Pontryagin numbers of a closed oriented manifold

Assume AC (The Axiom of Choice). The assumption is inherited from the Pontryagin class construction and is used only there, through Pontryagin classes by complexification its CW-type transport below, and the admissibility supplied by Smooth manifolds have CW homotopy type.

Let $M$ be a closed oriented smooth manifold of dimension $4k$ for an integer $k\ge 0$, with orientation $o$. Its fundamental class $[M]\in H_{4k}(M;\mathbb{Z})$ is the class determined by $o$ (Fundamental class of a compact oriented manifold). Its tangent Pontryagin classes, taken componentwise when $M$ is disconnected, are $p_i:=p_i(TM)\in H^{4i}(M;\mathbb{Z})(i\ge 0),$ with the conventions $p_0=1$ and $p_i=0$ whenever $2i>4k$.

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 for the characteristic-class construction: it is paracompact Hausdorff of CW homotopy type and its tangent bundle is a numerable smooth finite-rank real bundle (Smooth manifolds have CW homotopy type). For a partition $I=(i_1,\dots ,i_r)$ of $k$ with $i_j\ge 1$ (for $k=0$ the empty partition, with empty product $1$), the $I$-th Pontryagin number of $M$ is $p_I[M]:=⟨p_{i_1}\cdots p_{i_r},[M]⟩\in \mathbb{Z},$ the Kronecker evaluation of the cup product of the tangent Pontryagin classes on the fundamental class (Kronecker evaluation pairing); the value is well defined on cohomology and homology classes by The kronecker pairing is independent of cocycle and cycle representatives.

General closed oriented manifolds. Let $M$ be an arbitrary closed oriented manifold of dimension $4k$. By the componentwise statement for compact manifolds, $M$ has finitely many connected components $M_1,\dots ,M_s$; each component is open (Components of a topological manifold are open and at most countable) and closed in the compact $M$, hence compact, carries the restricted smooth structure as an open submanifold (An open subset of a smooth manifold has a canonical restricted smooth structure), and carries the orientation restricted to it; the family of these restricted orientations is the componentwise orientation of $o$ (Every manifold is F2-orientable and orientability is componentwise). Each $M_j$ is a closed oriented smooth $4k$-manifold and the connected case above applies. We set $p_I[M]:={\sum }_{j=1}^s{p}_I[M_j],$ the sum of the componentwise Pontryagin numbers; for connected $M$ this is exactly the single evaluation displayed above. A manifold whose dimension is not $4\mathrm{\mid }I\mathrm{\mid }$ is assigned the value $0$ by convention, and $p_I[M]$ for a partition $I$ with $i_j\ge 1$ never sees a class $p_i$ of index beyond the dimension.

The definition is independent of all choices: the complexification of $TM$ is determined up to canonical isomorphism, so the even Chern classes, hence the classes $p_i$, are determined; the fundamental class is determined by the orientation; and the Kronecker pairing descends through both quotients. The classes $p_i(TM)$ do not depend on the orientation. Replacing $o$ by $-o$ negates the fundamental class on every component (Fundamental class of a compact oriented manifold) and therefore negates every Pontryagin number, by linearity of the pairing in its second variable (The kronecker pairing is independent of cocycle and cycle representatives).

Naturality and stability on CW-type bases. The cited Pontryagin theorem is stated for path-connected CW complexes, whereas smooth manifolds here are only known to have CW homotopy type. The needed extension is as follows. For a numerable complex bundle $V$ over a path-connected paracompact Hausdorff CGWH base $B$ of CW type, choose homotopy inverse maps $h:K\to B$ and $g:B\to K$ with $K$ a path-connected CW complex. Homotopy invariance of bundle pullback gives $V\cong g^{\ast }{h}^{\ast }V$ (Homotopy invariance of vector-bundle pullback). The Chern naturality theorem permits a CW-type source and a CW target, so $c_j(V)=g^{\ast }{c}_j(h^{\ast }V)$ (Naturality, normalization, and Whitney sum for Chern classes, Chern classes from the projective-bundle relation). For $f:B^{\mathrm{\prime }}\to B$ between such bases, $f^{\ast }V\cong (g\circ f{)}^{\ast }{h}^{\ast }V$, so the same theorem with target $K$ gives $c_j(f^{\ast }V)=f^{\ast }{c}_j(V)$. Also $V\oplus {\epsilon }^r\cong g^{\ast }(h^{\ast }V\oplus {\epsilon }^r)$; Chern stability on $K$ and naturality along $g$ give $c_j(V\oplus {\epsilon }^r)=c_j(V)$. Complexification commutes with pullback and adjoining trivial summands, as seen from their transition matrices. The formula $p_i(E)=(-1{)}^i{c}_{2i}(E_{\mathbb{C}})$ therefore proves naturality and stability of Pontryagin classes on these CW-type bases too. For a finite disjoint union define the classes componentwise: every singular simplex lies in one component, so cohomology is the finite product of the component rings, with pullbacks and cup products computed componentwise. This also handles maps whose different source components land in the same target component. For the empty base all classes and evaluations have their unique zero values. AC is inherited by this transport from the stated bundle-homotopy and characteristic-class suppliers.

Behaviour under diffeomorphisms. Let $F:M^{\mathrm{\prime }}\to M$ be a diffeomorphism of closed oriented $4k$-manifolds (Diffeomorphisms and local diffeomorphisms of manifolds). Assume first that $M^{\mathrm{\prime }}$ and $M$ are connected and that $F$ is orientation-preserving (Orientation-preserving parametrizations). The differential of $F$ identifies $T{M}^{\mathrm{\prime }}$ with the pullback $F^{\ast }TM$, so naturality of Pontryagin classes gives $p_i(T{M}^{\mathrm{\prime }})=F^{\ast }{p}_i(TM)$ (by the CW-type derivation above). The pushforward $F_{\ast }[M^{\mathrm{\prime }}]$ restricts at every $y\in M$ to the image under $dF$ of the local generator of $o^{\mathrm{\prime }}$ at $F^{-1}(y)$, which is the local generator of $o$ at $y$ because $F$ is orientation-preserving; by the characterisation of the fundamental class through its pointwise restrictions (Fundamental class of a compact oriented manifold) this means $F_{\ast }[M^{\mathrm{\prime }}]=[M]$. Naturality of the Kronecker pairing (The kronecker pairing is independent of cocycle and cycle representatives) therefore gives $p_I[M^{\mathrm{\prime }}]=⟨p_I(T{M}^{\mathrm{\prime }}),[M^{\mathrm{\prime }}]⟩=⟨F^{\ast }{p}_I(TM),[M^{\mathrm{\prime }}]⟩=⟨p_I(TM),F_{\ast }[M^{\mathrm{\prime }}]⟩=⟨p_I(TM),[M]⟩=p_I[M]\mathrm{.}$ If $F$ is orientation-reversing, the same computation gives local generators that are negatives of those of $o$, so $F_{\ast }[M^{\mathrm{\prime }}]=-[M]$ and $p_I[M^{\mathrm{\prime }}]=-p_I[M]$. For disconnected $M^{\mathrm{\prime }}$ and $M$ the argument applies to each component, and the sum of the componentwise numbers transforms accordingly. In particular a nonzero Pontryagin number obstructs the existence of an orientation-reversing self-diffeomorphism. No choice beyond the AC stated above is used.