Geodesic
Quadratic mappings and smooth vector functions: Euler characteristics of level sets
Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya, Tome 35 (1989), pp. 179-239.

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Quadratic maps of $R^N$ into $R^K$ are studied. Explicit expressions are obtained for the Euler characteristics of level sets of such maps. The Euler characteristics of level sets of smooth vector-valued functions are also evaluated in terms of their values at critical points. 
@article{INTD_1989_35_a3,
     author = {A. A. Agrachev and R. V. Gamkrelidze},
     title = {Quadratic mappings and smooth vector functions: {Euler} characteristics of level sets},
     journal = {Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya},
     pages = {179--239},
     publisher = {mathdoc},
     volume = {35},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTD_1989_35_a3/}
}
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A. A. Agrachev; R. V. Gamkrelidze. Quadratic mappings and smooth vector functions: Euler characteristics of level sets. Itogi Nauki i Tekhniki. Seriya Sovremennye Problemy Matematiki. Noveishie Dostizheniya, Tome 35 (1989), pp. 179-239. http://geodesic.mathdoc.fr/item/INTD_1989_35_a3/