Spaces which look like quaternionic projective $n$-space
HTML articles powered by AMS MathViewer
- by C. A. McGibbon
- Trans. Amer. Math. Soc. 272 (1982), 569-587
- DOI: https://doi.org/10.1090/S0002-9947-1982-0662054-0
- PDF | Request permission
Abstract:
The projective $n$-spaces which correspond to the various multiplicative structures on the three sphere are studied. Necessary and sufficient conditions for a projective $n$-space to extend to a projective $n+1$-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 $n$-space. It is also shown that these exotic projective $n$-spaces do not occur as the finite skeleton of a classifying space for a group with the homotopy type of the three sphere.References
- J. F. Adams, On Chern characters and the structure of the unitary group, Proc. Cambridge Philos. Soc. 57 (1961), 189–199. MR 121795, DOI 10.1017/s0305004100035052
- J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8 —, Lectures on generalized cohomology, Lecture Notes in Math., vol. 99, Springer-Verlag, Berlin, 1969, pp. 1-138. —, Fake $HP^\infty$’s, manuscript dated 20 Feb. 1977.
- M. G. Barratt, I. M. James, and N. Stein, Whitehead products and projective spaces, J. Math. Mech. 9 (1960), 813–819. MR 0116331, DOI 10.1512/iumj.1960.9.59050
- A. L. Blakers and W. S. Massey, Products in homotopy theory, Ann. of Math. (2) 58 (1953), 295–324. MR 60820, DOI 10.2307/1969790
- Eldon Dyer, Chern characters of certain complexes, Math. Z. 80 (1962/63), 363–373. MR 148073, DOI 10.1007/BF01162393
- I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, DOI 10.2307/2007107
- 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, DOI 10.1073/pnas.46.7.978 C. A. McGibbon, Fake quaternionic projective spaces from a homotopy point of view, Thesis, Univ. of Wisconsin, 1977.
- John C. Moore, Some applications of homology theory to homotopy problems, Ann. of Math. (2) 58 (1953), 325–350. MR 59549, DOI 10.2307/1969791
- Robert A. Nowlan, $A_{n}$-actions on fibre spaces, Indiana Univ. Math. J. 21 (1971/72), 285–313. MR 288767, DOI 10.1512/iumj.1971.21.21023
- Gerald J. Porter, Higher-order Whitehead products, Topology 3 (1965), 123–135. MR 174054, DOI 10.1016/0040-9383(65)90039-X
- Gerald J. Porter, Higher products, Trans. Amer. Math. Soc. 148 (1970), 315–345. MR 256397, DOI 10.1090/S0002-9947-1970-0256397-4
- David L. Rector, Loop structures on the homotopy type of $S^{3}$, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle, Wash., 1971) Lecture Notes in Math., Vol. 249, Springer, Berlin, 1971, pp. 99–105. MR 0339153
- James F. Slifker, Exotic multiplications on $S^{3}$, Quart. J. Math. Oxford Ser. (2) 16 (1965), 322–359. MR 187242, DOI 10.1093/qmath/16.4.322
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112
- James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312. 108 (1963), 275-292; ibid. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
- James Stasheff, $H$-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372, DOI 10.1007/BFb0065896 D. Sullivan, Geometric topology. I. Localization, periodicity, and Galois symmetry, M.I.T. Notes, 1970.
- 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, DOI 10.1215/kjm/1250776579
- Hirosi Toda, On unstable homotopy of spheres and classical groups, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 1102–1105. MR 123329, DOI 10.1073/pnas.46.8.1102
- Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
- F. D. Williams, Higher Samelson products, J. Pure Appl. Algebra 2 (1972), 249–260. MR 303533, DOI 10.1016/0022-4049(72)90005-9
- Alexander Zabrodsky, Hopf spaces, North-Holland Mathematics Studies, Vol. 22, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1976. MR 0440542
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- 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