The Bredon-Löffler conjecture. (English) Zbl 0858.55012

Bruner, Robert; Greenlees, John

Exp. Math. 4, No. 4, 289-297 (1995).

Summary: We give a brief exposition of results of Bredon and others on passage to fixed points from stable \(C_2\) equivariant homotopy (where \(C_2\) is the group of order two) and its relation to Mahowald’s root invariant. In particular we give Bredon’s easy equivariant proof that the root invariant doubles the stem; the conjecture of the title is equivalent to the Mahowald-Ravenel conjecture that the root invariant never more than triples the stem. Our main result is to verify by computation that the algebraic analogue of this holds in an extensive range: this improves on results of M. E. Mahowald and P. Shick [Contemp. Math. 19, 227-231 (1983; Zbl 0528.55012)].

MSC:

55Q91

Equivariant homotopy groups

Keywords:

stable equivariant homotopy; Mahowald’s root

Citations:

Zbl 0528.55012

References:

[1]	Adams J. F., Algebraic topology pp 483– (1984)
[2]	Bredon G. E., Bull. Amer. Math. Soc. 73 pp 269– (1967) · Zbl 0152.21803 · doi:10.1090/S0002-9904-1967-11713-0
[3]	Bredon, G. E. ”Equivariant homotopy”. Proceedings of the Conference on Transformation Groups. 1967, New Orleans. Edited by: Paul Mostert, S. pp.281–292. New York: Springer. [Bredon 1967b]
[4]	Bruner R. R., Algebraic topology pp 71– (1993)
[5]	Greenlees J. P. C., Trans. Amer. Math. Soc. 310 pp 199– (1988)
[6]	Greenlees J. P. C., Proc. Edinburgh Math. Soc. 35 pp 473– (1992) · Zbl 0814.55006 · doi:10.1017/S0013091500005757
[7]	Greenlees J. P. C., Mem. Amer. Math. Soc. 543 (1995)
[8]	Iriye K., J. Math. Kyoto U. 29 pp 159– (1989) · Zbl 0706.55011 · doi:10.1215/kjm/1250520312
[9]	Jones J. D. S., Bull. London Math. Soc. 17 pp 479– (1985) · Zbl 0585.55012 · doi:10.1112/blms/17.5.479
[10]	Land weber P. S., Ann. Math. 89 pp 1250– (1969)
[11]	Lewis L. G., Equivariant stable homotopy theory (1986) · Zbl 0611.55001 · doi:10.1007/BFb0075778
[12]	Lin W. H., Math. Proc. Camb. Phil. Soc. 87 pp 449– (1980) · Zbl 0455.55007 · doi:10.1017/S0305004100056887
[13]	Lin W. H., Math. Proc. Camb. Phil. Soc. 87 pp 459– (1980) · Zbl 0462.55007 · doi:10.1017/S0305004100056899
[14]	Mahowald M. E., Topology 32 pp 865– (1993) · Zbl 0796.55008 · doi:10.1016/0040-9383(93)90055-Z
[15]	Mahowald M. E., Contemp. Math. 19 pp 227– (1983) · Zbl 0528.55012 · doi:10.1090/conm/019/711054
[16]	Miller H. R., Homotopy theory and related topics pp 210– (1990) · doi:10.1007/BFb0083705
[17]	Schultz R., Contemp. Math. 36 pp 513– (1985) · doi:10.1090/conm/036/780979
[18]	Shick P., Trans. Amer. Math. Soc. 301 pp 227– (1987)