𝜋_{𝑚𝑛-}2(𝑆ⁿ_{𝑚-}2) contains an element of order 𝑚

Shar, Albert

doi:10.1090/S0002-9939-1972-0292079-8

$\pi \sb{mn-}{}\sb{}{2}(S\sp{n}\sb{m-}{}\sb{}{2})$ contains an element of order

Author: Albert Shar
Journal: Proc. Amer. Math. Soc. 34 (1972), 303-306
MSC: Primary 55E40
DOI: https://doi.org/10.1090/S0002-9939-1972-0292079-8
MathSciNet review: 0292079
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Abstract: Let be the mth reduced product complex of ${S^n}$ with n an even integer greater than 2 and m any integer greater than 2. $S_m^n = S_{m - 1}^n \cup {e^{nm}}$ with attaching map $[i, \cdots ,i] \in {\pi _{nm - 1}}(S_{m - 1}^n)$ . Using a result of J. R. Hubbuck and a result of the author it is proven that the Whitehead product $[i,[i, \cdots ,i]] \in {\pi _{mn - 2}}(S_{m - 2}^n)$ is of order m.

[1] K. A. Hardie, Higher Whitehead products, Quart. J. Math. Oxford Ser. (2) 12 (1961), 241–249. MR 141110, https://doi.org/10.1093/qmath/12.1.241
[2] K. A. Hardie, A proof of the Nakaoka-Toda formula, Pacific J. Math. 14 (1964), 1249–1254. MR 172288
[3] J. R. Hubbuck, Finitely generated cohomology Hopf algebras, Topology 9 (1970), 205–210. MR 261600, https://doi.org/10.1016/0040-9383(70)90009-1
[4] I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, https://doi.org/10.2307/2007107
[5] I. M. James, On the suspension triad, Ann. of Math. (2) 63 (1956), 191–247. MR 77922, https://doi.org/10.2307/1969607
[6] I. M. James, Note on cup-products, Proc. Amer. Math. Soc. 8 (1957), 374–383. MR 91467, https://doi.org/10.1090/S0002-9939-1957-0091467-4
[7] Gerald J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123–135. MR 174054, https://doi.org/10.1016/0040-9383(65)90039-X
[8] Albert Shar, The homotopy groups of spaces whose cohomology is a 𝑍_{𝑝} truncated polynomial algebra, Proc. Amer. Math. Soc. 38 (1973), 172–178. MR 310877, https://doi.org/10.1090/S0002-9939-1973-0310877-X