Stiefel-Whitney numbers of a closed manifold

Assume AC (The Axiom of Choice). The assumption is inherited from the Stiefel-Whitney class construction and is used only there, through Stiefel–Whitney classes from the projective-bundle relation and the admissibility supplied by Smooth manifolds have CW homotopy type.

Let $M$ be a closed smooth $n$-manifold (Smooth manifolds and their smooth charts), with tangent Stiefel-Whitney classes $w_i:=w_i(TM)\in H^i(M;{\mathbb{F}}_2)(i\ge 0),$ so that $w_0=1$ and $w_i=0$ for $i>n$. Its canonical mod-two fundamental class is the fundamental class $[M]\in H_n(M;{\mathbb{F}}_2)$ of the canonical ${\mathbb{F}}_2$-orientation (Every manifold is F2-orientable and orientability is componentwise), in the sense of Fundamental class of a compact oriented manifold.

Consider a monomial $w^I=w_1^{r_1}\cdots w_n^{r_n}$ in the tangent classes, with non-negative exponents $r_1,\dots ,r_n$ and total degree $r_1+2r_2+\cdots +n{r}_n$. If the total degree equals $n$, the associated Stiefel-Whitney number of $M$ is the Kronecker evaluation $w^I[M]:=⟨w^I,[M]⟩\in {\mathbb{F}}_2$ (Kronecker evaluation pairing). A monomial of formal total degree $d=r_1+2r_2+\cdots +n{r}_n$ defines a class in $H^d(M;{\mathbb{F}}_2)$ by the graded cup product. Its formal degree is determined by the exponents, even when that class vanishes; membership of the zero class in a homogeneous summand does not determine the formal degree. Monomials of formal total degree different from $n$ are assigned the value $0$ by convention. The Stiefel-Whitney numbers of $M$ are the values attached to all monomials of total degree $n$. The evaluation is well defined on cohomology and homology classes by The kronecker pairing is independent of cocycle and cycle representatives.

Connected case. If $M$ is connected, then $M$ is path-connected: a manifold is locally path-connected (Topological manifolds are locally compact and locally path connected) and a connected locally path-connected space is path-connected (A connected, locally path-connected space is path-connected, because its path components are open). Hence $M$ is an admissible base and $TM$ is a numerable smooth finite-rank real bundle (Smooth manifolds have CW homotopy type), so the classes $w_i$ are those of the Stiefel-Whitney class construction; for $n=0$ the rank-zero conventions $w_0=1$, $w_i=0$ ($i>0$) are used. In particular the monomial of degree $0$ on a $0$-manifold is the empty product $1$, and $w^{\mathrm{\varnothing }}[M]=⟨1,[M]⟩$ is the parity of the cardinality of $M$ in ${\mathbb{F}}_2$.

General closed manifolds. Let $M$ be an arbitrary closed smooth $n$-manifold. By the componentwise statement for compact manifolds, $M$ has finitely many connected components $M_1,\dots ,M_s$ (Every manifold is F2-orientable and orientability is componentwise); each component is open (Components of a topological manifold are open and at most countable) and closed in the compact $M$, hence compact, and open components carry the restricted smooth structure (An open subset of a smooth manifold has a canonical restricted smooth structure). Each $M_j$ is a closed smooth $n$-manifold, and its canonical mod-two orientation is the restriction of the canonical orientation of $M$. The definition of the number is extended to $M$ by the componentwise sum $w^I[M]:={\sum }_{j=1}^s{w}^I[M_j],$ the sum of the componentwise Stiefel-Whitney numbers; for connected $M$ this is exactly the single evaluation displayed above.

The definitions are independent of all choices: the tangent bundle is determined by the smooth structure, its Stiefel-Whitney classes are determined by the bundle up to isomorphism (Naturality of Stiefel–Whitney classes), the canonical mod-two fundamental class is determined by the canonical mod-two orientation, and the pairing descends through both quotients. No orientation of $M$ is needed or used, and the numbers do not change when an orientation is supplied or reversed.

Behaviour under diffeomorphisms. Let $F:M^{\mathrm{\prime }}\to M$ be a diffeomorphism of closed smooth $n$-manifolds (Diffeomorphisms and local diffeomorphisms of manifolds). Assume first that $M^{\mathrm{\prime }}$ and $M$ are connected. The differential of $F$ identifies $T{M}^{\mathrm{\prime }}$ with the pullback $F^{\ast }TM$, so naturality of Stiefel-Whitney classes gives $w(T{M}^{\mathrm{\prime }})=F^{\ast }w(TM)$, hence $w^I(T{M}^{\mathrm{\prime }})=F^{\ast }{w}^I(TM)$ (Naturality of Stiefel–Whitney classes). The pushforward $F_{\ast }[M^{\mathrm{\prime }}]$ restricts at every $y\in M$ to the image under $dF$ of the canonical local generator at $F^{-1}(y)$; that local module is ${\mathbb{F}}_2$, so its unique nonzero element is carried to the unique nonzero element at $y$, which is the canonical local generator there. By the characterisation of the fundamental class through its pointwise restrictions (Fundamental class of a compact oriented manifold) this gives $F_{\ast }[M^{\mathrm{\prime }}]=[M]$. Naturality of the Kronecker pairing (The kronecker pairing is independent of cocycle and cycle representatives) therefore yields $w^I[M^{\mathrm{\prime }}]=⟨F^{\ast }{w}^I(TM),[M^{\mathrm{\prime }}]⟩=⟨w^I(TM),F_{\ast }[M^{\mathrm{\prime }}]⟩=w^I[M]$. For disconnected $M^{\mathrm{\prime }}$ and $M$ the argument applies to each component and the componentwise sums agree; thus the Stiefel-Whitney numbers are invariants of diffeomorphism. No choice beyond the AC stated above is used.