Among all the perfect covering surfaces which belong to a given surface there is one which is the “strongest.”²⁰ It is characterized by the statement: a curve on it is closed only if all curves on all perfect covering surfaces of which have the same trace as are closed. As a consequence of this property, the “universal covering surface” is normal; the group of its cover transformations, regarded as an abstract group, the so-called fundamental group of , is an analysis situs invariant of the base surface is simply connected. For if were a more than one-sheeted perfect covering surface over , then one could draw an unclosed curve on whose trace on would be closed. Since is also a perfect covering surface over , this would immediately contradict the characteristic property of .

The universal covering surface may be defined as follows. If is a fixed point of then every curve γ starting at defines a “point of which we say lies over the endpoint of γ. Two such curves γ, γ′ define the same point of if and only if on every perfect covering surface over every pair of curves which start at the same point and have traces γ, γ′ end at the same point. Let γ₀ be a curve on from to which defines the point on , and let be a neighborhood of on . If I attach to γ₀ all possible curves γ in which start at , then I say that the points of defined by all these curves γ₀ + γ form a “neighborhood” of . Since is simply connected, there is just one point of over each point of ; hence our concept of “neighborhood” satisfies the conditions stated in § 4.

The significance of covering surfaces for complex function theory results from the fact that every perfect covering surface of a Riemann surface is itself a Riemann surface without more ado. For let t() be a local parameter at a point of and assign to each point of the covering surface over p the same function value t(); thus one obtains a local parameter at each point of over Similarly from every function on , free of essential singularities, one obtains a function of the same character on Every function on an arbitrary perfect covering surface reappears as a uniform function on the universal covering surface ; so a study of the universal covering surface is capable of replacing, to some extent, the study of all other not so strong covering surfaces. Among the functions on , those which are functions f on the base surface are characterized by the property that they remain invariant under the group of cover transformations of ; that is, they satisfy the identity

where T : → T is any of these cover transformations.

The function theoretic exploitation of the simple connectivity of a surface rests on the following monodromy theorem.

If is a simply connected surface, if

is a function element on at , and if in the analytic continuation of z along arbitrary paths on one never meets critical points other than ordinary poles, then these continuations form a uniform (single-valued) function on , regular analytic except for poles.

Proof. One can define a perfect covering surface over in a manner analogous to the definition of the universal covering surface. A curve on this covering surface, whose trace on runs from to , shall be closed if and only if the continuation of the function element z along the trace leads back to the initial element. Because of the assumed simple connectivity of this covering surface must be one sheeted. Thus the proof of the monodromy theorem is already complete.

§ 9.    Differentials and line integrals. Homology

The Cauchy integral theorem is a special case of the monodromy theorem. To formulate the theorem generally we must discuss “differentials” on a Riemann surface. While a “function” is characterized by the fact that it has a definite value at each point of its domain of definition, a differential dz has no intrinsic value at a point ; rather, dz has a definite (complex) value, only in relation to the differential dt of any local parameter t at . If t and τ are local parameters at the same point , then

must always hold. A regular analytic function z in a domain has a differential dz for which

where is any point of and t is any local parameter at . Also, a harmonic function u gives rise to a differential dw, according to the formula

Multiplication of a differential dz by a function f yields a new differential dZ:

If at a point , for one local parameter t at , then this is the case for every local parameter at ; is then a zero of dz. If dZ and dz are two differentials defined in the same domain and if dz has no zeros, then f = dZ/dz is a function in ; for

is independent of the choice of the local parameter t.

Let t be a local parameter at the point and let |t| < r be an associated neighborhood corresponding to the domain on , which contains as an interior point. Suppose that r is small enough so that dz is defined in .

At each point of let the value of dz relative to the local parameter t − t() be denoted by dz/dt. If r may be chosen small enough so that dz/dt is a regular analytic function represented by a power series in t in |t| < r, then dz is called regular analytic at . Then it is also regular analytic at all points of a sufficiently small neighborhood of , and in changing to another local parameter τ at an equation of the type (9.1) is valid not only at the origin t = 0 but in a whole neighborhood |t| < r:

Thus the concept of a regular analytic differential at a point is independent of the choice of local parameter.

If dz/dt has a zero of order m at t = 0, then we also say that the differential dz has a zero of order m (or is of order m) at . If dz/dt has a pole of order n (then dz is defined in a neighborhood of except at itself), then we say that dz has a pole of order n (or is of order −n) at . These orders are independent of the choice of the local parameter t. The same is true of the coefficient A₋₁ in the development

This is most easily seen from the equation

where is any curve in the t-plane which lies in a sufficiently small neighborhood |t| < r and winds around the origin t = 0 once in the positive sense (without passing through the origin). When one formulates the conditions on this way, instead of taking to be a circle, it is clear that the left-hand side of our equation is in fact not affected by the choice of the local parameter t. Because of this equation I prefer to call 2πi A₋₁, not A₋₁ (as is the usual custom), the residue of dz at the point .²¹ If one wishes to prove the same fact by algebraic manipulation, then one writes the new local parameter τ in the form τ = ct η(t), where c ≠ 0 and η(t) has the power series development 1 + γ₁t + . One must show that the coefficient of t⁻¹ in the development of τ^{−(v + 1)} dτ/dt in powers of t has the value 1 for v = 0 and the value 0 for v = 1, 2, …. The first follows from the equation

The second follows from the fact that for v ≥ 1, − vc^v τ^{−(v + 1)} dτ/dt is the derivative with respect to t of

and hence the term in t⁻¹ is missing.

This last remark also shows that the differential of a function which is regular except for poles is itself regular analytic except for poles and has nowhere a nonzero residue. The converse of this theorem, as far as it is correct, is the content of the Cauchy integral theorem in its general formulation.

If the differential dz is regular except for poles in a simply connected domain and if dz has nowhere a residue ≠ 0, then there exists a uniform function in , regular except for poles, whose differential coincides with the given dz throughout .

Proof.  Let be any point of and let

for sufficiently small t, where t a local parameter at . But this is the derivative of the function element

which contains an arbitrary additive constant B. Thus the problem is solved locally at each point. If we start at a fixed point of , at which dz is regular, then it follows that we can continue the associated function element (normalized by B = 0) at ,

analytically along arbitrary paths γ in , starting at , without meeting any critical points except poles. (With the aid of this analytic continuation we define the integral The existence of a uniform function of the desired sort follows from the monodromy theorem.

We switch from Riemann surfaces to arbitrary smooth surfaces. Here the linear differential form corresponds to the differential. If two continuous real-valued functions f_x(x,y) and f_y(x, y) are given in a domain of the xy-plane, then one can regard these as the components of a vector or of a linear differential form

If is mapped onto a domain by a continuously differentiable topological transformation (x, y) → then (9.3) is to remain invariant under this transformation. That is, the components and of the same vector relative to are to be determined by the equations

It is clear how the concept is to be defined locally at any point of the smooth surface . Relative to an admissible local coordinate system (x, y) at , a vector df has two real components f_x and f_y at . The components in two admissible local coordinate systems (x, y) and are related by the equations (9.4), where now naturally the four derivatives are to be evaluated at the origin:

This definition is based on the fact that with a local continuously differentiable topological map (x, y) → at the origin O there is associated a linear transformation of the “differentials” (dx, dy)

such that composition of the maps is mirrored in the composition of the corresponding linear transformations. We will assume that the vector df is given not only at the point of the surface but is defined and continuous in a neighborhood. This is to say that there exists a square neighborhood |x| < ε, |y| < ε “belonging” to the admissible local coordinate system (x, y), whose image on the surface is with the following property. If (x₀, y₀) is any point of this neighborhood, then the components f_x and f_y at , relative to the admissible local coordinates (x − x₀, y − y₀) at , are continuous functions of x₀, y₀ in the square. (Here it is somewhat more convenient to use square or rectangular neighborhoods instead of circular ones; one can of course stick to circular neighborhoods.) If is another admissible local coordinate system at , then one can choose ε so small that and are defined and continuously differentiable functions of x, y in the square and have a nonvanishing functional determinant there; then for each point (x₀, y₀) of the square, is also an admissible local coordinate system at . The equations (9.4) between the components of df in one coordinate system and in another hold at all points of .

A complex differential dz = α dt, given in terms of a local parameter t = x + iy at a point of a Riemann surface, gives rise to two real linear differential forms Namely, if one writes the complex number α, separated into real and imaginary parts, in the form α = a − ib, then

The Cauchy-Riemann differential equations show that for every linear differential form a dx + b dy defined at the origin there is another (conjugate differential form) − b dx + a dy and that the relation between these two is invariant under conformal transformations.

We return to the linear differential forms on an arbitrary smooth surface . If is defined and everywhere continuously differentiable in a domain on and if it can be expressed as the function f(x, y) in a neighborhood of , where (x, y) is an admissible local coordinate system at , then the values of the derivatives f_x = ∂f/∂x, f_y = ∂f/∂y at the origin determine a vector at which is independent of the choice of the local coordinate system. Thus one obtains from a continuous vector field in for which the notation grad has become customary. The vector fields which arise in this way from continuously differentiable functions of points are called exact.²²

With analytic differentials on Riemann surfaces in mind we direct our attention to those continuous linear differential forms df on a smooth surface (or a subdomain of ) which are locally exact. That is, we assume that each point of the domain of definition has a neighborhood in which there is a continuously differentiable function f of points such that df = grad f (in coordinates, f_x = ∂f/∂x, f_y = ∂f/∂y) in that neighborhood. Such linear differential forms may be called closed differentials. In the case of Riemann surfaces we treated the step from an analytic differential dz to its integral along a curve γ rather cursorily; here we shall discuss the integration of closed differentials on smooth surfaces with painful precision. The integral of a continuous linear differential form along a path γ can be defined in the familiar way only when γ is a continuously differentiable (or more generally, a rectifiable) curve. If, however, we have an exact vector field grad , then the integial has the same value, for all rectifiable curves joining to . Thus this value may be assigned as the value of the integral along any arbitrary continuous curve γ from to , without demanding that γ be rectifiable. By partitioning the path γ into small pieces one extends this argument from exact to closed (that is, everywhere locally exact) differentials. The detailed development of this idea goes as follows.

If f is a continuously differentiable function in the square |x| < a, |y| < a, both of whose derivatives ∂f/∂x, ∂f/∂y vanish identically, then f is constant in the whole square. This familiar fact from the elements of analysis carries over to arbitrary open sets in the xy-plane as follows.

Lemma. If f (x, y) is a given continuously differentiable function in such that both ∂f/∂x and ∂f/∂y vanish identically in and if and are two points of , joined by a curve γ in , then f() = f().

The proof follows by enclosing each point of the curve in a square neighborhood and using an associated standard subdivision of the curve. Then one finds from the introductory remark that f has the same value at any two adjacent partition points.

Let now the differential df be defined and closed on and let be a curve on . For each point λ₀ of the curve, = (λ₀), we determine a smooth local coordinate system (x, y) and an associated square neighborhood |x| < a, |y| < a in which the components f_x and f_y of df = f_x dx + f_y dy are the derivatives of ∂f/∂x and ∂f/∂y of a function f(x, y) which is continuously differentiable in the square. Let be the image of the square on the surface . One can split γ into finitely many subarcs (i = 1, …, n) and choose a point on each γ_i such that γ_i is contained completely in The notations x_i, y_i, f_i are clear without more ado. Now we define

It must be proved that the result depends only on df and γ, not on the construction used.

If one carries out the process described in a second way by splitting γ into subarcs by partition points then one obtains the new value of The meaning of and is clear. By superposition of the two partitions one obtains a finer partition of γ into arcs γ′, each of which is the intersection of a γ_i with a For the moment set x = x_i, y = y_i; The small arc γ′ lies in the intersection In that intersection and are continuously differentiable functions of x and y with a nonvanishing functional determinant; the components f_x = ∂f/∂x, f_y = ∂f/∂y and are connected by the equations (9.4) and hence

in If γ′ runs from the point to the point then it follows from the Lemma that the increment of f^* along γ′, that is, is equal to the increment of f. But the increment Δ_if_i of f_i on γ_i is equal to the sum of its increments on the finer arcs γ′ into which γ_i is split; the increment along is equal to the sum of the increments of on the finer arcs γ′ into which is split. Thus, if γ_ij denotes the intersection of γ_i and

A curve γ₁ from to and a curve γ₂ from to can be joined to form a curve γ = γ₁ + γ₂ which runs from through to . Then always

For a closed path γ the integral is independent of the point at which one starts to trace the path. The exact differentials can now be characterized as the closed differentials for which the integral vanishes along every closed path γ.

Here it is possible, not only to ascend from Riemann surfaces to arbitrary smooth surfaces, but to extend the concepts to the full generality of topological surfaces. The resulting concept is that of integral function. A function of curves, F, on the surface is defined if with each curve γ on a (real) number F(γ) is associated. The function of curves is linear if F(γ₁ + γ₂) = = F(γ¹) + F(γ²) holds. F is called exact or cohomologous to zero, F ∼ 0, if F (γ) = 0 for every closed curve γ. In this case there exists a function of points on the surface such that F is the increment function of ; that is, for any curve γ from to the value of F(γ) is the increment of . I have called a linear function of curves an integral function (the concept was introduced in the first edition of this book) if each point of the surface has a neighborhood such that F(γ) = 0 for every closed curve γ in this neighborhood. This is the topological generalization of continuous closed differentials.

An integral function F on a simply connected surface necessarily exact. For choose a fixed point on and with each curve γ from to a point of associate a point over so that two curves γ₁ and γ₂ from to determine the same point if and only if F(γ₁) = F(γ₂). The result is a perfect covering surface over . If is simply connected, this covering must be one sheeted, and that says that F(γ₁) = F(γ₂) for any two curves γ₁ and γ₂ from to . If one denotes this value by , then F is the increment function of .

More generally, we can say that the covering surface which we have just constructed on the basis of a given integral function F, on which a curve with closed trace γ is closed if and only if F(γ) = 0, is a normal covering surface and its cover transformations S form an Abelian = commutative group . In fact, as on page 58, let be a point of over the point on and let α and γ be curves on which start and end at . The cover transformation TS composed of S = S_α and T = S_γ carries into the point obtained by starting at and following over the path α + γ. But since F(α + γ) = = F(γ + α), we reach the same point with the path γ + α, and hence TS = ST.

The concept of the residue may be extended to arbitrary closed differentials on smooth surfaces, and even to arbitrary integral functions on surfaces. Let a neighborhood of the point on be mapped topologically onto the interior E of the unit circle in the xy-plane so that corresponds to the origin O. Polar coordinates are introduced by the equations

where r ≥ 0 and we have set

If the origin O is punched out of E (“delected disc” ), then one obtains the infinite parallel strip 0 < r < 1 in the rϕ>-plane as a covering surface over with branch point O: the points (r, ϕ) and (r, ϕ + n) coincide on if n is any integer. A closed curve γ in the deleted neighborhood appears in the (r, ϕ)-image as a curve from the point (r, ϕ) to a point (r, ϕ + n). The integer n indicates how many times the image curve in goes around the origin. An integral function F defined in carries over into an integral function in the parallel strip. Since the strip is simply connected, F(γ) = 0 for every curve γ which is closed, not only in but also in the strip. Thus for all closed curves γ in , which have the same winding number n, F(γ) has the same value A⁽ⁿ⁾. For if one, γ₀, runs from (r₀, ϕ₀) to (r₀, ϕ₀ + n) in the strip, and another, γ₁, runs from (r₁, ϕ₁) to (r₁, ϕ₁ + n), then join the initial points (r₀, ϕ₀) and (r₁, ϕ₁) with a linear segment and join the terminal points with a parallel segment. Applying F(γ) = 0 to the closed curve consisting of γ₀, γ₁ and these two segments we obtain F(γ₀) = F(γ₁). Also it is clear that A⁽ⁿ⁾ = nA⁽¹⁾ where A⁽¹⁾ = A is the value of F for a circle in E with center O; for the circle traced n times winds about the point O n times. A is called the residue of F at .

Now we consider integral functions which are defined for all paths γ on the surface . They form a linear space; for multiplication of an integral function by a real constant results in another integral function, and similarly with the addition of two integral functions. The integral functions F₁, …, F_l are linearly independent (“in the sense of cohomology”) if there does not exist a set of real numbers c₁, …, c_l, not all zero, such that

If any h + 1 integral functions on are linearly dependent and if there exists a set of h linearly independent integral functions, then h is the dimension of the linear family of integral functions on ; we call h the degree (degree of connectivity) of the surface . Any h linearly independent integral functions F₁, …, F_h may be used as a basis for the linear family of integral functions in the sense that any integral function F is cohomologous to one and only one linear combination of F₁, …, F_h:

Naturally it may happen that the degree of a surface is infinite. A homology

with real coefficients c_i between closed paths γ_i means that

for each integral function F on . If one introduces formally such linear combinations of closed paths

as “streams” and if one does not distinguish between homologous streams, then the streams form a linear space which is dual to that of the integral functions if the inner product of the integral function F and the stream (9.6) is the number c₁ F(γ₁) + + c_l F(γ_l). Thus on a surface of degree h, any h + 1 closed paths satisfy a homology of the type (9.5) with not all coefficients c_i vanishing; on the other hand, there exist h closed paths which are linearly independent in the sense of homology and may be used as a basis of the closed paths.²³

For each curve γ one can form the curve traced in the opposite sense: −γ. If γ and γ′ are two closed curves, then one can form a closed curve γ + γ′ as follows: trace γ, starting at a point , trace a path β from to a point of γ′, then trace γ′ from to , and finally trace −β from back to . For every integral function F the relations

are valid for any closed paths γ and γ′. Thus one can also interpret any linear combination n₁ γ₁ + + n_ly_l of closed paths γ_i with integral coefficients n_i as a closed path instead of as a stream.

If, in a linear space of integral functions, one considers a linear subspace , then the degree of the family can decrease at most; it remains the same if each integral function F in is cohomologous to an F′ in : F − F′ ∼ 0. In particular, this fact is applied in the case of a smooth surface when, among all integral functions, we restrict ourselves to those which arise from integration of closed differentials. Similarly on a Riemann surface if, from the linear space of closed differentials, we single out the family of differentials which are everywhere harmonic (which are the real parts of analytic differentials). In both cases it turns out that, at least for closed surfaces, the degree does not decrease; the proof for the second case will be given in Chapter II. In the study of harmonic (or analytic) differentials on Riemann surfaces, the question arises as to whether we should depend on a consideration of closed differentials on arbitrary smooth surfaces or integral functions on surfaces. Generalization entails facilitation in that it simplifies the situation with which one has to deal; it is fruitful if essential features of the phenomena are not erased. Led by this point of view, we shall commence our study of functions and integrals on Riemann surfaces with a study of arbitrary smooth surfaces. Going back to topological surfaces would hardly yield further simplifications; and the smooth surfaces have the great advantage that one can use the linearizing processes of the differential calculus on them; for surfaces in general the intrinsically more complicated procedure of simplical approximation is indicated. Thus from now on we use as the definition of the degree h of a smooth surface ,the maximum number of linearly independent closed differentials on . In particular is called completely planar (schlicht) if h = 0, that is, if every closed differential on is exact. We have noted above that if is simply connected, then is completely planar. The homology (9.5) now means for us that the equation

holds for every closed differential df on .

A closed differential df is called compact if there exists a compact set on outside of which df = 0. If the dimension of the linear space of compact closed differentials is finite, it will be called the weak degree of connectivity of . It cannot exceed the degree of connectivity h.

The weak homology

means that the relation (9.7) is valid for every compact closed differential df on . For closed surfaces there is of course no distinction between homology and weak homology. But, for example, the interior of a circular annulus in the plane has degree of connectivity 1 and weak degree of connectivity 0. For if the annulus a < r < b, then any circle r = constant in the annulus is not homologous to 0 but is weakly homologous to 0. A surface on which all compact closed differentials are exact, that is, a surface whose weak degree of connectivity is 0, is called planar (schlichtartig). There is the following noteworthy theorem.

Every subdomain of a planar surface is also planar.

Proof. If df is a compact closed differential in , then df vanishes outside a compact subset of . If we set df = 0 at every point of outside that compact subset of , then we obtain a compact closed differential on . Since is planar, df must be exact on and, a fortiori, exact on .

One can construct a normal perfect covering surface over with the following property. A point in whose trace in follows a closed curve γ describes a closed curve on if and only if γ is weakly homologous to zero. We will call this surface the class surface of , for it is analogous to what is called the class field in the theory of algebraic number fields. To every closed path γ on there corresponds a definite cover transformation S = S_γ of which is unaltered if γ is replaced by another path weakly homologous to γ. If one follows over the path γ on , starting at a point , it leads to The group of cover transformations is Abelian, corresponding to the rule S_{α + γ} = S_αS_γ for any two closed paths α and γ, and is isomorphic to the additive group of closed paths if weakly homologous paths are identified.

Let df be a continuous closed differential on the surface obtained by deleting the finite set of points cti, from the surface ; let the residues of df be zero at each of these points. If α is a closed path which avoids the singular points and if α 0, then

The point of this theorem is that one can extend the formula (9.8) from differentials df which are everywhere continuous to differentials which have singularities without residues. If df is compact, then it is enough to assume that α is weakly homologous to zero. We supply the proof for a single singularity the extension of the proof to several singularities is trivial. We pick a disc : r ≤ a, in an admissible local coordinate system at . Since the residue is zero, df is the differential of a continuously differentiable function f in (that is, the disc with the origin deleted). We choose a smaller disc : r ≤ a₁, and a continuously differentiable function λ(r) which is identically 1 for r ≤ a₁, is identically 0 for r ≥ a, and which decreases from 1 to 0 in the interval a₁ ≤ r ≤ a. We call λ a smoothing function in the annulus (a₁, a). Then λf = η is a continuously differentiable function on the whole deleted surface which vanishes outside and coincides with f in . The “smoothed” differential df′ = df − dη, which differs from df only in , is also continuous at the point ; for it vanishes in a whole neighborhood of . If is chosen so small that the curve α does not penetrate , then clearly

and hence α 0 implies (9.8). In truth, (9.9) is valid for any closed curve α not passing through , since dη is an exact differential on ; this remark will be of importance for us later.

The degree of connectivity of a compact surface is finite.

The proof of this fundamental theorem utilizes the discs and the bridges which were used in § 8 to prove the axiom of countability for covering surfaces. But here we have the simplification that is covered with finitely many ; for is assumed to be compact. If F is any integral function on , then without altering the value F(γ) any closed path γ may be replaced by a closed path γ′ which consists of pieces Here consists of one piece in which joins the centers and of and the bridge and of a second piece in . Certainly then the integral function F is cohomologous to zero if the numbers all vanish; hence there cannot exist more than R integral functions which are linearly independent in the sense of cohomology.

§ 10.    Densities and surface integrals. The residue theorem

Let be an oriented smooth surface, a point on it, (x, y) an admissible local coordinate system at , and let given by

be a rectangular neighborhood of the origin which corresponds to a domain about on . Relative to such a local coordinate system a density ψ has a definite real value q at and the values q and q^* in two admissible local coordinate systems (x, y) and (x^*, y^*) are connected by the equation

where J is the (positive) functional determinant of the continuously differentiable transformation (x, y) → (x^*, y^*) at the origin. If (x₀, y₀) is a point of the rectangle (10.1), corresponding to the point of , and if the density ψ is defined in all of , then ψ has a value q(x₀, y₀) at relative to the admissible coordinate system x − x₀, y − y₀ at . We express this fact by the equation

The relation (10.2) remains valid for all points sufficiently close to , where J is understood to be the functional determinant

ψ is called continuous at the point if q(x, y) is continuous at the origin. The equation (10.2) shows that this condition is independent of the choice of the admissible local coordinate system.

Example of a density. If df and dg are two continuous vector fields in a neighborhood of a point on which, in terms of an admissible local coordinate system, are given by

then the “skew product”

is a continuous density. For the transformations (9.4) for f_x, f_y and g_x,g_y imply that the right-hand side of (10.3) satisfies the transformation (10.2).

Inside , (10.1), we pick a compact block B

with center (and coordinates x, y); here a and b are any two positive numbers which satisfy the conditions a < a′, b < b′. To begin with we assume that is closed (compact). I claim: with each density ψ which is defined and continuous on all of one can associate uniquely an integral which satisfies the following two axioms:

(ii)  If ψ vanishes outside a block B and if in B, ψ has the value q(x, y) relative to the block coordinates, then is equal to the Riemann integral

Proof. At the time of the first edition of this book one sought to prove this theorem by partitioning the surface into nonoverlapping pieces. Today, thanks to a simple artifice introduced by Dieudonné, it is possible to base the proof on a covering of the manifold with neighborhoods.

Enclose each point of the compact surface in a block B() = B as given by (10.4) for some admissible local coordinate system x, y at . Choose a positive number θ < 1 and replace B by the “shrunken” block B^θ given by |x| ≤ θa, |y| ≤ θb. We can choose a finite number of points such that the associated shrunken blocks cover completely. Let x_i and y_i be the coordinates belonging to . As will be shown shortly, one can construct continuous, and even differentiable, functions (Dieudonné factors) with the following properties: always 0 < μ_i < 1, μ_i vanishes outside B_i and the sum is identically 1. For a given continuous density ψ we take the Dieudonne decomposition

and compute the integral of ψ_i according to axiom (ii) by means of the coordinates x_i y_i belonging to B_i. Finally, according to axiom (i), we set

I claim that the functional computed thus for any continuous density ψ satisfies both axioms. The first is obvious, the second follows by virtue of the transformation law (10.2). For this says, for a ψ which vanishes outside the block B, that the integral of μ_iψ comes out the same whether we compute it with the coordinates belonging to B_i or with the coordinates x, y belonging to B. Also we have seen that the rules (i) and (ii) determine the value and hence this value is necessarily independent of the aids used to compute it: the blocks B_i the coordinates x_i y_i, and the Dieudonné decomposition by means of the factors μ_i. Besides the rules (i) and (ii), the following law holds:

We must still construct the functions μ_i. A continuous function λ() which satisfies the inequality 0 ≤ λ() ≤ 1 everywhere will be called a probability function. For an admissible block B, (10.4), one can easily construct a probability function λ() which vanishes outside of B and is identically 1 on the shrunken block B^θ. Namely, take a continuous probability function λ(x) of the real variable x, which is 1 for −θ ≤ x ≤ θ and which vanishes for |x| ≥ 1. We can see to it that this function is not only continuous but also continuously differentiable. If we set

then we have a function on of the type desired. In this fashion we construct a λ_i() for each block B_i. To construct the λ_i from these we use the following elementary operation of probability theory.

If α and β are the probabilities of two statistically independent events, 0 ≤ α, β ≤ 1, then the probability that one or the other of the two events occur is

Note that

The “probability sum” α ∨ β may be written as an ordinary sum α + β′, where β′ = β − αβ allies between 0 and β. The probability sum λ₁ ∨ λ₂ ∨ of the probability functions λ_i associated with the blocks B_i is identically 1 since at every point of the surface at least one λ_i = 1. This sum may be written as an ordinary sum μ₁ + μ₂ + by starting with μ₁ = λ₁ and Setting

These μ_i possess the required properties and are also continuously differentiable on the whole surface .

Instead of assuming that is compact, it is enough if the density ψ is compact; that is, ψ vanishes outside a compact subset G of . For determine a countable sequence so that every point of the surface is an interior point of some shrunken block Each point of the compact subset G has a neighborhood contained in one of the by covering G with a finite number of such neighborhoods one obtains a number n such that μ₁ + + μ_n = 1 is valid in all of G [and hence all higher μ_i(i > n) vanish in G]. Then by the axioms (i) and (ii) we are forced to

For μ_iψ vanishes everywhere for i > n: in G, where ψ_i = 0, and outside G, where ψ = 0. Since with this definition the axioms (i) and (ii) hold for all compact continuous densities, then, as above, the value of is independent of the construction.

Another situation in which may be defined unambiguously on a noncompact surface is the case of densities ψ which are nonnegative on all of . Of course we must allow oo as a value of the integral. In place of axiom (i) one demands here the stronger relation

for infinite sums ψ₁ + ψ₂ + of nonnegative densities ψ_i. From this it follows that is to be defined as Again it turns out that with this definition axioms (i^*) and (ii) are satisfied, and hence is independent of the construction.

If ψ₁ and ψ₂ are two nonnegative densities whose difference ψ₂ − ψ₁ = ψ is compact, then the equation

follows from (10.5); this implies that is finite or infinite according as is finite or infinite. A continuous density ψ may always be written as the difference ψ₂ − ψ₁ of two nonnegative densities ψ₁ and ψ₂. If one sets

then nonnegative ψ₁ and ψ₂ with the given difference ψ necessarily have the form

Therefore and can be finite only if are finite; then may be uniquely defined as the difference Along with the theorem that if a continuous density ψ on vanishes outside a block B, then the integral is equal to the Riemann integral over the block B, have the theorem for nonnegative continuous densities ψ ≥ 0 that It is enough to prove this for the shrunken block B^θ, since the block B can be generated by shrinking a somewhat larger one. If we introduce the probability function λ, used above to construct the Dieudonné factors μ_i, which is identically 1 in B^θ and vanishes outside B (“scarp function”), then, since ψ ≥ λψ everywhere, we obtain the chain of inequalities

Instead of using rectangular blocks one can use discs K : r ≤ a, relative to a local coordinate system x, y. Then it is appropriate to introduce polar coordinates r, ϕ in place of x, y. Let the density ψ be expressed as g(x, y) in terms of the coordinates x, y. We use the same symbol g also for the function g(r co ϕ, r si ϕ) of r and ϕ, although the density ψ in the coordinate system r, ϕ is 2πrg. Instead of the (z = x + iy)-disc K, I next consider the annulus R : b ≤ r ≤ a, where 0 < b < a. Then the following two theorems are valid.

(1) If ψ vanishes outside R, then equal to the Riemann integral

over the annulus.

(2) If ψ is a nonnegative density, then

The proof is in essence the same as for a block. Only in place of the block b ≤ r ≤ a, 0 ≤ ϕ d 1 in the r, ϕ-plane we have the strip b ≤ r ≤ a with the prescription that points (r, ϕ) whose ϕ coordinates differ by an integer are to be identified. By passing to the limit b = 0 one obtains the second theorem for the disc K. To extend the first theorem to a density vanishing outside K, choose an ε < b and introduce the quantity λψ where λ is a smoothing function for the annulus (ε, b). The integrals remain unchanged if is replaced by the square |x| ≤ b, |y| ≤ b and K by the disc r ≤ b. Both tend to zero with b as rapidly as b². By Theorem 1 for the annulus ϕ ≤ r ≤ a,

By passing to the limit b = 0 we do indeed obtain

If u(x, y) is a given continuous function in the disc K : |z| ≤ a and if G is a subset of K which possesses a Jordan content (for example, a rectangle or disc), then one can define the integral in the familiar Riemannian way. In particular,

If u ≥ 0 everywhere, then We shall make occasional use of this later.

By means of a smoothing function one can easily define the integral of a continuous density ψ which is defined and continuous only outside the hole : r < a on the surface . (The punched surface − has a border.) Choose then a radius b which is somewhat greater than a and a smoothing function λ for the annulus (a, b). To the integral of (1 − λ)ψ over add the integral, computed in polar coordinates, of λψ over the annulus a ≤ r ≤ b; the sum is to regarded as the integral of ψ over the punched surface. It is independent of the choice of b and λ. For two smoothing functions λ and λ′ for the annuli (a, b) and (a, b)′ may be regarded as smoothing functions in the same annulus (a, b) if b is the larger of the two numbers b and b′. The result then follows simply from the fact that the integral of the continuous density (λ − λ′)ψ, which vanishes outside the annulus (a, b), has the same value whether one integrates over the whole surface or only over the annulus. This remark is particularly useful if one wants to determine the “improper” integral of a density ψ which has a singularity at a point . One computes, in the way described, the integral of ψ over the punched surface and then lets the radius of the circular hole , with center , tend to zero.

The following lemma is of great consequence in the integration theory of densities.

Lemma L. If dη = grad η is an exact differential and if df is a closed continuous differential which is also compact, then

We give the proof for closed surfaces, and add at the end the modifications which must be made if the assumption of compactness is transferred from the surface to the differential df.

The notations and μ_i will be used as above; we shall assume that the Dieudonné factors μ_i are continuously differentiable. Then the left-hand side of (10.6) is the sum of the integrals

I claim that the individual integral (10.7) is equal to

Since μ_i vanishes outside B_i, the integral (10.8) is equal to the integral over all of of the density

The sum of these last integrals is 0, for grad μ_i = 0.

If one sets μη = λ, then to transform (10.7) into (10.8) one must show that

for a continuously differentiable X that vanishes outside of the block B, (10.4), and for a continuous closed df = f_xdx+ f_y dy in B. We choose a large natural number N = 1/ε and divide the block in 4N² small rectangles of dimensions a/N and b/N,

Since B is compact, N can certainly be chosen so large that the integral of the closed df around the perimeter of each small block B_ε vanishes. If one approximates λ by the first terms of its Taylor development about the vertex (x₀, y₀),

then the integral of λdf around the perimeter of B_ε is approximated by

with an error ε² o(ε), and uniformly for all the 4N² subrectangles B_ε. The first term drops out; in the second term naturally denote the values of f_x and f_y at the vertex (x₀, y₀). So we get for (10.10)

Summing over all rectangles and letting N → ∞ one obtains

where the left-hand integral is over the boundary of B. Since λ vanishes on the boundary of B, this gives the equation (10.9) which was to be proved. But the more general relation (10.11) (Green-Stokes formula) which does not assume that λ vanish on the boundary is important for us. An analogous equation is valid in an annulus R:

On the right is the difference of the integrals over the two boundaries of the annulus.

If the hypothesis that is compact is dropped and replaced by the hypothesis that df is compact, let G be a compact set outside of which df = 0. One must go far enough in the sequence of Dieudonné factors μ so that μ₁ + + μ^η = 1 in all of G in order to carry over the proof unchanged.

Punch out from the disc K₀: r ≤ a₀ about the point with admissible local coordinates x, y. Let η be a function which is defined and continuously differentiable on the remainder and let df be a continuous closed differential on We also assume that df vanishes identically outside a compact part of (which contains K₀). We speak then of a closed differential on − K₀ which is compact relative to . Choose two z-discs K and K′ with radii a and a′ such that a₀ < a < a′ and a smoothing function λ for the annulus (a, a′). By means of μ = 1 − λ, the integral of [grad η, df] over the punched surface with boundary k may be expressed as the sum of the integral of [grad μη, df] over the whole of and the integral of [grad λη, df] over the annulus (a, a′). The second of these integrals is to be computed in polar coordinates. By Lemma L, whose proof does not become invalid under the situation described, the first integral vanishes. By Stokes’ formula for the annulus, the second integral has the value Thus in place of (10.6) we have the formula

If are several points and if a circular hole with boundary is punched out about , then the equation

holds under analogous conditions. (Here it is assumed that the holes are disjoint, and of course we speak of a “circle” relative to some admissible local coordinate system at .)

A special case, obtained by setting η = 1, is the general residue theorem. This applies in the following situation. Let be a smooth oriented surface and let be the surface obtained by deleting the n points from . Let a closed differential df be given on and let df be compact relative to . The conclusion of the theorem is: the sum of the residues of df at the n “singularities” is equal to zero.

§ 11.    The intersection number

Put roughly, the intersection number ch (α, β) of two closed paths α and β on a smooth oriented surface is the algebraic sum of the crossings of α over βin this sum, a crossing from left to right contributes + 1, a crossing from right to left contributes – 1. The traffic rule on the “right of way” rests on the fact that a crossing of β by α from left to right is at the same time a crossing of α by β in the opposite sense. Thus the intersection number is skew symmetric in α and β

When one considers all the possibilities which the general concept of a continuous curve leaves open, it is clear that the definition in the above rough form is unusable. To arrive at a rigorous formulation we start with the following problem. For any two points and on which are “close together” and for a closed curve γ which does not pass through or , to define the number which tells “how many times in all γ passes between and in the positive sense.”

Let be a point of (x, y) admissible local coordinates at , and r < a₀ a corresponding disc neighborhood . It is no restriction to assume that a₀ = 1. Further let 1 and 2 be two points in and let ϕ₁(p) and ϕ₂(p) the angles which the rays and form with the positive x-axis; here p = (x, y) is any point of the xy-plane which is distinct from 1 and 2. The angles ϕ₁ ϕ₂ are only determined mod 1; but we can follow their continuous variation along arbitrary curves γ which do not pass through 1 or 2. The difference ψ = ϕ₂ – ϕ₁ measures the angle through which one must rotate the direction in the positive sense about p to carry it into Since the azimuth ϕ of the ray satisfies the equation

dψ is a continuous closed differential in the unit disc with the points 1 and 2 deleted; also dψ has a residue + 1 at the point 2 and a residue – 1 at the point 1. It is exact modulo 1 and is infinite of the first order at the points 1and 2.

We choose an a < 1 such that the concentric disc , r² = x² + y² < a², contained in , contains the points 1 and 2 in its interior. On the periphery of , ψ(p) is a uniform continuously differentiable function (which remains in absolute value < ); ψ may be extended to a function with the same properties on the punched surface – . In particular, let this be done in such a way that vanishes outside . One proceeds most simply as follows. Enclose in a concentric disc of radius ā and > a, and < 1, and form the function = λψ where λ is a (continuously differentiable) smoothing function for the annulus (a, ā) [where sticks out beyond ]. The function is defined outside . The differential dψ₁₂, which is equal to dψ in and equal to d outside , is a continuous closed differential on the surface with the points 1 and 2 deleted; it is exact mod 1. Thus one can form the integral along any curve γ on which does not pass through the points 1 or 2. For a closed curve γ its value is ≡ 0 (mod 1), that is, an integer. We are tempted to say that this number tells how many times the curve γ passes between 1 and 2. But we must beware the fact that besides the points 1 and 2 all sorts of constructions go into the definition.

If γ stays outside , then = 0.

Let df be a continuous closed compact differential on and let α₁₂ be any curve in joining the points 1 and 2. Then I claim that the formula

is valid. Because of the singularities of dψ₁₂ at 1 and 2, the integral on the left-hand side is an improper integral. For the proof we can restrict ourselves to the interior of , for ψ₁₂ = vanishes outside . Remove small discs and of radius ε about 1 and 2, and let denote the punched unit disc (that is, with the two small discs removed). On , df is the differential of a uniform continuously differentiable function f, and the formula (10.13) yieldS

From passage to the limit ε → 0 we obtain

which is in fact (11.2).

The next step is to piece together an arbitrarily extended curve α with small arcs such as α₁₂. Let α be a curve from to on . After choosing an admissible local coordinate system at each point p of α and an associated disc neighborhood (p), we obtain a standard subdivision of α into arcs each of which satisfies the hypotheses on α₁₂. Thus for each of these arcs there is an admissible coordinate system x, y and an associated disc containing the arc. Using these coordinates, we form the differentials dψ₁₂, dψ₂₃, …for each individual arc. It follows from (11.2) that the sum

satisfies the equation

But it is disturbing that dψ^* has singularities not only at the endpoints of α but also at the division points 2,3, …. . We shall eliminate these by a simple smoothing process .²⁴ The point = 2 is covered by the coordinates x,y associated with α₁₂ as well as by the coordinates x′,y′ for α₂₃. Let and have the same meaning for x′,y′ as and ϕ₂ for x, y. We choose a neighborhood of which is contained in both and and which is a disc about in the local coordinates x – x₀, y – y₀ at . In , dψ₁₂ = dϕ₂ – dϕ₁ and in , The problem is to smooth the differential at . Because of the invariance of the order under topological maps, for every closed curve γ in which does not pass through .

Therefore is a uniform continuously differentiable function in (the disc with removed). By means of a smoothing function λ which vanishes outside and is identically 1 in a smaller concentric disc , we obtain from a function ω = λ. The smoothed differential d – dω is then regular at , for it vanishes in a whole neighborhood of . But by subtracting dω, the value of the integral has not been changed for any closed curve β which does not pass through 1, 2, or 3. Also the value of the surface integral of [dψ₁₂ + dψ₂₃ + df] is unchanged, since ∫[dω, df] = 0. For if we operate in the disc and use the polar coordinates associated with x – x₀, y – y₀, then the value of that integral over an annulus, bounded by and a smaller concentric circle of radius ε, is df by Stokes’ formula (10.12) for an annulus. It is easy to see that ω remains bounded in a neighborhood of ; but it suffices to observe that dω is at most of first order at and hence that at worst ω itself becomes logarithmically infinite. Then it follows that the integral tends to zero at least as rapidly as ε log(l/ε), as ε → 0.

After this smoothing we obtain dψ in place of dψ^* and the two differ by an exact differential dω. More exactly, ω is a function which is continuously differentiable except at a finite number of singular points (here the points 2,3, … ,n – 1) and whose differential dω has an infinity of at most first order at each point. Such a function will be called an ω-function. This is enough to draw the conclusion ∫[dω,df] = 0 for every closed compact df. In the case of a closed curve α, the smoothing is also to be carried out at the initial = terminal point, 1 = n. Finally we allow the addition to dψ of the differential of an arbitrary ω-function with singularities at the endpoints and of the curve α (for the case of a closed curve: the differential of an arbitrary ω-function without singularities). Denote the resulting differential by ds Then

To simplify the terminology we shall understand by (α) a curve α together with the following constructions: a subdivision of α into arcs, each of which is contained in a unit disc relative to some definite local coordinates x, y. Then ds = ds(α) is uniquely determined by (α) in the following sense: any two differentials ds(α) and ds^* (α) corresponding to (α) satisfy the formula

of “modified cohomology.” It means that the difference ds(α) – ds^* (α) is the differential of an ω-function whose singularities are at most the endpoints and of α. [A closed curve is without endpoints, and the qualification (, ) behind formula (11.4) disappears.] If α is a curve from to and β is a curve from to , then one can form not only α + β in the obvious way from α and β, but also (α + β) from (α) and (β). Obviously holds.

Our undertaking, to arrive at an invariant definition of the intersection number, would be hopeless if ds(α) were not also independent, in the sense of cohomology, of the constructions associated with α. Fortunately the following theorem is valid.

If α and α′ are two paths joining the same pair of points and and if the closed path γ = α – α′ is weakly homologous to zero - for which we write simply α ∼ α′ -, then always holds.

In particular this is the case if α′ and α coincide. Thus for every closed path β which does not pass through or , the integer

is uniquely determined by α and β in such a fashion that a may be replaced by any α′ ∼ α without changing this integer. Then it is reasonable to call (11.6) the intersection number of the closed curve β with the curve α.

To prove our theorem we form (α) – (α′) = (γ). This (γ) belongs to the closed cruve γ which, by assumption, is weakly homologous to zero. Let β be a second closed curve, which by the related constructions gives a (β) and an associated differential ds(β) that is everywhere continuous and closed, and is also compact. We form the density integral ∫[ds(γ), ds(β)]. By our fundamental formula (11.3) this integral has the value ∫_γ ds(β) and hence is zero, since γ = α – α′ ∼ 0. On the other hand, by interchanging β and γ we obtain for the same integral the value

The vanishing of the integral on the right-hand side for every closed path β amounts to ds(γ) ∼ 0. Thus (11.5) follows.

From the observation that ∫_γ dψ₁₂ = 0 in case the closed curve γ lies outside (p. 81), and by partitioning a into sufficiently fine arcs α₁₂, ... we obtain the following result. The intersection number ch (β, α) is always zero if the closed curve β does not meet the curve α.

Now we can write ds_α in place of ds(α), indicating thereby that ds_α is determined uniquely, in the sense of the modified cohomology (11.4), by the path α. In this sense ds_α remains unchanged even if α is replaced by a path α′ ∼ α.

The intersection number (11.6) of two closed curves α and β is obviously unchanged if β is replaced by a path weakly homologous to β Also, from the result obtained, it remains unaltered under a similar replacement of α. From the equation

which has already been used, comes the law of skew symmetry (11.1). The goal set at the beginning of this paragraph, a rigorous definition of the intersection number of closed curves, is thus attained.

Before we turn to closed curves finally, we shall generalize our formulas to include the case where not only α but also β is open. Let and be two points on and let df be a continuous closed differential, on the surface with the points and deleted, which is compact relative to . Let the residues of df at and be – A and + A, respectively. Let be a curve which does not pass through or . We can choose the partition of β into arcs α₁₂, α₂₃, … such that the points and lie outside all of the “discs” enclosing the arcs of α. Then we can use the formula (11.2) unchanged for the individual arcs α₁₂, for dψ₁₂ vanishes outside and everything takes place in the unit disc , in which df has no singularity. Let and be joined by a curve β which does not meet the points 1 and 2. One can also find a curve β′ which joins and and does not penetrate the interior of . Namely, note the first and last intersections of β with the periphery of . If those points are and , then replace that portion of β from to by the arc of from to . Then for ψ₁₂ vanishes outside . Thus ∫_β dψ₁₂ is equal to the integral of dψ₁₂ over the closed path β – β′ and hence is an integer. Summation and subsequent smoothing yields the relations

Now we may take β to be an arbitrary path from to which does not pass through the endpoints and of α; it is no longer necessary that β avoid the division points 2, 3, … . A ds_α arises from dψ by the addition of the differential dω of an ω-function which has singularities only at the points and . On the one hand,

On the other hand, the formula (10.13) applied to = ω and two circles shrinking down to and yields

Thus the equation

follows. From these considerations themselves it follows that if df and β are given, then the left-hand side of (11.8) remains unchanged if the construction of ds_α is altered, so long as α remains fixed. For the difference of two such ds_α is the differential of an ω-function which has singularities only at and !

If we take now for df the differential ds_β associated with a curve β from to , then we see that

is an integer which for a fixed choice of ds_β is uniquely determined by α. Because of the skew symmetry of this expression in α and β, it depends only on β and not on the special construction of ds_β. Therefore the integer (11.9), uniquely determined by α and β, may be called the intersection number of the two open paths α and β. It is 0 if α and β are disjoint. If β is closed, and therefore ds_β has no singularity, then, by virtue of (11.9) gives the former value ∫_β ds_α.

If, instead of df one applies the formula (11.8) to the differential df – A ds_β, which has the residue 0 at and , then

from which the congruence (11.8) mod A is turned into an equation. The difference ∫_β ds_α – ch (β, α) remains unaltered if β is replaced by a path β^*joining the same two points and , and therefore will be written in abbreviated form as Naturally, this must be so, for the expression ∫[ds_α, df] – ∫_αdf contains df but not a path β joining its two singularities and .

With an eye to the function theoretic applications, we supplement equation (11.10) with the following. Let u_α be a continuously differentiable function everywhere except at the endpoints and of α, and let du_α have infinities of at most the first order at and . Then the equation

is valid, where df is a differential satisfying the conditions specified for (11.10). Here denotes the difference This is nothing but the equation (11.7) in different notation. If df possesses not just two but several singularities ₁, … , _n with the residues A₁ … , A_n, then choose a point not on α Instead of (11.10) and (11.11) one obtains the more general equation

The remainder of this paragraph is concerned only with closed curves. Let α be one such. It is not only true that the weak homology α ∼ 0 implies sh(α, β) = 0 for every closed curve β, but the important converse is valid.

Theorem T. If the closed path α satisfies the equation ch(α, β) = 0 for every closed path β, then α is weakly homologous to zero.

Since that equation can be written in the form ∫_β ds_α = 0, the assertion may be expressed as follows : the weak homology α ∼ 0 and the cohomology ds_α ∼ 0 are equivalent. It is apposite to formulate this theorem more generally as follows.

If α₁, … ,α_l are closed paths and if c₁ … ,c_l are real constants, then I weak homology

and the cohomology

are equivalent.

Proof. The relation (11.12) is by definition equivalent to the statement that the equation

holds for every closed compact differential df But this may be written in the form

or

So we must show that ds ∼ 0 implies (11.13) for every closed compact differential df; and conversely, that if (11.13) holds for every such differential, then ds ∼ 0. The first part is identical with Lemma L. One gets the converse by picking df to be ds_β and then concluding, by virtue of ∫[ds, ds_β] = 0, that

for each closed path β, and therefore ds ∼ 0.

If we regard closed streams

as equal if they are weakly homologous to each other, then according to our theorem there corresponds to each such stream σ a unique (in the sense of cohomology) differential

Now we shall assume that the weak degree of the maximal number of closed paths which are linearly independent in the sense of weak homology, is finite; we shall now denote this number by h. If γ_l, … , γ_h are any h closed paths which are linearly independent in the specified sense, then every closed path α is weakly homologous to a linear combination with real coefficients x_i. Thus if df is any closed compact differential on then the equation

holds. Thus df ∼ 0 if the h equations are satisfied.

By the general theorem we have just proved, the h differentials ds_i = ds_γi are linearly independent in the sense of cohomology. That is, if the equation

holds for all closed paths α, then we must have Thus the h linear equations

for h unknows c_j, with integral coefficients have only the trivial solution In other words, the determinant Δ of the s_ij is not zero; or, the skew-symmetric bilinear form (characteristic form)

which gives the value of

for the two streams

is nondegenerate.

Since the equations

always have a unique solution, no matter what real numbers a_i are given, the periods ∫ _γi df of a closed compact differential may be specified arbitrarily ; then df is uniquely determined in the sense of cohomology.

For a closed path α,

the equations

hold and may be used to compute the x_j. Since the left-hand members and the coefficients are all integers, it follows that the x_i are necessarily rational numbers with denominator Δ.

But a skew-symmetric bilinear form can be nondegenerate only if the dimension number h is even: if the weak degree h is finite, it is even. For the transposed matrix has the same determinant Δ as then also Δ = (– 1)^hΔ, which is compatible with Δ ≠ 0 only if h is even. For closed oriented smooth surfaces we set h = 2p and call p the genus of the surface.

Relative to the basis every closed curve α is, by (11.15), characterized by the vector x = (x₁, …, x_h) in an h-dimensional vector space. These vectors form a discontinuous lattice G, i.e., a group under addition. In other words: the zero vector belongs to the lattice; if x: belongs to G, then –x belongs to G; if two vectors x and x′ belong to G, so does their sum x + x′. The coordinates x_i of a lattice vector are rational numbers with the universal denominator Δ. On the other hand, the unit lattice consisting of all vectors with integral coordinates x_i is contained in G. By a construction which Minkowski called the “adaptation of a number lattice relative to a sublattice,”²⁵ one can choose the basis γ₁ … γ_h such that for every closed path α the coefficients x_i in the weak homology (11.15) are integers ; that is, so that G coincides with the unit lattice. We speak then of an integral basis for the closed paths. The passage from one such basis to another is accomplished by a unimodular transformation; that is, a linear transformation A with integral coefficients and determinant ± 1. [Since A^–1 also has integral coefficients, the last follows from the equation det(A) · det(A^–1) = 1.]

The construction of an integral basis goes as follows. Among the finitely many vectors of the lattice of the form

be that one for which r₁ has the smallest value. Among the finitely many lattice vectors of the form

be that one for which r₂ is as small as possible. Among all lattice vectors

let

be that one for which r₃ is as small as possible. And so on. Then the vectors γ₁..., γ_h form an integral basis for the lattice. For if α ∼ (r_l …, r_h) is an arbitrary vector of the lattice, then one can determine the integers n_h, n_h-1 … , n₁ successively so that for

the inequalities

hold. But then one concludes from the definition of γ_h, γ_h-1

and the end result establishes our claim:

If S_l, … ,S_h are those cover transformations of the class surface of which correspond to the paths γ₁ … , γ_h of the integral basis, then the Abelian group of cover transformations of consists of all transformations of the form where the n_i are arbitrary integers. Thus this group is a free Abelian group with h generators.

If we compute the characteristic form (11.14) relative to an integral basisγ₁ … , γ_h, then the form is determined to within a unimodular simultaneous transformation of the variables x_i and y_i. If we call two forms equivalent and put them in the same class in case they are related by such a transformation, then the class to which the characteristic form belongs is determined by the surface. But, at least for closed surfaces, the characteristic form can always be put in the normal form

by an appropriate choice of the integral basis. Thus the form leads to no topological invariants beyond h = 2p. An integral basis relative to which the characteristic form is given by (11.16) is called a canonical basis. The proof of this theorem consists of a topological part and an arithmetical part. The topological part says that the determinant Δ of the characteristic form (which, as for every skew-symmetric form, must be positive) is equal to 1. The easily proved arithmetic part says that a skew-symmetric form with integral coefficients and determinant 1 may be transformed unimodularly into (11.16). The topological part may be expressed in two ways.

(1) If n₁, … , n_h are arbitrary integers, then there exists a closed path α such that

(2) To each primitive closed path α there corresponds a closed path β such that ch(α, β) = 1. Here α is called primitive if there does not exist a closed path α′ such that α is homologous to a multiple nα′ of α′ (with n = 2, or 3, or 4, or … ).

We shall not prove these facts for closed surfaces at this point; nor shall we prove the other theorem, that a completely planar closed surface is necessarily simply connected. Both theorems are relatively easy results in combinatorial topology; but this is not on our route, since we would have to introduce the assumption that the surface is triangulable. We prefer to prove these results under the assumption that the surface is a Riemann surface. In this form they will automatically fall into our laps as corollaries of the theory of Riemann surfaces.

Riemann based the topological investigation of a closed oriented surface of genus p on the construction of p canonical pairs of cuts [Rückkehrschnittpaaren] π₁, ; … ; π_p, . These are simple closed curves; two curves belonging to different pairs do not intersect; π_i and intersect in exactly one point , at which crosses π_i from left to right. Thus we have a canonical basis with special properties. Furthermore Riemann joined a point of the surface, not on any of the canonical pairs, to each of the points by simple disjoint paths σ_i not intersecting the canonical pairs, and showed that was turned into a simply connected domain by cutting it along the “reins” σ_i and the canonical pairs π_i and . To make this procedure rigorous one must start with a triangulation of the surface and specialize all the curves to polygons on the triangulated surface. I proceeded in that way in the first edition of this book. By means of this dissection one can show that the class surface is the strongest of all normal perfect covering surfaces for which the group of cover transformations is Abelian. For a curve on any such covering surface whose trace on is closed and homologous to zero is necessarily closed.²⁶

As Möbius and C. Jordan have proved,²⁷ the genus is the only topological invariant of closed oriented smooth surfaces. Any two such surfaces of the same genus are equivalent, not merely topologically, but as smooth surfaces : one may be mapped onto the other by a one-to-one transformation which, together with its inverse, is continuously differentiable. It is easy to give closed oriented surfaces in three-dimensional Euclidean space which have a specified genus p. For example, a sphere with p handles has genus p. The incisive significance of the genus of a closed Riemann surface for the theory of functions on this surface will become sufficiently clear from the function theoretic theorems of the next chapter.

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¹) The term “subset“ (of a set) is to be understood to include both the whole set and the void set which contains no element.

²) See the paper of Weierstrass, Definition analytischer Funktionen einer Veränderlichen vermittelst algebraischer Differentialgleichungen, which was written in 1842 but was first published in his Mathematische Werke, 1 (1894) 83-84. See also the first pages in Riemann’s Theorie der Abelschen Funktionen (1857), [Werke, 2nd edition, pp. 88-89]. See also the article by Osgood in Encyklopädie der math. Wiss. II B 1, no. 13.

³) See Weierstrass, Vorlesungen über die Theorie der Abelschen Transzendenten (edited by G. Hettner and I. Knoblauch), Werke, 4, 16-19.

⁴) This concept is something more than that of the point set which consists of all points passed in the motion. We are concerned with the same distinction as, in the case of a pedestrian, that between the path traced (which, as long as he walks, is in statu nascendi) and the path (long since existing) on which he walks.

⁵) Poincaré, Rendiconti del Circolo matematico di Palermo, 2 (1888) 197-200. Volterra, Atti della Reale Academia dei Lincei, ser 4, IV₂, 355.

⁶) For this reason part (a) includes statements that may be derived from the requirements in part (b).

⁷) “Domain-continuous” [“gebiets-stetig”] in the first edition.

⁸) For some time the term bicompact was used, since the term compact was used for a wider class of sets. Today the terminology accepted here has been generally agreed upon, at least in America.

⁹) See, for example, Lebesgue, Leçons sur l’intégration, Paris 1904, 104–105.

¹⁰) With this word [verstreut] I have called to mind a beautiful poem, Sommer-abend, of Hermann Hesse, containing tKe lines :

                    Sommemacht hat ihre diinnen Sterne verstreut,

                    Jugendgedachtnis duftet im mondhellen Laub...

¹¹) Möbius, Werke, 2, 484–485 and 519–521.

¹²) The surface represented by these equations is by no means a developable.

¹³) The term “paddle motion” (“Paddelbewegung”, a Low German colloquialism) is meant to indicate that the group is obtained as follows: one repeatedly flips the plane over about an axis, alternately to the left and to the right, and at each flip pushes the plane forward (through a distance 2π) in the direction of that axis.

¹⁴) Leipzig 1882. See also Klein, Neue Beiträge zur Riemannschen Funktionentheorie, Math. Ann, 21 (1883), §1–3 [pp. 146–151]. Surfaces, closed by boundary identifications, as carriers of analytic functions appear earlier: Riemann, Art. 12 of Theorie der Abelschen Funktionen, Werke, p. 121; H. A. Schwarz in his fundamental paper of 1870 on the integration of the partial differential equation ∂²u/∂x² + ∂²u/∂y² = 0, Gesammelte mathematische Abhandlungen, II, 161; Dedekind, Jour. f. Math., 83 (1877) 274 ff. Surfaces imbedded in space were first used, though only for investigations in analysis situs, by Tonelli (1875, Atti dei Lincei, ser II, v. 2) and Clifford (1876, Mathematical Papers, 249 ff). Klein himself, as he relates in the preface to his monograph, Über Riemann’s Theorie (p. IV), derived the initial idea for his formulation from a chance oral remark of Prym (1874). Klein considers only closed surfaces. The most general concept is probably found explicitly first in Koebe’s work; see, for example, Göttinger Nachrichten (1908), 338–339, footnote.

¹⁵) Über die Hypothesen, welche der Geometrie zugrunde liegen, Werke, 2nd edition, 272–287; edited with a commentary by H. Weyl, 3rd edition, Berlin 1923.

¹⁶) That there exists, in a certain sense, an invariant measure of length, is a deep fact in uniformization theory. See § 21 of this book.

¹⁷) See L. Lichtenstein, Beweis des Satzes, dass jedes hinreichend kleine, im wesentlichen stetig gekrümmte, singularitätenfreie Flächenstück auf einem Teil einer Ebene zusammenhängend und in den kleinsten Teilen ähnlich abgebildet werden kann, Abhandlungen der Preussischen Akademie der Wissenschaften vom Jahre 1911, Anhang.

¹⁸) This definition singles out that property of simply connected surfaces which is decisive in function theoretic applications.

^l9) On a surface 1 ocated in space we distinguish the two sides of the surface at a point by the two directions which may be assigned to the normal to the surface at this point. A sense of rotation on the surface may be joined with the direction of the normal so that together they constitute a “left-handed screw.” If we replace the direction of the normal by this sense of rotation on the surface, we become independent of the imbedding of the surface in space. Felix Klein accomplished this step in the definition of one sided and two sided: Math. Ann., 9 (1876) p. 479.

²⁰) Poincaré, Bulletin de la société mathématique de France, 11 (1883) 113−114.

²¹) One must always bear in mind that a residue is something associated with a differential, not a function.

²²) The use of this term varies in the literature; frequently “exact” is used in the sense of “closed.”

²³) One may well say that this definition of the degree exposes the real kernel of the method used by Weierstrass, Hensel and Landsberg, among others, in the theory of algebraic functions: to investigate the behavior of integrals first, and to draw conclusions therefrom about the paths of integration. The passage from closed paths (or streams) to the dual linear space of integral functions has been developed extensively and systematized during the last decade as cohomology theory in topology. In that case manifolds and paths of arbitrary dimension are considered.

²⁴) On a Riemann surface, where we may always take z = x + iy to be a local parameter, these singularities are certainly not present. In that case the smoothing process is superfluous.

²⁵) See H. Minkowski, Diophantische Approximation, Leipzig 1907, pp.90–95.

²⁶) See page 176of the second edition (1923).

²⁷) Möbius, Theorie der elemantaren Verwandtschaft (1863), Werke II, 435-471; C. Jordan, Journal de Mathématiques, ser.1, 11 (1886), 105. Also see W. Dyck, Math. Ann.,32 (1888) 457.