Odd primary analogs of real orientations

Abstract

We define, in C p -equivariant homotopy theory for p > 2, a notion of p -orientation analogous to a C 2 -equivariant Real orientation. The definition hinges on a C p -space CP 1p , which we prove to be homologically even, in a sense generalizing recent C 2 -equivariant work on conjugation spaces.We prove that the height p 1 Morava E-theory is p -oriented and that tmf.2/ is 3 -oriented. We explain how a single equivariant map v p 1 W S 2 ! † 1 CP 1 p completely generates the homotopy of E p 1 and tmf.2/, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.