Let \( f: M^{n-k} \rightarrow N^n\) be a self-transverse immersion of the manifold \(M\) into \(N\). For each integer \(r \geq 1\), one associates immersions \(\psi_r\) and \(\mu_r\) of certain \((n-rk)\)-dimensional manifolds into \(N\) and \(M\), respectively. These so-called self-intersection immersions cover the \(r\)-fold self-intersection sets of \(f\) in \(N\) and \(M\), respectively. The immersions \(\psi_r\) and \(\mu_r\) define homology classes in \(N\) and \(M\), respectively, whose Poincaré dual classes are denoted \(n_r\) and \(m_r\), respectively. A formula of Herbert states that
\[ f^\ast n_r = m_{r+1} + e \cup m_r \in H^{rk}(M; {\mathbb Z}_2) \]
where \(e\) is the Euler class of the normal bundle of \(f\).
The content of this paper can now be summarized in the authors’ own words:
We prove a geometrical version of Herbert’s formula by considering the self-intersection immersions of a self-transverse immersion up to bordism. This clarifies the geometry lying behind Herbert’s formula and leads to a homotopy commutative diagram of Thom complexes. It enables us to generalize the formula to other homology theories. The proof is based on Herbert’s but uses the relationship between self-intersections and stable Hopf invariants and the fact that bordism of immersions gives a functor on the category of smooth manifolds and proper immersions.