Abstract
We evaluate the cohomology obstructions to the existence of fiber-preserving unital embedding of a locally trivial bundle A k → X whose fiber is a complex matrix algebra M k (ℂ) in a trivial bundle with fiber M kl (ℂ) under the assumption that (k, l) = 1. It is proved that the first obstruction coincides with the obstruction to the reduction of the structure group PGL k (ℂ) of the bundle A k to SL k (ℂ), which coincides with the first Chern class c 1(ξ k ) reduced modulo k under the assumption that A k ≌ End(ξ k ) for some vector ℂ k -bundle ξ k → X. If the first obstruction vanishes, then A k ≌ End(\(\tilde \xi _k \)) for some vector ℂk bundle ξ k → X with structure group SL k (ℂ), and the second obstruction is c 2(\(\tilde \xi _k \))modk ∈ H 4(X, ℤ/kℤ). Further, the higher obstructions are defined using a Postnikov tower, and each of the obstructions is defined on the kernel of the previous obstruction.
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Original Russian Text © A. V. Ershov, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 4, pp. 521–540.
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Ershov, A.V. Obstructions to embeddings of bundles of matrix algebras in a trivial bundle. Math Notes 94, 482–498 (2013). https://doi.org/10.1134/S0001434613090198
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DOI: https://doi.org/10.1134/S0001434613090198