Applications of Homotopy Theory

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AbstractAward: DMS-0707014Principal Investigator: James E. McClureThe principal investigator plans to work in four quite differentareas where homotopy-theoretic methods can be applied. The firstarea is string topology. This part of the project is joint withRalph Kaufmann; Kaufmann and McClure will combine their previouswork on string topology in order to obtain a lifting of stringtopology to the chain level for arbitrary genus. The second areais surgery spectra. This is joint with Gerd Laures; Laures andMcClure will complete their proof that certain symmetricL-spectra have E-infinity ring structures and that the symmetricsignature is an E-infinity ring map. They will also extend theirwork in order to show that a certain map from algebraic K-theoryto symmetric L-theory defined by Weiss and Williams is anE-infinity ring map. The third area is intersection homology.This is joint with Greg Friedman and Scott Wilson; Friedman,McClure and Wilson will work to develop homotopy theoreticstructure on intersection chains which should (for example) leadto the development of a rational homotopy theory of stratifiedspaces. The fourth area is C* algebras. This is joint withMarius Dadarlat; Dadarlat and McClure will work to show that theclassical work of Dixmier and Douady on bundles whose fibre isthe algebra of compact operators can be extended to bundles whosefibre is a Kirchberg algebra.The original goal of homotopy theory was to develop methods whichcould be applied to geometric problems (including problems indimensions higher than three). Since its origin in the earlytwentieth century, homotopy theory has developed into anautonomous discipline, but the connection with geometry hasalways been a central concern. Since about 1960, it has becomeapparent that homotopy theory has applications to other thingsbesides geometry, for example to analysis, algebra, and morerecently to physics. The work of the principal investigator andhis coauthors is intended to take the latest developments inhomotopy theory (for example, the theory of symmetric spectra)and apply them to other areas, including geometric problems (theclassification of higher-dimensional shapes), and problems inmathematical physics (string topology is part of the programbeing carried on by many people to develop a clearer mathematicalunderstanding of ideas from string theory).