Geodesic
On multiplication in a~complex $K$-functor
Izvestiya. Mathematics , Tome 5 (1971) no. 3, pp. 641-667.

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A model is calculated in this paper of the geometric bundle $\varkappa\in K_C^0(X)$ stably equivalent to the product $\xi\cdot\eta$ of two given bundles $\xi,\eta\in K^1(X)$, i.e. the mapping $X\stackrel\varkappa\to BU$. The result is obtained in an explicit form. 
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     author = {O. V. Manturov},
     title = {On multiplication in a~complex $K$-functor},
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O. V. Manturov. On multiplication in a~complex $K$-functor. Izvestiya. Mathematics , Tome 5 (1971) no. 3, pp. 641-667. http://geodesic.mathdoc.fr/item/IM2_1971_5_3_a7/

[1] Atya M. F., Lektsii po $K$-teorii, Mir, M., 1967 | MR

[2] Khyuzmoller D., Rassloennye prostranstva, Mir, M., 1970

[3] Mosher R., Tangora M., Kogomologicheskie operatsii i ikh prilozheniya v teorii gomotopii, Mir, M., 1970 | MR | Zbl

[4] Shvarts Dzh., Differentsialnaya geometriya i topologiya, Mir, M., 1970 | Zbl

[5] Hodgkin L., “On the $K$-theory of Lie groups”, Topology, 6:1 (1967), 1–36 | DOI | MR | Zbl