Using generating functions, we derive a multiple point formula for every generic immersion between even dimensional oriented manifolds. This produces explicit formulas for the signature and Pontrjagin numbers of the multiple point manifolds. The formulas take a particular simple form in many special cases, eg when the immersion is nullhomotopic, we recover Szűcs’s formulas in [Proc. Amer. Math. Soc. 126 (1998) 1873-1882]. They also include Hirzebruch’s virtual signature formula in Topological methods in algebraic geometry.
"The cobordism class of the multiple points of immersions." Algebr. Geom. Topol. 8 (1) 581 - 601, 2008. https://doi.org/10.2140/agt.2008.8.581