Mathematical Sciences: Homotopy Theory and Its Applications

点击翻译按钮获取中文摘要

9422413 Ravenel Neisendorfer's project is to compute the homology of certain spaces that are inverse limits, for example, a union of infinitely many spheres of radii converging to zero and with one point in common. Using the new algebraic result that an inverse limit of connected free Lie algebras is free, methods of rational homotopy theory can be applied. Deeper results seem to be obtainable using the geometric methods of Pontrjagin and Thom. Ravenel is studying problems in stable homotopy theory related to the telescope conjecture. This appears to be the best approach to understanding the stable homotopy groups of finite complexes, which is one of the oldest problems in algebraic topology. In more general terms, algebraic topology is the study of geometric problems (often in more than 3 dimensions) in their qualitative rather than quantitative aspects, classifying and counting structures rather than measuring their sizes. For example an essential difference between a 2-dimensional plane and a 3-dimensional space is that removing a line separates the former but not the latter into two separate pieces. Topologists have devised methods called homotopy and homology for keeping track of such properties systematically in very complicated geometric objects (objects than can arise, for example, as the spaces of solutions to sytems of ordinary or differential equations). Neisendorfer and Ravenel's research is concerned with refining and extending these methods. ***