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The author, a leading figure in algebraic topology, provides a modern treatment of a long established set of questions in this important research area. The book's principal objective--and main result--is the classification theorem on k-variants and boundary invariants, which supplement the classical picture of homology and homotopy groups, along with computations of types that are obtained by applying this theorem. Research mathematicians in algebraic topology will be interested in this new attempt to classify homotopy types of simply connected CW-complexes.