All characteristic numbers vanish on null-cobordant manifolds

Unoriented case. Suppose $M$ is null-cobordant, so by [F1] there is a bordism from $M$ to the empty $n$-manifold. The empty manifold is a closed smooth $n$-manifold, and by [F3] each of its Stiefel-Whitney numbers is the empty componentwise sum $0$. Applying the invariance theorem [F2] to the pair $(M,\mathrm{\varnothing })$ gives $w^I[M]=w^I[\mathrm{\varnothing }]=0\in {\mathbb{F}}_2$ for every monomial $w^I$ of total degree $n$.

Oriented case. Suppose now that $(M,o)$ is a closed oriented $4k$-manifold that is null-cobordant in the oriented theory. By [F1] there is an oriented bordism from $(M,o)$ to the empty oriented $4k$-manifold, whose Pontryagin numbers are the empty sums $0$ by [F3]. Applying the oriented half of [F2] gives $p_J[M]=p_J[\mathrm{\varnothing }]=0\in \mathbb{Z}$ for every partition $J$ of $k$; the same comparison gives the vanishing of the Stiefel-Whitney numbers of $M$ as well.

Equivalence and conventions. Taking contrapositives, a nonzero $w^I[M]$ obstructs unoriented null-cobordism of $M$, and a nonzero $p_J[M]$ obstructs oriented null-cobordism; this is the stated equivalence, since a manifold is null-cobordant precisely when its class is zero in the corresponding bordism group [F1]. If $M$ is presented as the boundary of a compact $W$, the collar data of the null-cobordism definition give a bordism from $M$ to $\mathrm{\varnothing }$, so steps 1.1 and 1.2 re-derive the published boundary-vanishing propositions [F4]. In degree zero, a closed $0$-manifold is a finite set of signed points and its only Stiefel-Whitney number is the cardinality mod $2$; null-cobordism forces an even cardinality by [F4], matching step 1.1, and in the oriented theory the signed count is zero, matching step 1.2. For $n=0$ the oriented case $k=0$ and the empty manifold are both covered by the empty-sum convention; no choice beyond the cited suppliers is used, since only bordism data and the componentwise sums are compared.