Thom's counterexamples for the Steenrod problem

The present paper deals with integral classes $\xi_p\in H_{2p+1}(L^{2p+1}\times L^{2p+1})$ which are counterexamples for the Steenrod realization problem, where $L^{2p+1}$ is the $(2p+1)$-dimensional lens space and $p\geq 3$ is a prime number. For $p=3$, this is 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.