Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T03:33:37.420Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Bredon, G. E., Introduction to compact transformation groups (Academic Press, New York, 1972).Google Scholar
2. Durfee, A. and Kauffman, L., Periodicity of branched cyclic covers, Math. Ann. 218 (1975), 157–174.Google Scholar
3. Kervaire, M. A., Les noeuds de dimensions supérieures, Bull. Soc. Math. France 98 (1965), 225–271.Google Scholar
4. Levine, J., Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244.Google Scholar
5. Levine, J., An algebraic classification of some knots of codimension two, Comment. Math. Helv. 45 (1970), 185–198.Google Scholar
6. Liang, C. C., Knots fixed by Zp-actions, and periodic links, Math. Ann. 233 (1978), 49–54.Google Scholar
7. Liang, C. C., Z(2)-knot cobordism in codimension two, and involutions on homotopy spheres, Trans. Amer. Math. Soc. 256 (1979), 89–97.Google Scholar
8. Liang, C. C., Some exotic dihedral group actions on homotopy spheres, Indiana Univ. Math. Jour. 28 (1979), 127–130.Google Scholar
9. Serre, J.-P., Linear presentations of finite groups (Springer-Verlag, New York-Heidelberg-Berlin, 1977).Google Scholar
10. Stoltzfus, N. W., Unraveling the integral knot concordance group, Mem. Amer. Math. Soc. 192 (1977).Google Scholar
11. Van der Waerden, B. L., Group theory and quantum mechanics, Die Grund. der Math. 214 (Springer-Verlag, New York-Heidelberg-Berlin, 1974).Google Scholar