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.