By Selman Akbulut, Henry King (auth.)

In the autumn of 1975 we began a joint venture with the final word objective of topo logically classifying genuine algebraic units. This has been an extended chuffed collaboration (c.f., [K2)). In 1985 whereas vacationing M.S.R.1. we prepared and offered our type effects as much as that time within the M.S.R.1. preprint sequence [AK14] -[AK17]. seeing that those effects are interdependent and require a few necessities in addition to familiarity with genuine algebraic geometry, we determined to lead them to self contained through offering them as part of a e-book in genuine algebraic geometry. even if we haven't arrived to our ultimate target but we believe that it's time to introduce them in a self contained coherent model and exhibit their use through giving a few purposes. bankruptcy I offers the evaluate of the category application. bankruptcy II has the entire useful history for the remainder of the e-book, which as a result can be utilized as a path in genuine algebraic geometry. It begins with the undemanding houses of actual algebraic units and ends with the hot answer of the Nash Conjecture. bankruptcy III and bankruptcy IV enhance the idea of answer towers. solution towers are easy topologically outlined gadgets generalizing the proposal of manifold.