The classification of stunted projective spaces by stable homotopy type
HTML articles powered by AMS MathViewer
- by S. Feder and S. Gitler PDF
- Trans. Amer. Math. Soc. 225 (1977), 59-81 Request permission
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.References
- J. F. Adams, On the groups $J(X)$. I, Topology 2 (1963), 181–195. MR 159336, DOI 10.1016/0040-9383(63)90001-6
- J. F. Adams, On the groups $J(X)$. II, Topology 3 (1965), 137–171. MR 198468, DOI 10.1016/0040-9383(65)90040-6
- J. F. Adams and G. Walker, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 61 (1965), 81–103. MR 171285, DOI 10.1017/S0305004100038688
- M. F. Atiyah and J. A. Todd, On complex Stiefel manifolds, Proc. Cambridge Philos. Soc. 56 (1960), 342–353. MR 132552, DOI 10.1017/s0305004100034642
- Raoul Bott, A note on the $KO$-theory of sphere-bundles, Bull. Amer. Math. Soc. 68 (1962), 395–400. MR 153019, DOI 10.1090/S0002-9904-1962-10819-2
- S. Feder and S. Gitler, Stable homotopy types of stunted complex projective spaces, Proc. Cambridge Philos. Soc. 73 (1973), 431–438. MR 315703, DOI 10.1017/s0305004100076994
- S. Feder and S. Gitler, Mappings of quaternionic projective spaces, Bol. Soc. Mat. Mexicana (2) 18 (1973), 33–37. MR 336740
- Samuel Feder and Samuel Gitler, Stable homotopy types of Thom complexes, Quart. J. Math. Oxford Ser. (2) 25 (1974), 143–149. MR 362304, DOI 10.1093/qmath/25.1.143
- 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 348736, DOI 10.1090/S0002-9904-1974-13584-6
- Samuel Gitler and James D. Stasheff, The first exotic class of $BF$, Topology 4 (1965), 257–266. MR 180985, DOI 10.1016/0040-9383(65)90010-8
- R. P. Held and D. Sjerve, On the stable homotopy type of Thom complexes, Canadian J. Math. 25 (1973), 1285–1294. MR 339148, DOI 10.4153/CJM-1973-135-5
- Kee Yuen Lam, Fiber homotopic trivial bundles over complex projective spaces, Proc. Amer. Math. Soc. 33 (1972), 211–212. MR 293654, DOI 10.1090/S0002-9939-1972-0293654-7
- Daniel Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR 279804, DOI 10.1016/0040-9383(71)90018-8
- B. J. Sanderson, Immersions and embeddings of projective spaces, Proc. London Math. Soc. (3) 14 (1964), 137–153. MR 165532, DOI 10.1112/plms/s3-14.1.137
- François Sigrist and Ulrich Suter, Cross-sections of symplectic Stiefel manifolds, Trans. Amer. Math. Soc. 184 (1973), 247–259. MR 326728, DOI 10.1090/S0002-9947-1973-0326728-8
- James Stasheff, A classification theorem for fibre spaces, Topology 2 (1963), 239–246. MR 154286, DOI 10.1016/0040-9383(63)90006-5
- C. T. C. Wall, Poincaré complexes. I, Ann. of Math. (2) 86 (1967), 213–245. MR 217791, DOI 10.2307/1970688
Additional Information
- © Copyright 1977 American Mathematical Society
- 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