Stable classification of homotopy equivalences between two manifolds deals with the following question: If $f : M \to N$ is a homotopy equivalence between two manifolds, when is $f \times id\sb{\IR\sp{k}} : M \times \IR\sp{k} \to N \times \IR\sp{k}$ homotopic to a homeomorphism? Due to Mazur's result, we know that this is true as long as $k \ge\ {\rm dim} M + 2$ and f is tangential. Therefore, we try to minimize the value of k by imposing additional conditions. It is already known that if for each Sylow subgroup H of $\pi\sb1 M,$ where $\pi\sb1 M$ is finite, and the liftings $f\sb{H} : M\sb{H}\to N\sb{H}$ are homotopic to a homeomorphism, then $f\times id\sb{\IR\sp3} : M\times \IR\sp3 \to N \times \IR\sp3$ is homotopic to a homeomorphism. The next step would be to ask if under some additional conditions it is possible to obtain stabilization of f on multiplication by $\IR\sp2$. This turns out to be the most intricate case and this dissertation is devoted to its analysis The central ideas for the proof come from the inductive detection phenomenon and surgery theory of compact and non-compact manifolds. Heavy use of computational results for L-groups of cyclic groups of order $p,\ \doubz\sb{p},$ as well as analysis of the surgery exact sequence are required at several stages