Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Hasse invariant of a vector bundle


Author: Richard R. Patterson
Journal: Trans. Amer. Math. Soc. 150 (1970), 425-443
MSC: Primary 55.50; Secondary 16.00
DOI: https://doi.org/10.1090/S0002-9947-1970-0268893-4
MathSciNet review: 0268893
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space $ X$. A homomorphism is defined between them analogous to the generalized Hasse invariant. Upon evaluation, the Witt group is seen to be $ \tilde KO(X)$, the graded Brauer group $ 1 + {H^1}(X;{Z_2}) + {H^2}(X;{Z_2})$ with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: $ 1 + {w_1} + {w_2}$.


References [Enhancements On Off] (What's this?)

  • [1] M. Arkowitz and C. R. Curjel, On the number of multiplications of an $ H$-space, Topology 2 (1963), 205-209. MR 27 #2985. MR 0153016 (27:2985)
  • [2] M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), suppl. 1, 3-38. MR 29 #5250. MR 0167985 (29:5250)
  • [3] H. Bass, Topics in algebraic $ K$-theory, Mimeographed Notes, Tata Institute, Bombay, India, 1967.
  • [4] A. Delzant, Définition des classes de Stiefel-Whitney d'un module quadratique sur un corps de caractéristique différente de 2, C. R. Acad. Sci. Paris 255 (1962), 1366-1368. MR 26 #175. MR 0142606 (26:175)
  • [5] A. Dold, Halbexakte Homotopiefunktoren, Lecture Notes in Math., No. 12, Springer-Verlag, Berlin, 1966. MR 0198464 (33:6622)
  • [6] P. Donovan and M. Karoubi, Graded Brauer groups and $ K$-theory with local coefficients, (to appear).
  • [7] -, Groupe de Brauer et coefficients locaux en $ K$-théorie, C. R. Acad. Sci. Paris 269 (1969), 387-389.
  • [8] A. Grothendieck, Algèbres d'Azumaya et interprétations diverse, Séminaire Bourbaki, 17e année (1964/65), exposé 290, reprint, Benjamin, New York, 1964/65. MR 33 #5420L.
  • [9] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [10] W. C. Hsiang, A note on free differentiable actions of $ {S^1}$ and $ {S^3}$ on homotopy spheres, Ann. of Math. (2) 83 (1966), 266-272. MR 33 #731. MR 0192506 (33:731)
  • [11] J. Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430-436. MR 17, 1120. MR 0077932 (17:1120a)
  • [12] -, Microbundles. I, Topology 3 (1964), suppl. 1, 53-80. MR 28 #4553b. MR 0161346 (28:4553b)
  • [13] W. Scharlau, Quadratische Formen und Galois-Cohomologie, Invent. Math. 4 (1967), 238-264. MR 37 #1442. MR 0225851 (37:1442)
  • [14] M. Spivak, Spaces satisfying Poincaré duality, Topology 6 (1967), 77-101. MR 35 #4923. MR 0214071 (35:4923)
  • [15] N. Steenrod, The topology of fibre bundles, Princeton Math. Series, vol. 14, Princeton Univ. Press, Princeton, N. J., 1951. MR 12, 522. MR 0039258 (12:522b)
  • [16] C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/64), 187-199. MR 29 #4771. MR 0167498 (29:4771)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55.50, 16.00

Retrieve articles in all journals with MSC: 55.50, 16.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0268893-4
Keywords: Vector bundles, Witt group, graded Brauer group, Hasse invariant, Clifford bundles
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society