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**Automorphisms and embeddings of surfaces and quadruple points of regular homotopies.**
*(English)*
Zbl 1032.57030

The author studies a regular homotopy invariant \(Q(\cdot,\cdot)\) of immersions. Let \(i,\;i':F\to \mathbb{R}^3\) be two generic immersions of an orientable closed surface \(F\) to 3-dimensional Euclidean space \(\mathbb{R}^3\), having a generic regular homotopy between them. Then the number \(Q(i,i') \mod 2\) of quadruple points of this regular homotopy is well-defined by Theorem 3.9 of the author’s paper [Topology 39, 1069-1088 (2000; Zbl 0962.57015)]. This article contains two main theorems (Theorems 5.6 and 7.3) concerning the invariant \(Q(i,i')\).

The first one is the following. For any generic immersion \(i:F\to \mathbb{R}^3\) of an orientable closed surface \(F\) to \(\mathbb{R}^3\) and any diffeomorphism \(h\) of \(F\) such that \(i\) and \(i\circ h\) are generically regularly homotopic, it holds that \(Q(i,i\circ h)=\text{rank }(h_*-Id)+(g(F)+1)\epsilon (h) \mod 2\), where \(h_*\) is the isomorphism of \(H_1(F;\mathbb{Z}_2)\) defined by \(h\), \(g(F)\) denotes the genus of \(F\) and \(\epsilon (f)=0\) or \(1\) according \(h\) being orientation-preserving or not.

The second one gives an explicit formula for \(Q(e,e')\) for two generically regularly homotopic embeddings \(e,e':F\to \mathbb{R}^3\). The expression of the formula is really simple. But its calculation may not be as easy as that of the first one, because to determine the first term of the right hand side of the formula, readers are required to construct so called totally singular decompositions of \(H_1(F;\mathbb{Z}_2)\), associated with the embeddings \(e\) and \(e'\), and to determine the rank of the endomorphism \(T-Id\) of \(H_1(F;\mathbb{Z})_2\), where \(T\) is some orthogonal map with respect to the quadratic form induced from the embedding \(e\) and with properties connected to the above decompositions (see §6).

The first one is the following. For any generic immersion \(i:F\to \mathbb{R}^3\) of an orientable closed surface \(F\) to \(\mathbb{R}^3\) and any diffeomorphism \(h\) of \(F\) such that \(i\) and \(i\circ h\) are generically regularly homotopic, it holds that \(Q(i,i\circ h)=\text{rank }(h_*-Id)+(g(F)+1)\epsilon (h) \mod 2\), where \(h_*\) is the isomorphism of \(H_1(F;\mathbb{Z}_2)\) defined by \(h\), \(g(F)\) denotes the genus of \(F\) and \(\epsilon (f)=0\) or \(1\) according \(h\) being orientation-preserving or not.

The second one gives an explicit formula for \(Q(e,e')\) for two generically regularly homotopic embeddings \(e,e':F\to \mathbb{R}^3\). The expression of the formula is really simple. But its calculation may not be as easy as that of the first one, because to determine the first term of the right hand side of the formula, readers are required to construct so called totally singular decompositions of \(H_1(F;\mathbb{Z}_2)\), associated with the embeddings \(e\) and \(e'\), and to determine the rank of the endomorphism \(T-Id\) of \(H_1(F;\mathbb{Z})_2\), where \(T\) is some orthogonal map with respect to the quadratic form induced from the embedding \(e\) and with properties connected to the above decompositions (see §6).

Reviewer: Tsutomu Yasui (Kagoshima)

### MSC:

57R42 | Immersions in differential topology |

57M99 | General low-dimensional topology |

57R40 | Embeddings in differential topology |