The locally constant homotopy functors of the title are: (1) L-theory, (2) the homotopy fiber of the cyclotomic trace map from algebraic K-theory K to topological cyclic homology TC. These are functors from very general rings to spectra, and may be viewed as functors on spaces by composition with a well known functor from spaces to very general rings. "Locally constant" means (in this case) that they don't distinguish between spaces with the same fundamental groups. L-theory arises in the classification of closed manifolds in a fixed homotopy type, and algebraic K-theory arises in the classification "up to product with [0,1]" of compact manifolds in a fixed homotopy type. TC has achieved fame as a good and highly computable, if mysterious, approximation to algebraic K-theory. K and TC are not locally constant in their own right. I want to present and defend my belief that there should be a homotopy fiber sequence of (locally constant) functors L --> t(Z/2;K) --> t(Z/2;TC). Here t(Z/2; ) is "Tate cohomology of the group Z/2", which acts on algebraic K-theory and TC by a suitable duality involution. Put differently, L should be the same as t(Z/2; ) of the homotopy fiber of the cyclotomic trace K --> TC (at least for `easy' rings).
Monday April 10th, 10.00-11.45 (NB: STRANGE NEW TIME), room 656