In this talk I will supply some details, including fairly complete proofs, of the results I stated in my colloquium on the previous day. Namely, I'll explain how to construct all "rational homotopy periods" of simply-connected spaces. Note that having attended the colloquium isn't a prerequisite. We'll establish basic properties of these generlized Hopf invariants. To do so at some point we'll need to (again) treat the homology and cohomology of configuration spaces. We then apply loopspace machinery to show relate this homology and cohomology to our homotopy functionals and Whitehead products. We will also show how Boardman-Steer-Hopf invariants are included in this approach. We'll also compare with approaches of Sullivan and Hain. We'll end with a number of ideas for further work - the Hopf invariant one problem for general spaces, analogues at characteristic p, "homotopy Jacobians" for varieties, and rational mapping space invariants. |