Recherche en théorie de l'homotopie

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The major theme of this proposal is research in homotopy theory. My program will explore two main directions: (I) topological complexity and L.-S.-category; and (II) non-simply connected spaces. ***(I) Topological complexity The image that one should keep in mind is the one of a robotic arm positioned somewhere fixed in an assembly plant X. The problem is: is it possible to find a continuous path that the arm will follow between any two given points that the arm can reach, i.e., does the diagonal map d:X -> X x X admit an appropriate continuous section? Unfortunately the only assembly plant X for which the map d admits a global section is one with no obstruction (no pillard to hold up the roof, not even the robot itself!). Hence we have to settle for the next best thing: continuous local sections, i.e., the topological complexity, TC(X), of a space X is the least natural number k such that there exists a covering of X x X formed by k+1 open sets U1,,Uk+1, admitting continuous local sections si:Ui->XI of the classical evaluation fibration XI->X x X associated to the diagonal map d. In fact TC(X) is a special case of the sectional category of a fibration within the Lusternick-Schnirelmann category theory framework. The study of these L.S.-type inavariants have attracted massive amount of resources over the last 40 years and continues to this date to be a very dynamic subject with applications to homotopy theory, differential geometry, dynamical systems and engineering. It is ideal for HQP's in both pure and/or applied mathematics. My goal for the next five years is to pursue the study of these invariants and to incorporate to my research at least one Ph.D. student and two Master's students. ***(II) Non-simply connected spaces - Quillen and Sullivan constructed theories to algebraically model the rational homotopy type of simply-connected spaces. Unfortunately, both can't handle spaces like RP2. To remedy this situation, Bousfield and Kan proposed the following: Let X be a space admitting a universal cover UX and consider the associated fibration UX->X->B(pi1(X)) and apply fibrewise rationalization to obtain a new fibration UXo->X'->B(pi1(X)), where UXo is the usual rationalization of the simply-connected space UX. The total space X' of that fibration is the model for the rationalization of the original space X. The next step is to find an algebraic category to characterize X' in the spirit of Quillen and Sullivan. Gomez-Tato, Halperin and Tanré constructed an algebraic category of local systems that admitted minimal models and realization functors very much like Sullivan. The main drawback of their theory is that it only applies to spaces having a finite type universal cover. My goal is to expand these ideas to all universal covers by using the Quillen vision to model UXo and build those local systems. This direction is ideal to incorporate a post-doctoral fellow and at least one Ph.D. student.