The local oriented intersection sign

Let $M$ be an oriented smooth $n$-manifold and let $f:X^x\to M$ and $g:Z^z\to M$ be smooth maps from oriented smooth manifolds, with $x+z=n$ and $f,g$ transverse in the sense of Transverse smooth maps. Let $a\in X$, $b\in Z$ satisfy $f(a)=g(b)=y$. Then $T_{a}X\oplus T_{b}Z$ and $T_{y}M$ have the same dimension, and transversality says that

$(d{f}_a,d{g}_b):T_{a}X\oplus T_{b}Z⟶T_{y}M,(v,w)\mapsto d{f}_a(v)+d{g}_b(w),$

is surjective, hence a linear isomorphism (Transverse linear subspaces); the sum is direct because the dimensions add up. The ordered direct sum $T_{a}X\oplus T_{b}Z$ carries the product orientation with the first factor $T_{a}X$ and the second factor $T_{b}Z$ (Product orientations), and the three tangent spaces carry their manifold orientations (Oriented smooth manifolds and oriented charts).

The local oriented intersection sign $\epsilon (a,b)\in \{+1,-1\}$ is $+1$ when this isomorphism carries the product orientation of $T_{a}X\oplus T_{b}Z$ to the orientation of $T_{y}M$, and $-1$ otherwise; equivalently it is the orientation sign of the induced isomorphism of determinant lines (Determinant-line orientations of finite-dimensional real vector spaces, Orientation of a finite-dimensional real vector space).

For transverse oriented embedded submanifolds $A^a,B^b\subseteq M$ with $a+b=n$ one takes $f$ and $g$ to be the inclusion maps (Transverse embedded submanifolds); the sign at $p\in A\cap B$ then compares $T_{p}A\oplus T_{p}B\to T_{p}M$ with $A$ first. The empty intersection is allowed and carries no signs. In dimension zero the sign compares the two orientation rays directly, and for $x=z=n=0$ it is ${\epsilon }_X(a){\epsilon }_Z(b){\epsilon }_M(y)$, the product of the two source point signs and the ambient point sign. The factor order is part of the definition: with the two factors exchanged the signs are multiplied by $(-1{)}^{ab}$, exactly the graded-commutativity sign measured later on this page by the factor-interchange theorem. No compactness, closedness hypothesis or choice axiom is used in this local definition.