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Topology of quadratic maps and Hessians of smooth maps. (English. Russian original) Zbl 0719.58006
J. Sov. Math. 49, No. 3, 990-1013 (1990); translation from Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 26, 85-124 (1988).
Let K be a cone in $${\mathbb{R}}^ k$$ and $$K^*=\{\omega \in ({\mathbb{R}}^ k)^*:\omega$$ (x)$$\leq 0$$ for all $$x\in K\}$$ its dual cone. Consider a symmetric bilinear map p on $${\mathbb{R}}^{N+1}$$ with values in $${\mathbb{R}}^ k$$. For $$\omega \in K^*\setminus \{0\}$$ denote by $$\omega$$ P the operator on $${\mathbb{R}}^{N+1}$$ satisfying $$\omega p(x,y)=((\omega P)x,y)$$ for $$x,y\in {\mathbb{R}}^{N+1}$$ and assume that for $$\omega \in K^*\setminus \{0\}$$ we have that ($$\omega$$ P)x$$\neq 0$$ whenever $$x\in {\mathbb{R}}^{N+1}\setminus \{0\}$$ is such that p(x,x)$$\in K$$. Call $$S^{k- 1}$$ and $$S^ N$$ the unit spheres in $$({\mathbb{R}}^ k)^*$$ and $${\mathbb{R}}^{N+1}$$, let $$\Omega:=K^*\cap S^{k-1}$$ and $$B:=\{(\omega,x)\in \Omega \times S^ N:\omega p(x,x)>0\}$$ and denote by $$\beta_ 1: B\to \Omega$$ the projection onto the first factor. For $$\omega\in \Omega$$, order the eigenvalues $$\lambda_ n(\omega P)$$ of $$\omega$$ P in increasing order and let $$\Omega_ n=\{\omega \in \Omega:\lambda_{n+1}(\omega P)\geq 0\}.$$
The author proves that for an abelian group $${\mathcal A}$$ we have that $$H^ i(\Omega_{N-j},\Omega_{N-j-1};{\mathcal A})=E^{ij}({\mathcal A})$$, where $$(E_ r({\mathcal A}),d_ r)$$ is the Leray spectral sequence for $$\beta_ 1$$. Moreover, for $${\mathcal A}={\mathbb{Z}}_ 2$$, he computes the differential $$d_ 2$$ in terms of cup products. Similar results are obtained when $${\mathbb{R}}^{N+1}$$ is replaced by a separable Hilbert space. Finally, these results are applied to second derivatives of differentiable mappings.

##### MSC:
 58D15 Manifolds of mappings 58C25 Differentiable maps on manifolds