Bordism groups of immersions and classes represented by self-intersections. (English) Zbl 1136.57016

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.


57R42 Immersions in differential topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
57R90 Other types of cobordism
Full Text: DOI arXiv


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