Spaces which look like quaternionic projective 𝑛-space

McGibbon, C. A.

doi:10.1090/S0002-9947-1982-0662054-0

Spaces which look like quaternionic projective -space

Author: C. A. McGibbon
Journal: Trans. Amer. Math. Soc. 272 (1982), 569-587
MSC: Primary 55R35
DOI: https://doi.org/10.1090/S0002-9947-1982-0662054-0
MathSciNet review: 662054
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The projective -spaces which correspond to the various multiplicative structures on the three sphere are studied. Necessary and sufficient conditions for a projective -space to extend to a projective -space are described. At each odd prime, an infinite family of exotic projective spaces is constructed. These exotic spaces are not homotopy equivalent, at the prime in question, to the classical quaternionic projective -space. It is also shown that these exotic projective -spaces do not occur as the finite skeleton of a classifying space for a group with the homotopy type of the three sphere.

[1] J. F. Adams, On Chern characters and the structure of the unitary group, Proc. Cambridge Philos. Soc. 57 (1961), 189–199. MR 121795, https://doi.org/10.1017/s0305004100035052
[2] J. F. Adams, On the groups 𝐽(𝑋). IV, Topology 5 (1966), 21–71. MR 198470, https://doi.org/10.1016/0040-9383(66)90004-8
[3] -, Lectures on generalized cohomology, Lecture Notes in Math., vol. 99, Springer-Verlag, Berlin, 1969, pp. 1-138.
[4] -, Fake $HP^\infty$ 's, manuscript dated 20 Feb. 1977.
[5] M. G. Barratt, I. M. James, and N. Stein, Whitehead products and projective spaces, J. Math. Mech. 9 (1960), 813–819. MR 0116331, https://doi.org/10.1512/iumj.1960.9.59050
[6] A. L. Blakers and W. S. Massey, Products in homotopy theory, Ann. of Math. (2) 58 (1953), 295–324. MR 60820, https://doi.org/10.2307/1969790
[7] Eldon Dyer, Chern characters of certain complexes, Math. Z. 80 (1962/63), 363–373. MR 148073, https://doi.org/10.1007/BF01162393
[8] I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, https://doi.org/10.2307/2007107
[9] Arunas Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 978–981. MR 132543, https://doi.org/10.1073/pnas.46.7.978
[10] C. A. McGibbon, Fake quaternionic projective spaces from a homotopy point of view, Thesis, Univ. of Wisconsin, 1977.
[11] John C. Moore, Some applications of homology theory to homotopy problems, Ann. of Math. (2) 58 (1953), 325–350. MR 59549, https://doi.org/10.2307/1969791
[12] Robert A. Nowlan, 𝐴_{𝑛}-actions on fibre spaces, Indiana Univ. Math. J. 21 (1971/72), 285–313. MR 288767, https://doi.org/10.1512/iumj.1971.21.21023
[13] Gerald J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123–135. MR 174054, https://doi.org/10.1016/0040-9383(65)90039-X
[14] Gerald J. Porter, Higher products, Trans. Amer. Math. Soc. 148 (1970), 315–345. MR 256397, https://doi.org/10.1090/S0002-9947-1970-0256397-4
[15] David L. Rector, Loop structures on the homotopy type of 𝑆³, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971) Springer, Berlin, 1971, pp. 99–105. Lecture Notes in Math., Vol. 249. MR 0339153
[16] James F. Slifker, Exotic multiplications on 𝑆³, Quart. J. MAth. Oxford Ser. (2) 16 (1965), 322–359. MR 0187242, https://doi.org/10.1093/qmath/16.4.322
[17] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
[18] James Dillon Stasheff, Homotopy associativity of 𝐻-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, https://doi.org/10.1090/S0002-9947-1963-0158400-5
[19] James Stasheff, 𝐻-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372
[20] D. Sullivan, Geometric topology. I. Localization, periodicity, and Galois symmetry, M.I.T. Notes, 1970.
[21] Hirosi Toda, 𝑝-primary components of homotopy groups. IV. Compositions and toric constructions, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1959), 297–332. MR 111041, https://doi.org/10.1215/kjm/1250776579
[22] Hirosi Toda, On unstable homotopy of spheres and classical groups, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 1102–1105. MR 123329, https://doi.org/10.1073/pnas.46.8.1102
[23] Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
[24] F. D. Williams, Higher Samelson products, J. Pure Appl. Algebra 2 (1972), 249–260. MR 303533, https://doi.org/10.1016/0022-4049(72)90005-9
[25] Alexander Zabrodsky, Hopf spaces, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1976. North-Holland Mathematics Studies, Vol. 22; Notas de Matemática, No. 59. MR 0440542