The classification of stunted projective spaces by stable homotopy type

Feder, S.; Gitler, S.

doi:10.1090/S0002-9947-1977-0423338-2

The classification of stunted projective spaces by stable homotopy type

Authors: S. Feder and S. Gitler
Journal: Trans. Amer. Math. Soc. 225 (1977), 59-81
MSC: Primary 55D15
DOI: https://doi.org/10.1090/S0002-9947-1977-0423338-2
MathSciNet review: 0423338
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A complete classification of stable homotopy types of complex and quaternionic stunted projective spaces, denoted by ${\mathbf{C}}P_n^k$ and ${\mathbf{Q}}P_n^k$ respectively, is obtained. The necessary conditions for such equivalences are found using K-theory and various characteristic classes introduced originally by J. F. Adams. As a by-product one finds the J-orders of the Hopf bundles over ${\mathbf{C}}{P^n}$ and ${\mathbf{Q}}{P^n}$ respectively. The algebraic part is rather involved. Finally a homotopy theoretical argument yields the constructions of such homotopy equivalences as are allowed by the fulfillment of the necessary conditions.

[1] J. F. Adams, On the groups 𝐽(𝑋). I, Topology 2 (1963), 181–195. MR 0159336, https://doi.org/10.1016/0040-9383(63)90001-6
[2] J. F. Adams, On the groups 𝐽(𝑋). II, Topology 3 (1965), 137–171. MR 0198468, https://doi.org/10.1016/0040-9383(65)90040-6
[3] J. F. Adams and G. Walker, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 61 (1965), 81–103. MR 0171285
[4] M. F. Atiyah and J. A. Todd, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 56 (1960), 342–353. MR 0132552
[5] Raoul Bott, A note on the 𝐾𝑂-theory of sphere-bundles, Bull. Amer. Math. Soc. 68 (1962), 395–400. MR 0153019, https://doi.org/10.1090/S0002-9904-1962-10819-2
[6] S. Feder and S. Gitler, Stable homotopy types of stunted complex projective spaces, Proc. Cambridge Philos. Soc. 73 (1973), 431–438. MR 0315703
[7] S. Feder and S. Gitler, Mappings of quaternionic projective spaces, Bol. Soc. Mat. Mexicana (2) 18 (1973), 33–37. MR 0336740
[8] Samuel Feder and Samuel Gitler, Stable homotopy types of Thom complexes, Quart. J. Math. Oxford Ser. (2) 25 (1974), 143–149. MR 0362304, https://doi.org/10.1093/qmath/25.1.143
[9] S. Feder and S. Gitler, Stunted projective spaces and the 𝐽-order of the Hopf bundle, Bull. Amer. Math. Soc. 80 (1974), 748–749. MR 0348736, https://doi.org/10.1090/S0002-9904-1974-13584-6
[10] Samuel Gitler and James D. Stasheff, The first exotic class of 𝐵𝐹, Topology 4 (1965), 257–266. MR 0180985, https://doi.org/10.1016/0040-9383(65)90010-8
[11] R. P. Held and D. Sjerve, On the stable homotopy type of Thom complexes, Canad. J. Math. 25 (1973), 1285–1294. MR 0339148, https://doi.org/10.4153/CJM-1973-135-5
[12] Kee Yuen Lam, Fiber homotopic trivial bundles over complex projective spaces, Proc. Amer. Math. Soc. 33 (1972), 211–212. MR 0293654, https://doi.org/10.1090/S0002-9939-1972-0293654-7
[13] Daniel Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR 0279804, https://doi.org/10.1016/0040-9383(71)90018-8
[14] B. J. Sanderson, Immersions and embeddings of projective spaces, Proc. London Math. Soc. (3) 14 (1964), 137–153. MR 0165532, https://doi.org/10.1112/plms/s3-14.1.137
[15] François Sigrist and Ulrich Suter, Cross-sections of symplectic Stiefel manifolds, Trans. Amer. Math. Soc. 184 (1973), 247–259. MR 0326728, https://doi.org/10.1090/S0002-9947-1973-0326728-8
[16] James Stasheff, A classification theorem for fibre spaces, Topology 2 (1963), 239–246. MR 0154286, https://doi.org/10.1016/0040-9383(63)90006-5
[17] C. T. C. Wall, Poincaré complexes. I, Ann. of Math. (2) 86 (1967), 213–245. MR 0217791, https://doi.org/10.2307/1970688