Real Johnson-Wilson Theories and Computations

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Johns Hopkins University
Our main result is a computation of ER(n)^*(CP^\infty), the Real Johnson-Wilson cohomology of CP^\infty, for all n. We apply techniques from equivariant stable homotopy theory to the Bockstein spectral sequence. We produce permanent cycles, compute differentials, and solve extension problems to give an explicit description of the ring ER(n)^*(CP^\infty). In the case n=1, our results reproduce KO^*(CP^\infty) as computed by Sanderson, Fujii, Yamaguchi, and Bruner and Greenlees. In the case n=2, our result yields the TMF_0(3)-cohomology of CP^\infty after a suitable completion. This thesis forms part of a program to compute the ER(n)-cohomology of basic spaces. We conclude with a discussion of work in progress with Kitchloo and Wilson on the ER(n)-cohomology of CP^k, classifying spaces of various groups, and Eilenberg-MacLane spaces, as well as future directions and possible applications to topology and geometry. We include an appendix which proves some lemmas in equivariant stable homotopy theory used in our computations.
Algebraic topology, Homotopy Theory