain. However, if f has domain S and g has domain T, both subsets of X, then f and g are germ equivalent at x in X if first S and T are germ equivalent at x, say \scriptstyle S\,\cap\,U\;=\;T\,\cap\,U, and then moreover \scriptstyle f|_{S\cap V} = g|_{T\cap V}, for some smaller neighbourhood V with \scriptstyle x\in V \subset U. This is particularly relevant in two settings:

  1. f is defined on a subvariety V of X, and
  2. f has a pole of some sort at x, so is not even defined at x, as for example a rational function, which would be defined off a subvariety.

Basic properties

If f and g are germ equivalent at x, then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc. Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x.

Moreover, if the target Y is a vector space, then it makes sense to add germs: to define [f]x + [g]x, first take representatives f and g, defined on neighbourhoods U and V respectively, then [f]x + [g]x is the germ at x of the map f + g (where f + g is defined on \scriptstyle U\cap V). (In the same way one can define more general linear combinations.)

The set of germs at x of maps from X to Y does not have a useful topology, except for the discrete one. It therefore makes little or no sense to talk of a convergent sequence of germs. However, if X and Y are manifolds, then the spaces of jets \scriptstyle J_x^k(X,Y) (finite order Taylor series at x of map(-germs)) do have topologies as they can be identified with finite-dimensional vector spaces.

Relation with sheaves

The idea of germ is behind the definition of sheaves and presheaves. A presheaf \scriptstyle\mathcal{F} on a topological space X is an assignment of an Abelian group \scriptstyle\mathcal{F}(U) to each open set U in X. Typical examples of Abelian groups here are: real valued functions on U, differential forms on U, vector fields on U, holomorphic functions on U (when X is a complex space), constant functions on U and differential operators on U.

If \scriptstyle V \subset U then there is a restriction map \scriptstyle\mathrm{res}_{VU}:\mathcal{F}(U)\to \mathcal{F}(V), satisfying certain compatibility conditions. For a fixed x, one says that elements \scriptstyle f\in\mathcal{F}(U) and \scriptstyle g\in \mathcal{F}(V) are equivalent at x if there is a neighbourhood \scriptstyle W\subset U\cap V of x with resWU(f) = resWV(g) (both elements of \scriptstyle\mathcal{F}(W)). The equivalence classes form the stalk \scriptstyle\mathcal{F}_x at x of the presheaf \scriptstyle\mathcal{F}. This equivalence relation is an abstraction of the germ equivalence described above.

Examples

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

  • If X,Y are both topological spaces, the subset
C^0(X,Y) \subset \mbox{Hom}(X,Y)\,
of continuous functions defines germs of continuous functions.
  • If both X and Y admit a differentiable structure, the subset
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 a complex structure (for instance, are subsets of complex vector spaces), holomorphic functions between them can be defined, and therefore spaces of germs of holomorphic 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.

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.

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.

See also

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 Narasimhan (1973). Analysis on Real and Complex Manifolds (2nd ed. 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".
  • Ian R. Porteous (2001) Geometric Differentiation, page 71, Cambridge University Press ISBN 0-521-00264-8 .
  • 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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