Let \(M_{2n}\) and \(N_{4n}\) be closed and orientable Riemanian manifolds of dimension \(2n\) and \(4n\), respectively. Numerical bounds on the Euler-Poincaré characteristic \(\chi(M_{2n})\) and the Pontryagin numbers \(P_I(N_{4n})\) are presented in terms of the sectional curvature \(k\leq\text{sec}\leq K\) and radius \(r\). Here the letter \(I\) appearing in \(P_I(N_{4n})\) indicates a partition \(I= k_1,\dots,k_s\) of unity. They are summarized as follows: \(|\chi (M_{2n})|\) is bounded by \(\frac{(2n)!^2}{n!^2}\frac{(Kr^2)^n}{2^{3n}}\) if \(k=0\), and if \(k<0\) by
\[ \frac{(2n)!^2C^n(-1)^{n-1}}{2^{3n-1}n!(n-1)!(-k)^n}\sum^{n-1}_{m=0}(-1)^m{n-1\choose m}\left(\frac {\cosh^{2m +1}(r\sqrt{-k})-1}{2m+1} \right) \] where \(C\) is given by \[ |R_{ij1m}|<C=\max(-k,K,\tfrac 23(K-K)). \] Next, \(|P_I(N_{4n})|\) is bounded by
\[ \frac{(4n)!^{s+1}}{(2n)!\prod^s_{i=1}(4n-2k_i)!} \frac{(Kr^2)^{2n}}{2^{4n}} \]
if \(k=0\), and by
\[ A(n)\frac{C^{2n}} {(-k)^{2n}} \sum^{2n-1}_{m=0}(-1)^m{2n-1\choose m}\left(\frac{\cosh^{2m+1}(r\sqrt{-k})-1}{2m+2} \right) \]
where
\[ A(n)=\frac {(4n)!^{s+1}(-1)^{2n-1}}{(2n-1)!2^{4m-1}\prod^s_{i=1} (4n-2K_i)!} \]
if \(k<0\).