The mod-two degree of a map to a sphere

Let $M$ be a closed smooth $m$-manifold with $m\ge 1$, so that $M$ is compact and has empty boundary (Smooth manifolds and their smooth charts, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right), and let $f:M\to S^m$ be smooth, where $S^m\subseteq {\mathbb{R}}^{m+1}$ is the Euclidean unit sphere (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$). A point $y\in S^m$ is a regular value of $f$ when every point of $f^{-1}(y)$ is a regular point of $f$ (Regular and critical points and values). For a regular value $y$ of $f$ define ${\mathrm{deg}}_2(f):=\mathrm{\mid }{f}^{-1}(y)\mathrm{\mid }<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\in \mathbb{Z}\mathrm{/}2\mathbb{Z},$ the mod-two degree of $f$, the parity of the number of points of the regular fibre, computed in the quotient set $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ (The congruence class $[a{]}_n$ and the quotient set $\mathbb{Z}\mathrm{/}n$). The empty fibre is allowed and contributes the parity of the empty set, namely $0$; in particular a constant map has mod-two degree $0$, since a value different from its image has empty fibre and is vacuously regular.

No orientation of $M$ is required, and no orientation of $S^m$ is used: only the number of points of the fibre enters. This is what distinguishes the invariant from the integer degree of Degree of a proper smooth map by compact-support cohomology, which is defined for oriented source and target and counts points with signs. When $M$ is nonempty, connected and oriented and $m\ge 1$, the signed count of the same fibre is the integer degree of $f$ by Regular-value formula for degree, and reducing that identity modulo two gives ${\mathrm{deg}}_2(f)\equiv \mathrm{deg}(f)(\mathrm{m}\mathrm{o}\mathrm{d}2)$.

Two things are not asserted by this definition and are proved in The mod-two degree is well defined and homotopy invariant ↗, which is why that lemma is recorded in justified_by. First, the parity of the fibre is independent of the regular value chosen, so the notation ${\mathrm{deg}}_2(f)$ denotes a single element of $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ rather than a value attached to a pair $(f,y)$; until that is known, the displayed formula defines a candidate value for each supplied regular value. Second, homotopic maps have equal mod-two degree, so ${\mathrm{deg}}_2$ descends to free homotopy classes.

The fibre is finite whenever $y$ is a regular value, so the parity is a cardinality of a finite set and no cardinal arithmetic is involved. Indeed, every $p\in f^{-1}(y)$ is a regular point, and since $\dim M=\dim S^m=m$ the differential $d{f}_p$ is an isomorphism of tangent spaces; the inverse function theorem for smooth maps of manifolds (The smooth inverse function theorem on manifolds) then makes $f$ a local diffeomorphism at $p$, so $f$ is injective on some neighbourhood of $p$ and the fibre is discrete in the sense that each of its points is isolated in it. The fibre is closed because $f$ is continuous (Smooth maps are continuous) and $\{y\}$ is closed, and a closed discrete subset of the compact space $M$ is finite: the family consisting of $M\setminus f^{-1}(y)$ and of all open neighbourhoods meeting the fibre in exactly one point is an open cover of the compact space $M$, and a finite subcover selects finitely many of those neighbourhoods, each meeting the fibre in exactly one point, so the fibre is finite. Assuming ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), regular values exist by Sard's theorem for smooth manifolds (Morse-Sard for smooth manifolds), whose statement in this library assumes the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$; this definition itself makes no choice and uses no orientation, and the choice assumption enters only through the existence of regular values and through the well-definedness lemma.