##
**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.

\[ f^\ast n_r = m_{r+1} + e \cup m_r \in H^{rk}(M; {\mathbb Z}_2) \]

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.

Reviewer: James Hebda (St. Louis)

### MSC:

57R42 | Immersions in differential topology |

55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |

57R90 | Other types of cobordism |

PDF
BibTeX
XML
Cite

\textit{P. J. Eccles} and \textit{M. Grant}, Algebr. Geom. Topol. 7, 1081--1097 (2007; Zbl 1136.57016)

### References:

[1] | M F Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961) 200 · Zbl 0104.17405 |

[2] | M G Barratt, P J Eccles, \(\Gamma^+\)-structures III: The stable structure of \(\Omega^{\infty}\Sigma^{\infty}A\), Topology 13 (1974) 199 · Zbl 0304.55010 |

[3] | C Biasi, O Saeki, On the Betti number of the image of a codimension-\(k\) immersion with normal crossings, Proc. Amer. Math. Soc. 123 (1995) 3549 · Zbl 0855.57023 |

[4] | G Braun, G Lippner, Characteristic numbers of multiple-point manifolds, Bull. London Math. Soc. 38 (2006) 667 · Zbl 1102.57017 |

[5] | J Caruso, F R Cohen, J P May, L R Taylor, James maps, Segal maps, and the Kahn-Priddy theorem, Trans. Amer. Math. Soc. 281 (1984) 243 · Zbl 0549.55005 |

[6] | P J Eccles, M Grant, Bordism classes represented by multiple point manifolds of immersed manifolds, Tr. Mat. Inst. Steklova 252 (2006) 55 · Zbl 1351.57031 |

[7] | T Ekholm, A Sz\Hucs, Geometric formulas for Smale invariants of codimension two immersions, Topology 42 (2003) 171 · Zbl 1024.57026 |

[8] | M Grant, Bordism of immersions, PhD thesis, University of Manchester (2006) |

[9] | R J Herbert, Multiple points of immersed manifolds, Mem. Amer. Math. Soc. 34 (1981) · Zbl 0493.57012 |

[10] | K Jänich, On the classification of \(O(n)\)-manifolds, Math. Ann. 176 (1968) 53 · Zbl 0153.53801 |

[11] | M Kamata, On multiple points of a self-transverse immersion, Kyushu J. Math. 50 (1996) 275 · Zbl 0882.57023 |

[12] | R C Kirby, The topology of 4-manifolds, Lecture Notes in Mathematics 1374, Springer (1989) · Zbl 0668.57001 |

[13] | U Koschorke, B J Sanderson, Self-intersections and higher Hopf invariants, Topology 17 (1978) 283 · Zbl 0398.57030 |

[14] | D Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971) · Zbl 0214.50502 |

[15] | F Ronga, On multiple points of smooth immersions, Comment. Math. Helv. 55 (1980) 521 · Zbl 0457.57013 |

[16] | C P Rourke, B J Sanderson, The compression theorem I, Geom. Topol. 5 (2001) 399 · Zbl 1002.57057 |

[17] | C P Rourke, B J Sanderson, The compression theorem III: Applications, Algebr. Geom. Topol. 3 (2003) 857 · Zbl 1032.57029 |

[18] | R E Stong, Notes on cobordism theory, Mathematical notes, Princeton University Press (1968) · Zbl 0181.26604 |

[19] | R M Switzer, Algebraic topology-homotopy and homology, Die Grundlehren der mathematischen Wissenschaften 212, Springer (1975) · Zbl 0305.55001 |

[20] | A Sz\Hucs, Cobordism of singular maps, |

[21] | A Sz\Hucs, Cobordism groups of \(l\)-immersions I: (A) Homotopy representability of the cobordism group of immersions with a given multiplicity of self-intersections, Acta Math. Acad. Sci. Hungar. 27 (1976) 343 · Zbl 0356.57024 |

[22] | A Sz\Hucs, Cobordism groups of \(l\)-immersions II, Acta Math. Acad. Sci. Hungar. 28 (1976) 93 · Zbl 0364.57001 |

[23] | A Sz\Hucs, On the multiple points of immersions in Euclidean spaces, Proc. Amer. Math. Soc. 126 (1998) 1873 · Zbl 0896.57019 |

[24] | M Takase, A geometric formula for Haefliger knots, Topology 43 (2004) 1425 · Zbl 1060.57021 |

[25] | R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17 · Zbl 0057.15502 |

[26] | P Vogel, Cobordisme d’immersions, Ann. Sci. École Norm. Sup. \((4)\) 7 (1974) · Zbl 0302.57011 |

[27] | R Wells, Cobordism groups of immersions, Topology 5 (1966) 281 · Zbl 0145.20202 |

[28] | H Whitney, On the topology of differentiable manifolds, University of Michigan Press (1941) 101 · Zbl 0063.08233 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.