Within the Fall of 1975 we begun a joint venture with the final word target of topo­ logically classifying actual algebraic units. This has been an extended satisfied collaboration (c.f., [K2)). In 1985 whereas traveling M.S.R.1. we equipped and offered our type effects as much as that time within the M.S.R.1. preprint sequence [AK14] -[AK17]. given that those effects are interdependent and require a few necessities in addition to familiarity with genuine algebraic geometry, we made up our minds to cause them to self contained by way of offering them as part of a publication in actual algebraic geometry. although we haven't arrived to our ultimate objective but we suppose that it's time to introduce them in a self contained coherent model and show their use by way of giving a few purposes. bankruptcy I provides the review of the category software. bankruptcy II has all of the invaluable history for the remainder of the e-book, which as a result can be utilized as a direction in actual algebraic geometry. It begins with the straight forward houses of actual algebraic units and ends with the new resolution of the Nash Conjecture. bankruptcy III and bankruptcy IV strengthen the idea of answer towers. answer towers are simple topologically outlined items generalizing the proposal of manifold.

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