The Gottlieb group of finite linear quotients of odd-dimensional spheres

Broughton, S. Allen

doi:10.1090/S0002-9939-1991-1041012-1

The Gottlieb group of finite linear quotients of odd-dimensional spheres
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by S. Allen Broughton PDF

Proc. Amer. Math. Soc. 111 (1991), 1195-1197 Request permission

Abstract:

Let $G$ be a finite, freely acting group of homeomorphisms of the odd-dimensional sphere ${S^{2n - 1}}$. John Oprea has proven that the Gottlieb group of ${S^{2n - 1}}/G$ equals $Z(G)$, the centre of $G$. The purpose of this short paper is to give a considerably shorter, more geometric proof of Oprea’s theorem in the important case where $G$ is a linear group.

References

D. H. Gottlieb, A certain subgroup of the fundamental group, Amer. J. Math. 87 (1965), 840–856. MR 189027, DOI 10.2307/2373248
Daniel Henry Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729–756. MR 275424, DOI 10.2307/2373349
T. Ganea, Cyclic homotopies, Illinois J. Math. 12 (1968), 1–4. MR 220283

Characters of finite groups

I. M. Isaacs, Real representations of groups with a single involution, Pacific J. Math. 71 (1977), no. 2, 463–464. MR 444758
George E. Lang Jr., Evaluation subgroups of factor spaces, Pacific J. Math. 42 (1972), 701–709. MR 314043
Jerry Malzan, On groups with a single involution, Pacific J. Math. 57 (1975), no. 2, 481–489. MR 382420
John Oprea, Finite group actions on spheres and the Gottlieb group, J. Korean Math. Soc. 28 (1991), no. 1, 65–78. MR 1107213
Jingyal Pak, On the fibered Jiang spaces, Fixed point theory and its applications (Berkeley, CA, 1986) Contemp. Math., vol. 72, Amer. Math. Soc., Providence, RI, 1988, pp. 179–181. MR 956490, DOI 10.1090/conm/072/956490