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stable equivalence of vector bundles

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Definition

Definition

(stable equivalence of topological vector bundles)

Let XX be a topological space. Define an equivalence relation stable\sim_{stable} on topological vector bundles over XX by declaring two vector bundles E 1E 2Vect(X)E_1 E_2 \in Vect(X) to be equivalent if there exists a trivial vector bundle X×k nX \times k^n of some rank nn such that after tensor product of vector bundles with this trivial bundle, both bundles become isomorphic

(E 1 stableE 2)n(E 1 X(X×k n)E 2 X(X×k n)). \left( E_1 \sim_{stable} E_2 \right) \;\Leftrightarrow\; \underset{n \in \mathbb{N}}{\exists} \left( E_1 \otimes_X (X \times k^n) \;\simeq\; E_2 \otimes_X (X \times k^n) \right) \,.

If E 1 stableE 2E_1 \sim_{stable} E_2 we say that E 1E_1 and E 2E_2 are stably equivalent vector bundles.

Created on May 26, 2017 at 08:40:52. See the history of this page for a list of all contributions to it.