Let Mn denote a closed
manifold with smooth involution, and Mn→ Mn the corresponding principal
Z2-bundle ξ classified by w1(ξ) in H1(Mn;Z2). The existence of a nested sequence
Ñn−k+1⊂Ñn−k+2⊂⋯⊂Ñn−1⊂Ñn=Mn of characteristic sub-manifolds
(corresponding to principal Z2-bundles ξn−k+1⊂ ξn−k+2⊂⋯⊂ ξn−1⊂ ξn= ξ)
satisfying (w1(ξi))n−k+i= 0, for 1 ≤ k ≤ n and all i with n − k + 1 ≤ i ≤ n,
provides a necessary and sufficient condition for imbedding ξ in vector bundle
Mn× Rk→ Mn (preserving fibers).
For closed n-manifolds, the condition (w1(ξ))k= 0 allows ξ to be imbedded in the
vector bundle Mn×Rk→ Mn “up to free Z2-cobordism.”
