  A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a
double point self-intersection set which is the image of an immersion of a
smooth surface, the double point self-intersection surface. We prove that this
surface may have odd Euler characteristic if and only if k is congruent to 1
modulo 4 or k+1 is a power of 2. This corrects a previously published result by
Andras Szucs.
  The method of proof is to evaluate the Stiefel-Whitney numbers of the double
point self-intersection surface. By earier work of the authors these numbers
can be read off from the Hurewicz image h(\alpha ) in H_{2k+2}\Omega ^{\infty
}\Sigma ^{\infty }MO(k) of the element \alpha in \pi _{2k+2}\Omega ^{\infty
}\Sigma ^{\infty }MO(k) corresponding to the immersion under the
Pontrjagin-Thom construction.
