3-primary 𝑣₁-periodic homotopy groups of 𝐹₄ and 𝐸₆

Bendersky, Martin; Davis, Donald M.

doi:10.1090/S0002-9947-1994-1231069-4

$3$-primary $v_ 1$-periodic homotopy groups of $F_ 4$ and $E_ 6$
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by Martin Bendersky and Donald M. Davis

Trans. Amer. Math. Soc. 344 (1994), 291-306

DOI: https://doi.org/10.1090/S0002-9947-1994-1231069-4

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Abstract:

We compute the 3-primary ${v_1}$-periodic homotopy groups of the exceptional Lie groups ${F_4}$ and ${E_6}$. The unstable Novikov spectral sequence is used for the most delicate part of the analysis.

References

Martin Bendersky, The derived functors of the primitives for $\textrm {BP}_\ast (\Omega S^{2n+1})$, Trans. Amer. Math. Soc. 276 (1983), no. 2, 599–619. MR 688964, DOI 10.1090/S0002-9947-1983-0688964-7
Martin Bendersky, The $v_1$-periodic unstable Novikov spectral sequence, Topology 31 (1992), no. 1, 47–64. MR 1153238, DOI 10.1016/0040-9383(92)90063-N
Martin Bendersky, Unstable towers in the odd primary homotopy groups of spheres, Trans. Amer. Math. Soc. 287 (1985), no. 2, 529–542. MR 768724, DOI 10.1090/S0002-9947-1985-0768724-0
Martin Bendersky, Some calculations in the unstable Adams-Novikov spectral sequence, Publ. Res. Inst. Math. Sci. 16 (1980), no. 3, 739–766. MR 602467, DOI 10.2977/prims/1195186928
Martin Bendersky, The BP Hopf invariant, Amer. J. Math. 108 (1986), no. 5, 1037–1058. MR 859769, DOI 10.2307/2374595
M. Bendersky, E. B. Curtis, and H. R. Miller, The unstable Adams spectral sequence for generalized homology, Topology 17 (1978), no. 3, 229–248. MR 508887, DOI 10.1016/0040-9383(78)90028-9
M. Bendersky, E. B. Curtis, and D. Ravenel, $EHP$ sequences in BP theory, Topology 21 (1982), no. 4, 373–391. MR 670742, DOI 10.1016/0040-9383(82)90018-0
Martin Bendersky and Donald M. Davis, $2$-primary $v_1$-periodic homotopy groups of $\textrm {SU}(n)$, Amer. J. Math. 114 (1992), no. 3, 465–494. MR 1165351, DOI 10.2307/2374767
Martin Bendersky, Donald M. Davis, and Mark Mahowald, $v_1$-periodic homotopy groups of $\textrm {Sp}(n)$, Pacific J. Math. 170 (1995), no. 2, 319–378. MR 1363868, DOI 10.2140/pjm.1995.170.319
Martin Bendersky, Donald M. Davis, and Mamoru Mimura, $v_1$-periodic homotopy groups of exceptional Lie groups: torsion-free cases, Trans. Amer. Math. Soc. 333 (1992), no. 1, 115–135. MR 1116310, DOI 10.1090/S0002-9947-1992-1116310-9
Armand Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris 230 (1950), 1378–1380 (French). MR 34768
F. R. Cohen, J. C. Moore, and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. (2) 110 (1979), no. 3, 549–565. MR 554384, DOI 10.2307/1971238
Lawrence Conlon, An application of the Bott suspension map to the topology of $EIV$, Pacific J. Math. 19 (1966), 411–428. MR 203727, DOI 10.2140/pjm.1966.19.411
Donald M. Davis, The $v_1$-periodic homotopy groups of $\textrm {SU}(n)$ at odd primes, J. London Math. Soc. (2) 43 (1991), no. 3, 529–544. MR 1113391, DOI 10.1112/jlms/s2-43.3.529
Donald M. Davis and Mark Mahowald, Some remarks on $v_1$-periodic homotopy groups, Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 176, Cambridge Univ. Press, Cambridge, 1992, pp. 55–72. MR 1232199, DOI 10.1017/CBO9780511526312.011

Three contributions to the homotopy theory of the exceptional Lie groups

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John R. Harper, The $\textrm {mod}\ 3$ homotopy type of $F_{4}$, Localization in group theory and homotopy theory, and related topics (Sympos., Battelle Seattle Res. Center, Seattle, Wash., 1974) Lecture Notes in Math., Vol. 418, Springer, Berlin, 1974, pp. 58–67. MR 0365558
John R. Harper, $H$-spaces with torsion, Mem. Amer. Math. Soc. 22 (1979), no. 223, viii+72. MR 546361, DOI 10.1090/memo/0223
Bruno Harris, Suspensions and characteristic maps for symmetric spaces, Ann. of Math. (2) 76 (1962), 295–305. MR 149479, DOI 10.2307/1970276
Mamoru Mimura, The homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ. 6 (1967), 131–176. MR 206958, DOI 10.1215/kjm/1250524375
R. Michael F. Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970), 283–310. MR 266216, DOI 10.1007/BF01129978
Robert D. Thompson, The $v_1$-periodic homotopy groups of an unstable sphere at odd primes, Trans. Amer. Math. Soc. 319 (1990), no. 2, 535–559. MR 1010890, DOI 10.1090/S0002-9947-1990-1010890-8
Hirosi Toda, On the double suspension $E^2$, J. Inst. Polytech. Osaka City Univ. Ser. A 7 (1956), 103–145. MR 92968
Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217