Nonimmersions of $RP^n$ implied by tmf, revisited

Abstract

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 first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for $RP^n$ with $n$ as small as 48, and yet it was stated there that the first new result occurred when $n = 1536$. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map $f$ must satisfy $f^*(X) = X_1 + X_2$. 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 the 2002 paper. Finally, we completely determine $tmf^{8*}(RP^\infty × RP^infty)$ and $tmf^*(CP^\infty × CP^\infty)$ in positive dimensions.