## 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.

### MSC:

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

### Keywords:

immersions; bordism; cobordism; Herbert’s formula
Full Text:

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