Two remarks on fiber homotopy type

Abstract

Section 1 of this note considers the normal sphere bundle of a compact, connected, orientable manifold Mn (without boundary) differentiably imbedded in euclidean space Rn+k. (These hypotheses on Mn will be assumed throughout § 1.) It is shown that if k is sufficiently large then the normal sphere bundle has the fiber homotopy type of a product bundle if and only if there exists an S-map from Sn to Mn of degree one (i.e. for some p there exists a continuous map of degree one from Sn+P to the p-old suspension of Mn). The proof is based on the fact that the Thorn space of the normal bundle is dual in the sense of Spanier- Whitehead [8] to the disjoint union of Mn and a point. © 1960 by Pacific Journal of Mathematics.