The cobordism class of the multiple points of immersions (Q2426844)

Let \(f:M^ m \rightarrow N^ n\) be a generic immersion between oriented closed even-dimensional smooth manifolds. Then the \(k\)-tuples \((x_1,\dots, x_k)\in M\times\cdots\times M\) such that \(f(x_1)=\cdots =f(x_k)\) and \(x_i\neq x_j\) if \(i\neq j\) form a manifold \(\widetilde\Delta{_ k}(f)\). The symmetric group \(S_k\) on \(k\) elements acts freely on this manifold in the obvious way; let \(\Delta{_ k}(f)\) denote the manifold \(\widetilde\Delta{_ k}(f)/S_k\). The author expresses the Pontryagin numbers of \(\Delta{_ k}(f)\) in terms of certain cohomology invariants of \(M\), \(N\), and \(f\). He derives formulae which generalize, for instance, the virtual signature formula (see 9.3(4') in [\textit{F. Hirzebruch}, Topological methods in algebraic geometry. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. 1978. Classics in Mathematics. (Berlin): Springer-Verlag. (1995; Zbl 0843.14009)]) or formulae of \textit{A. Szűcs} [Proc. Am. Math. Soc. 126, 1873-1882 (1998; Zbl 0896.57019)]. To outline the technical background, let \(i_k: \widetilde\Delta{_ k}(f)\rightarrow M\times\cdots\times M\) (\(k\) factors in the latter) denote the inclusion and \(j_k: \widetilde\Delta{_ k}(f)\rightarrow M\) the projection to the first factor. The author solves a recursion (implied by \textit{F. Ronga}'s theorem on clean intersections [Comment. Math. Helv. 55, 521--527 (1980; Zbl 0457.57013)]) on cohomology classes using power series. This enables him to deduce an explicit formula for \({j_ k}_ !\circ i_ k^\ast\) (where \({j_ k}_ !\) is the cohomology Umkehr homomorphism corresponding to \(j_ k\)) as the key to obtaining explicit formulae for characteristic numbers.