Relations of smooth Kervaire classes over the mod 2 Steenrod algebra

In the surgery theory of smooth smooth manifolds, it is often difficult to determine the existence or the non-existence of a smooth normal map with nontrivial surgery obstructions. We present and prove a simple relation over the mod 2 Steenrod algebra between two smooth Kervaire classes in different dimensions. This formula enables us to compute the Kervaire surgery invariants for various manifolds.