Thom's counterexamples for the Steenrod problem

The present paper deals with integral classes ξₚ∈ H₂ₚ₊₁(L₁²ᵖ⁺¹× L₂²ᵖ⁺¹) which are counterexamples for the Steenrod realization problem, where L₁²ᵖ⁺¹ is the $(2p+1)$-Lens space and p≥ 3 is a prime number. For $p=3$, this is the Thom's famous counterexample. We give a geometric description of this class using the theory of stratifolds. As a consequence, we obtain a geometric interpretation of the obstruction to realizability, in terms of the Atiyah-Hirzebruch spectral sequence.