nLab stable equivalence of vector bundles

Definition

Definition

(stable equivalence of topological vector bundles)

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

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

If $E_1 \sim_{stable} E_2$ we say that $E_1$ and $E_2$ are stably equivalent vector bundles.

Related concepts

• topological K-theory