  In the quantum Hall regime, the longitudinal resistivity $\rho_{xx}$ plotted
as a density--magnetic-field ($n_{2D}-B$) diagram displays ringlike structures
due to the crossings of two sets of spin split Landau levels from different
subbands [e.g., Zhang \textit{et al.}, Phys. Rev. Lett. \textbf{95}, 216801
(2005)]. For tilted magnetic fields, some of these ringlike structures "shrink"
as the tilt angle is increased and fully collapse at $\theta_c \approx
6^\circ$. Here we theoretically investigate the topology of these structures
via a non-interacting model for the 2DEG. We account for the inter Landau-level
coupling induced by the tilted magnetic field via perturbation theory. This
coupling results in anti-crossings of Landau levels with parallel spins. With
the new energy spectrum, we calculate the corresponding $n_{2D}-B$ diagram of
the density of states (DOS) near the Fermi level. We argue that the DOS
displays the same topology as $\rho_{xx}$ in the $n_{2D}-B$ diagram. For the
ring with filling factor $\nu=4$, we find that the anti-crossings make it
shrink for increasing tilt angles and collapse at a large enough angle. Using
effective parameters to fit the $\theta = 0^\circ$ data, we find a collapsing
angle $\theta_c \approx 3.6^\circ$. Despite this factor-of-two discrepancy with
the experimental data, our model captures the essential mechanism underlying
the ring collapse.
