Editing Intersection number (section)

== Definition for Riemann surfaces ==
{{main|Differential forms on a Riemann surface#Duality between 1-forms and closed curves}}
Let ''X'' be a [[Riemann surface]]. Then the intersection number of two closed curves on ''X'' has a simple definition in terms of an integral. For every closed curve ''c'' on ''X'' (i.e., smooth function <math>c : S^1 \to X</math>), we can associate a [[differential form]] <math>\eta_c</math> of compact support, the [[Poincaré duality|Poincaré dual]] of ''c'', with the property that integrals along ''c'' can be calculated by integrals over ''X'':

:<math>\int_c \alpha = -\iint_X \alpha \wedge \eta_c = (\alpha, *\eta_c)</math>, for every closed (1-)differential <math>\alpha</math> on ''X'',

where <math>\wedge</math> is the [[wedge product]] of differentials, and <math>*</math> is the [[Hodge star]]. Then the intersection number of two closed curves, ''a'' and ''b'', on ''X'' is defined as

:<math>a \cdot b := \iint_X \eta_a \wedge \eta_b = (\eta_a, -*\eta_b) = -\int_b \eta_a</math>.

The <math>\eta_c</math> have an intuitive definition as follows. They are a sort of [[dirac delta]] along the curve ''c'', accomplished by taking the differential of a [[unit step function]] that drops from 1 to 0 across ''c''. More formally, we begin by defining for a simple closed curve ''c'' on ''X'', a function ''f<sub>c</sub>'' by letting <math>\Omega</math> be a small strip around ''c'' in the shape of an annulus. Name the left and right parts of <math>\Omega \setminus c</math> as <math>\Omega^{+}</math> and <math>\Omega^{-}</math>. Then take a smaller sub-strip around ''c'', <math>\Omega_0</math>, with left and right parts <math>\Omega_0^{-}</math> and <math>\Omega_0^{+}</math>. Then define ''f<sub>c</sub>'' by

:<math>f_c(x) = \begin{cases} 1, & x \in \Omega_0^{-} \\ 0, & x \in X \setminus \Omega^{-} \\ \mbox{smooth interpolation}, & x \in \Omega^{-} \setminus \Omega_0^{-} \end{cases}</math>.

The definition is then expanded to arbitrary closed curves. Every closed curve ''c'' on ''X'' is [[Homology (mathematics)|homologous]] to <math>\sum_{i=1}^N k_i c_i</math> for some simple closed curves ''c<sub>i</sub>'', that is,

:<math>\int_c \omega = \int_{\sum_i k_i c_i} \omega = \sum_{i=1}^N k_i \int_{c_i} \omega</math>, for every differential <math>\omega</math>.

Define the <math>\eta_c</math> by

:<math>\eta_c = \sum_{i=1}^N k_i \eta_{c_i}</math>.