The oriented intersection number

Let $X$ be a compact oriented smooth $x$-manifold without boundary, let $M$ be an oriented smooth $n$-manifold without boundary, and let $Z\subseteq M$ be a closed oriented embedded $z$-submanifold with $x+z=n$ (Oriented smooth manifolds and oriented charts). For a smooth $f:X\to M$ transverse to $Z$, the oriented intersection number is the finite sum

$I(f,Z):={\sum }_{p\in f^{-1}(Z)}\epsilon (p)\in \mathbb{Z},$

with the local signs of The local oriented intersection sign; the sum is finite by Compact transverse complementary intersections are finite. For an arbitrary smooth $g:X\to M$, choose a transverse $f$ homotopic to $g$ (The transversality homotopy theorem, under Countable Choice; the homotopy is a smooth family in the sense of Smooth families of maps and their evaluation maps) and set $I(g,Z):=I(f,Z)$; independence of the choice is The oriented intersection number is homotopy invariant ↗, not assumed here. For compact oriented complementary-dimensional submanifolds $A,B\subseteq M$, where one is compact and the other closed, one sets $I(A,B):=I(i_A,B)$ with $i_A$ the inclusion, so the first factor is the submanifold $A$. If the source is not compact, or the target submanifold is not closed, the sum may fail to be finite or invariant; the safe extension by properness with compact trace is recorded in the remark on properness later on this page. Countable Choice is assumed for selecting transverse representatives and for the classification used to prove independence of that selection; the signs, the finite sum and the empty case (which contributes $0$) are choice-free.

For transverse maps $f:X^x\to M$, $g:Z^z\to M$ with compact oriented boundaryless sources and $x+z=\dim M$, define $I(f,g)$ as the sum of the local signs of The local oriented intersection sign over $X{\times }_{M}Z$. This set is closed in the compact product because $M$ is Hausdorff, and discrete by Transverse complementary-dimensional intersection sets, so the sum is finite. If $g$ is an inclusion this recovers $I(f,Z)$. Ambient compactness is unnecessary in either construction.