By Milgram R.J.

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Where the exterior of the circle represents E1 (η × ζ) and the exterior of the square represents E1 (η) × E(ζ) ∪ E(η) × E1 (ζ). In particular, there is an evident radial deformation retraction from E1 (η×ζ) to E1 (η)×E(ζ)∪E(η)× E1 (ζ) and T (η × ζ) E(η × ζ)/(E1 (η) × E(ζ) ∪ E(η) × E1 (ζ) = T (η) ∧ T (ζ). where the base point of T (η) is ∗ = {E1 (η)}, so that the ∧ makes sense.

In particular BGLk (R) BOk (R). Classification of stable vector bundles. Let BGL(R) = lim− → n BGLn (R) and BO = lim− → n BOn (R). As above BO BGL(R). The stable bundles over a finite CW complex X are in one-one correspondence with the homotopy classes of maps X −−→ BO to the classifying space BO. Remark . The homotopy groups π∗ (BO) are 8-periodic by the Bott periodicity theorem, and are given by: n(mod 8) 0 1 2 3 4 5 6 7 πn (BO) Z Z2 Z2 0 Z 0 0 0 Proposition 8. If k > m then two k-plane vector bundles η, η over an m-dimensional finite CW complex X are isomorphic if and only if they are stably isomorphic.