Slicing knots and homology cobordism

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Among the most active areas of research in low dimensional topology over the past 15 years takes origin in the early work of Fox and Milnor, in the 1950s. A knot is slice if it bounds an embedded disk in a four dimensional ball, a geometric condition which determines an abelian group of equivalence classes, the concordance group of knots. Understanding this relation, both as stated above, and a higher dimensional version, dominated much of the discussion on embedding theory throughout the 1960's and early 1970's.Two modern advances in this area have once again re-energized mainstream knot theory. New approaches using Whitney towers and related geometric underpinnings, along with the introduction of L2-Index theory, have clarified a possible path toward the classification of topological concordance. The dynamic new Heegaard-Floer invariants have revealed secrets of smoothing theory in low dimensions.These new tools draw us to the more general problem of classifying homology cobordism of manifolds. We discuss the problem, some early triumphs, and how these toolspotentially crack adamant problems in manifold theory, while discussing recent breakthroughs, and the hope for a future classification.