Vol. 89, No. 1, 1980

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Imbedding smooth involutions in trivial bundles

Kenneth Joseph Prevot

Vol. 89 (1980), No. 1, 163–168
Abstract

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 Ñnk+1 Ñnk+2 Ñn1 Ñn = Mn of characteristic sub-manifolds (corresponding to principal Z2-bundles ξnk+1 ξnk+2 ξn1 ξn = ξ) satisfying (w1(ξi))nk+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.”

Mathematical Subject Classification 2000
Primary: 55R25
Milestones
Received: 4 May 1979
Published: 1 July 1980
Authors
Kenneth Joseph Prevot