Nonimmersions of RP^n implied by tmf, revisited
Donald M. Davis and Mark Mahowald
dmd1@lehigh.edu
mark@math.northwestern.edu
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 original paper.
Finally, we present a complete determination of
tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty)
in positive dimensions.