Let $k\ge 0$ be an integer, and let $M_0$ and $M_1$ be closed oriented smooth manifolds of dimension $4k$. Suppose that $M_0$ and $M_1$ are oriented cobordant: there exist a compact oriented smooth manifold $W$ of dimension $4k+1$ and an orientation-preserving diffeomorphism from the boundary $\partial W$, with its boundary orientation, onto the disjoint union $(-M_0)\sqcup M_1$.

For every sequence of nonnegative integers $(a_1,\dots,a_k)$ satisfying $\sum_{j=1}^{k} j a_j = k$, with the empty sequence allowed when $k=0$, the corresponding Pontryagin numbers agree:

\begin{align*} \left\langle p_1(TM_0)^{a_1}\cdots p_k(TM_0)^{a_k},[M_0]\right\rangle = \left\langle p_1(TM_1)^{a_1}\cdots p_k(TM_1)^{a_k},[M_1]\right\rangle. \end{align*}

Here $p_j(TM_i)\in H^{4j}(M_i;\mathbb R)$ denotes the $j$-th Pontryagin class of the smooth real tangent bundle $TM_i\to M_i$, $[M_i]\in H_{4k}(M_i;\mathbb R)$ denotes the orientation fundamental class, and for $k=0$ the monomial is the empty product $1\in H^0(M_i;\mathbb R)$.