Odd primary 𝑏𝑜 resolutions and classification of the stable summands of stunted lens spaces

González, Jesús

doi:10.1090/S0002-9947-99-02284-9

Odd primary $bo$ resolutions and classification of the stable summands of stunted lens spaces
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Trans. Amer. Math. Soc. 352 (2000), 1149-1169 Request permission

Abstract:

The classification of the stable homotopy types of stunted lens spaces and their stable summands can be obtained by proving the triviality of certain stable classes in the homotopy groups of these spaces. This is achieved in the 2-primary case by Davis and Mahowald using classical Adams spectral sequence techniques. We obtain the odd primary analogue using the corresponding Adams spectral sequence based at the spectrum representing odd primary connective $K$-theory. The methods allow us to answer a stronger problem: the determination of the smallest stunted space where such stable classes remain null homotopic. A technical problem prevents us from giving an answer in all situations; however, in a quantitative way, the number of cases missed is very small.

References

J. F. Adams, Lectures on generalised cohomology, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR 0251716
J. Hejtmanek, Scattering theory of the linear Boltzmann-operator, Methoden und Verfahren der mathematischen Physik, Band 9, Bibliographisches Institut, Mannheim, 1973, pp. 183–201. MR 0364905
Donald M. Davis, Odd primary $b\textrm {o}$-resolutions and $K$-theory localization, Illinois J. Math. 30 (1986), no. 1, 79–100. MR 822385
Kh. N. Inasaridze, Characterization of exact bifunctor homology theories, Soobshch. Akad. Nauk Gruzin. SSR 121 (1986), no. 2, 241–243 (Russian, with English and Georgian summaries). MR 863395
Donald M. Davis and Mark Mahowald, The image of the stable $J$-homomorphism, Topology 28 (1989), no. 1, 39–58. MR 991098, DOI 10.1016/0040-9383(89)90031-1
S. Feder, S. Gitler, and K. Y. Lam, Composition properties of projective homotopy classes, Pacific J. Math. 68 (1977), no. 1, 47–61. MR 515518
J. González. The regular complex in the ${B}{P}\langle 1\rangle$ Adams spectral sequence. Trans. Amer. Math. Soc. 350 (1998), 2629–2664.
J. González. A vanishing line in the ${B}{P}\langle 1\rangle$ Adams spectral sequence. To appear in Topology.
Jesus Gonzalez, Classification of the stable homotopy types of stunted lens spaces for an odd prime, Pacific J. Math. 176 (1996), no. 2, 325–343. MR 1434994
David Copeland Johnson and W. Stephen Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327–353. MR 334257, DOI 10.1016/0040-9383(73)90027-X
Richard M. Kane, Operations in connective $K$-theory, Mem. Amer. Math. Soc. 34 (1981), no. 254, vi+102. MR 634210, DOI 10.1090/memo/0254
Wolfgang Lellmann, Operations and co-operations in odd-primary connective $K$-theory, J. London Math. Soc. (2) 29 (1984), no. 3, 562–576. MR 754942, DOI 10.1112/jlms/s2-29.3.562
Wolfgang Lellmann and Mark Mahowald, The $b\textrm {o}$-Adams spectral sequence, Trans. Amer. Math. Soc. 300 (1987), no. 2, 593–623. MR 876468, DOI 10.1090/S0002-9947-1987-0876468-1
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