Stiefel-Whitney numbers detect (unoriented) bordism classes and together with Pontryagin numbers, they determine oriented bordism classes.

• Stiefel-Whitney numbers detect (unoriented) bordism classes and together with Pontryagin numbers, they determine oriented bordism classes.
• Ranicki's total surgery obstruction of a finite $n$-dimensional Poincaré complex vanishes if and only the space is homotopy equivalent to a closed topological manifold of the same dimension.