The mod 2 intersection number

Let $X$ be a compact smooth manifold without boundary, $M$ a smooth $n$-manifold, $Z\subseteq M$ a closed embedded submanifold, and let $f:X\to M$ be smooth and transverse to $Z$ with $\dim X+\dim Z=n$ (Transverse smooth maps, Transverse complementary-dimensional intersection sets). The mod 2 intersection number is

$I_2(f,Z):=\mathrm{\#}{f}^{-1}(Z)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\in \mathbb{Z}\mathrm{/}2\mathbb{Z},$

the cardinality of the finite transverse intersection reduced modulo two (Compact transverse complementary intersections are finite, The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$).

For an arbitrary smooth $g:X\to M$ choose a smooth map $f$ homotopic to $g$ with $f⋔Z$ (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 define $I_2(g,Z):=I_2(f,Z)$. For compact complementary-dimensional transverse submanifolds $A,B\subseteq M$, where one is compact and the other closed, $I_2(A,B):=I_2(i_A,B)$ with $i_A$ the inclusion (Transverse embedded submanifolds). No orientability of $X$, $M$ or $Z$ is assumed; the count lives in $\mathbb{Z}\mathrm{/}2\mathbb{Z}$. Well-definedness of the extension to arbitrary maps is established by The mod 2 intersection number is homotopy invariant ↗, not assumed here. Countable Choice is assumed for selecting transverse representatives and for the classification used to prove independence of that selection; the transverse finite count and the empty case (which contributes $0$) are choice-free.

For compact boundaryless sources $X,Z$ and complementary-dimensional smooth maps $f:X\to M$, $g:Z\to M$, set $I_2(f,g):=I_2(f\times g,{\mathrm{\Delta }}_M)$. The diagonal is a closed embedded $n$-submanifold of $M\times M$ (The diagonal is an embedded submanifold); closedness follows from Hausdorffness. Modulo its tangent diagonal, the differential of $f\times g$ is $(v,w)\mapsto df(v)-dg(w)$, so transversality is exactly that of $f,g$. In the transverse case this number is $\mathrm{\#}(X{\times }_{M}Z)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$. Homotopies of either or both maps give product homotopies, hence this number is invariant by the fixed-submanifold homotopy theorem. The arbitrary-map extension is under ${\mathrm{A}\mathrm{C}}_{\omega }$ as above.