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The ordinary cross product in R3 is a widespread tool in mathematics and other sciences. It has applications in many areas such as several variable calculus, abstract algebra, geometry, and physics. In this thesis, we investigate in which Euclidean spaces R𝑛 there exist cross products. Based on the properties of the cross product in R3, we introduce two different notions of a cross product in R𝑛. Our first definition is based on the Pythagorean property and the perpendicular property of the cross product in R3. By direct calculation, we show that there is exactly one cross product in R1, no cross product in R2, and exactly two cross products in R3. We also show that if R𝑛 has a cross product, then 𝑛 = 1, 3, or 7. Our second definition uses the following self-selected properties of the cross product in  R3: the triple property, and the nondegeneracy property, leading to the notion of a semi-crossproduct. By direct computation, we discover that R3 has exactly two semi-cross products, which coincide with its cross products, moreover, there does not exist any semi-cross product in R1 or R2. The main result of the thesis is that there are no semi-cross products in R𝑛 for 𝑛 ≥ 4. As far as we know, the results of this chapter are new.

On associe à chaque variété algébrique définie sur R un complexe de cochaînes filtré, qui calcule la cohomologie à supports compacts et coefficients dans Z_2 de ses points réels. Ce complexe filtré est additif pour les inclusions fermées et acyclique pour la résolution des singularités, et est unique à quasi-isomorphisme filtré près. Il est représenté par la filtration duale de la filtration géométrique sur les chaînes semi-algébriques à supports fermés définie par McCrory and Parusiński, et induit une suite spectrale qui calcule la filtration par le poids sur la cohomologie à supports compacts. Cette suite spectrale est un invariant naturel qui contient les nombres de Betti virtuels. On montre que le produit cartésien de deux variétés nous permet de comparer le produit de leurs complexe de poids et suite spectrale respectifs avec ceux du produit, et on prouve que les produits cap et cup en cohomologie et homologie sont filtrés par rapport à ces filtrations par le poids réelles. / We associate to each algebraic variety defined over R a filtered cochain complex, which computes the cohomology with compact supports and Z_2-coefficients of the set of its real points. This filtered complex is additive for closed inclusions and acyclic for resolution of singularities, and is unique up to filtered quasi-isomorphism. It is represented by the dual filtration of the geometric filtration on semialgebraic chains with closed supports defined by McCrory and Parusiński, and leads to a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the additive virtual Betti numbers. We then show that the cross product of two varieties allows us to compare the product of their respective weight complexes and spectral sequences with those of their product, and prove that the cup and cap products in cohomology and homology are filtered with respect to the real weight filtrations.