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σ4-Actions On Homotopy Spheres

Published online by Cambridge University Press:  20 November 2018

Chao-Chu Liang
Chao-Chu Liang*
Affiliation:
University of Kansas, Lawrence, Kansas
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Let σ4 denote the group of all permutations of {a, b, c, d}. It has 24 elements, partitioned into five conjugacy classes: (1) the identity 1; (2) 6 transpositions: (ab), …, (cd); (3) 8 elements of order 3: (abc), …, (bcd); (4) 6 elements of order 4: (abcd), …, (adcb); (5) 3 elements of order 2: x = (ab)(cd), y = (ac)(bd), z = (ad)(bc).

In this paper, we study the differentiate actions of σ4 on odd-dimensional homotopy spheres modelled on the linear actions, with the fixed point set of each transposition a codimension two homotopy sphere.

A simple (2n – l)-knot is a differentiate embedding of a homotopy sphere K2n–l into a homotopy sphere Σ2n+1 such that πj(Σ – K) = πj(S1) for j < n.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 275 - 281
Copyright
Copyright © Canadian Mathematical Society 1981

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