NONIMMERSIONS OF RP n IMPLIED BY tmf, REVISITED

In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The rst is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RPn with n as small as 48, and yet it was stated there that the rst new result occurred when n = 1536. Here we give a simple proof of those overlooked results. Secondly, we ll in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f (X) = X1+X2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in [6]. Finally, we present a complete determination of tmf (RP1 RP1) and tmf (CP1 CP1) in positive dimensions.