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Self-maps of projective spaces


Author: C. A. McGibbon
Journal: Trans. Amer. Math. Soc. 271 (1982), 325-346
MSC: Primary 55P40; Secondary 55S25
DOI: https://doi.org/10.1090/S0002-9947-1982-0648096-X
MathSciNet review: 648096
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Abstract: The classical projective $ n$-spaces (real, complex, and quaternionic) are studied in terms of their self maps, from a homotopy point of view. Self maps of iterated suspensions of these spaces are also considered. The goal in both cases is to classify, up to homology, all such maps. This goal is achieved in the stable case. Some partial results are obtained in the unstable case. The results from both cases are used to compute the genus groups and the stable genus groups of the classical projective spaces. Applications to other spaces are also given.


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  • [1] J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603-632. MR 0139178 (25:2614)
  • [2] -, On the groups $ J(X)$. IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • [3] -, Infinite loop spaces, Ann. of Math. Studies, no. 90, Princeton Univ. Press, Princeton, N.J., 1978. MR 505692 (80d:55001)
  • [4] M. Arkowitz and C. R. Curjel, On the maps of $ H$-spaces, Topology 6 (1967), 137-148. MR 0230306 (37:5868)
  • [5] M. F. Atiyah, Thom complexes, Proc. London Math. Soc. 11 (1961), 291-310. MR 0131880 (24:A1727)
  • [6] J. C. Becker, Characteristic classes and $ K$-theory, Lecture Notes in Math., vol. 428, Springer-Verlag, Berlin and New York, 1974, pp. 132-143. MR 0377877 (51:14046)
  • [7] F. R. Cohen, J. C. Moore and J. A. Neisendorfer, Torsion in homotopy, Ann. of Math. (2) 109 (1979), 121-168. MR 519355 (80e:55024)
  • [8] D. Davis, $ BP$-operations and mappings of stunted complex projective spaces, Lecture Notes in Math., vol. 741, Springer-Verlag, Berlin and New York, 1979, pp. 373-393. MR 557178 (80m:55009)
  • [9] E. Dror, A generalization of the Whitehead theorem, Lecture Notes in Math., vol. 249, Springer-Verlag, Berlin and New York, 1971, pp. 13-22. MR 0350725 (50:3217)
  • [10] E. Dyer, Chern characters of certain complexes, Math. Z. 80 (1963), 363-373. MR 0148073 (26:5582)
  • [11] S. Feder and S. Gitler, Mappings of quaternionic projective space, Bol. Soc. Mat. Mexicana 18 (1973), 33-37. MR 0336740 (49:1513)
  • [12] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies, no. 15, North-Holland, Amsterdam, 1976. MR 0478146 (57:17635)
  • [13] I. M. James, Cross-sections of Stiefel manifolds, Proc. London Math. Soc. 8 (1958), 536-547. MR 0100840 (20:7268)
  • [14] -, The topology of Stiefel manifolds, London Math. Soc. Lecture Notes Series, vol. 24, Cambridge Univ. Press, Cambridge, 1976. MR 0431239 (55:4240)
  • [15] H. J. Marcum and D. Randall, A note on self-mappings of quarternionic projective spaces, An. Acad. Brasil, Ciênc. 47 (1975), 7-9. MR 0436136 (55:9086)
  • [16] C. A. McGibbon, Multiplicative properties of power maps. II, Trans. Amer. Math. Soc. (to appear). MR 675065 (84d:55008)
  • [17] -, Stable properties of rank $ 1$ loop structures, Topology 20 (1981), 109-118. MR 605651 (82e:55018)
  • [18] R. J. Milgram and E. Rees, On the normal bundle to an embedding, Topology 10 (1971), 299-308. MR 0290391 (44:7572)
  • [19] M. Mimura and H. Toda, On $ p$-equivalences and $ p$-universal spaces, Comment Math. Helv. 46 (1971), 87-97. MR 0285007 (44:2231)
  • [20] J. A. Neisendorfer, $ 3$-primary exponents, Proc. Cambridge Philos. Soc. 90 (1981), 63-83. MR 611286 (82e:55026)
  • [21] J. A. Neisendorfer and P. Selick, Some examples of spaces with or without exponents, Proceedings of Current Trends in Algebraic Topology, London, Ontario, 1981 (to appear). MR 686124 (84b:55017)
  • [22] D. L. Rector, Loop structures on the homotopy type of $ {S^3}$, Lecture Notes in Math., vol. 249, Springer-Verlag, Berlin and New York, 1971, pp. 99-105. MR 0339153 (49:3916)
  • [23] E. Rees, Embeddings of real projective spaces, Topology 10 (1971), 309-312. MR 0288778 (44:5974)
  • [24] J. P. Serre, A course in arithmetic, Graduate Texts in Mathematics, no. 7, Springer-Verlag, Berlin and New York, 1970. MR 0344216 (49:8956)
  • [25] G. Segal, The stable homotopy of complex projective space, Quart. J. Math. Oxford Ser. (2), 24 (1973), 1-5. MR 0319183 (47:7729)
  • [26] P. Selick, Odd primary torsion in $ {\pi _k}({S^3})$, Topology 17 (1978), 407-412. MR 516219 (80c:55010)
  • [27] V. P. Snaith, Algebraic cobordism and $ K$-theory, Mem. Amer. Math. Soc., no. 221 (1979). MR 539791 (80k:57060)
  • [28] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [29] N. E. Steenrod (with D. B. A. Epstein), Cohomology operations, Ann. of Math. Studies, no. 50, Princeton Univ. Press, Princeton, N.J., 1962. MR 0145525 (26:3056)
  • [30] D. Sullivan, Geometry topology: Part I, localization, periodicity, and Galois symmetry, MIT Notes, Cambridge, 1970.
  • [31] H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, no. 49, Princeton Univ. Press, Princeton, N.J., 1962. MR 0143217 (26:777)
  • [32] A. Zabrodsky, $ p$-equivalences and homotopy type, Lecture Notes in Math., vol. 418, Springer-Verlag, Berlin and New York, 1974, pp. 161-171. MR 0377867 (51:14036)
  • [33] -, Hopf spaces, North-Holland Math. Studies, no. 22, North-Holland, Amsterdam, 1976. MR 0440542 (55:13416)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0648096-X
Keywords: Projective $ n$-space, suspension, Adams $ e$-invariant, Hopf bundle, Vandermonde determinant, Maclaurin's series, genus of a nilpotent homotopy type, Lefschetz number, order of the identity
Article copyright: © Copyright 1982 American Mathematical Society

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