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Signature formulae for topological invariants. (English) Zbl 1059.58027

This monograph is an exposition on an algebraic method of computing topological invariants of real varieties and maps by using the signature formula which expresses the topological degree of a smooth map-germ \(f:(\mathbb{R}^n,0)\to(\mathbb{R}^n, 0)\) in terms of the signature of a quadratic form on the local algebra \(Q(f)= {\mathcal O}_n/f^*{\mathcal O}_n\). This formula is the counterpart in the real category for the multiplicity of complex analytic-germs \(f:(\mathbb{C}^n,0)\to\mathbb{C}^n,0)\), and is very effective in computation. The proof presented here is based on [G. N. Khimshiashvili, Soobshch. Akad. Nauk Gruz. SSR 85, 309–312 (1977; Zbl 0346.55008)]. The readers are also referred to [D. Eisenbud and H. I. Levine, Ann. Math. (2) 106, 19–44 (1977; Zbl 0398.57020)]. Many applications are introduced, for instance, the Petrovskiĭ-Oleĭnik inequalities [V. I. Arnol’d, Funkts. Anal. Prilozh. 12, No. 1, 1–14 (1978; Zbl 0398.57031)], computing Euler characteristics of real varieties [e.g., Z. Szafraniec, Topology 25, No. 4, 411–414 (1986; Zbl 0611.32007)] topological invariants of real singularities of map-germs [e.g., T. Nishimura, T. Fukuda and K. Aoki, Arch. Ration. Mech. Anal. 108, No. 3, 247–265 (1989; Zbl 0699.58054)] and invariants of totally real surfaces in a complex manifold [e.g., G. Ishikawa and T. Ohmoto, Ann. Global Anal. Geom. 11, No. 1, 125–133 (1993; Zbl 0831.57019)].

MSC:

58K65 Topological invariants on manifolds
57R45 Singularities of differentiable mappings in differential topology
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