CLASSIFICATION OF CERTAIN HIGHER-DIMENSIONAL KNOTS OF CODIMENSION TWO

M Sh Farber
1980 Russian Mathematical Surveys  
An «-dimensional knot is a pair (S n + 2 , k") consisting of an oriented sphere S" + 2 and a smooth closed oriented submanifold k that is homotopyequivalent to an «-dimensional sphere. Two «-dimensional knots (S n + 2 , k v ) (y = 1 or 2) are equivalent (or of the same isotopy type) if there is an orientation-preserving isotopy of S n + 2 taking k x to k 2 . In this lecture we consider the problem of describing the set of isotopy types of «-dimensional knots. We use terminology of differential
more » ... gy of differential topology. 1. Homotopy Seifert pairings. Let Vbe a connected compact oriented (« + 1 )-dimensional submanifold of the sphere S n + 2 with a non-empty boundary 9 V. Let Υ be the closure of the complement of an open tubular neighbourhood of V in S n + 2 . We denote by u: V f\Y^-S n + 1 the canonical pairing of Spanier-Whitehead duality. Let i + : V-*-Υ be the map given by a small shift along the field of positive normals to V in S n + 2 . A homotopy Seifert pairing of the manifold V is the composition Q: V /\V ^i V AY-^ S n +\
doi:10.1070/rm1980v035n03abeh001823 fatcat:dgdzvapbhrcz5gh2x3yx3hzphu