Chichilnisky, Graciela (1972): Group Actions on Spin Manifolds. Published in: Transactions of the American Mathematical Society , Vol. 172, (October 1972): pp. 307315.

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Abstract
A generalization of the theorem of V. Bargmann concerning unitary and ray representations is obtained and is applied to the general problem of lifting group actions associated to the extension of structure of a bundle. In particular this is applied to the Poincare group 'P' of a Lorentz manifold 'M'. It is shown that the topological restrictions needed to lift an action in 'P' are more stringent than for actions in the proper Poincare group 'P'. Similar results hold for the Euclidean group of a Riemannian manifold.
Item Type:  MPRA Paper 

Original Title:  Group Actions on Spin Manifolds 
Language:  English 
Keywords:  Spin Manifolds; Manifold; V. Bargmann; unitary representations; ray representations; topological; topology; 
Subjects:  C  Mathematical and Quantitative Methods > C0  General 
Item ID:  7906 
Depositing User:  Graciela Chichilnisky 
Date Deposited:  25. Mar 2008 05:56 
Last Modified:  19. Feb 2013 09:02 
References:  V. Bargmann, On unitary ray representation of continuous groups, Ann. of Math. (2) 59 (1954), 146. MR 15, 397 K. Birchteler, Global existance of spin structures for gravitational fields, J. Mathematical Phys. 9 (1968), 813815 P. Chernoff and J. Marsden, Hamiltonian systems and quantum mechanics (in preparation). G. Chichilnisky, Group actions on spin manifolds, Thesis, Berkeley, 1970. A. Crumeyrolle, Structures spinorielles, Ann. Inst. H. Poincare Sect. A 11 (1969), 1955. MR 42 #6737 D. G. Ebin , The manifolf of Riemannian metrics, Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc., Providence, R. I., 1970, pp. 1140. MR 42 # 2506 H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040 R. Geroch, Spinor structure of spacetimes in general relativity. I, J. Mathematical Phys. 9 (1968), 17391744. MR 38 #3019 M. J. Greenberg, Lectures on algabraic topology, Benjamin, New York, 1967. MR 35 #6137. A. Haefliger, Sur l'extension du groupe structual d'un espace fibre, C. R. Acad. Sci. Paris 243 (1956), 558560. MR 18, 920. N. Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962. MR 26 #1345. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. 1, Interscience Tracts in Pure and Appl. Math., no 15, Interscience, New York, 1969. MR 38 # 6501 J. Marsden, Hamiltonian systems with spin, Canad. Math. Bull. 12 (1969), 203208. MR 40#1109. J. Milnor, The representaion rings of some classical groups, Mimeographed notes, Princeton University, Princeton, N. J., 1963. Notes on characteristic Classes, Mimeographed notes, Princeton University, Princeton, N. J., 1963. Spin structures on manifolds, Enseignement Math. (2) 9 (1963), 198203. MR 28 #622 S.B. Myers and N. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. (2) 40 (1939), 400416 R.S. Palais, Seminar on the AtiyahSinger index theorem, Ann. of Math. Studies, no. 57, Princeton Univ. Press, Princeton, N.J., 1965 MR 33#6649 D. J. Simms, Lie groups and quantum mechanics, Lecture Notes in Math., no. 52, SpringerVerlag, Berlin and NEw York, 1968. MR 12, 522. E. H. Spanier, Algebraic topology, McGrawHill, New York, 1977. MR 35 #1007. N. E. Steenrod, The topology of fibre bundles, Princeton Math. Series, vol. 14, Princeton Univ. Press, Princeton, N. J., 1951. MR 12, 522. E. P. Wigner, On unitary representations of the inhomogeneous group, Ann. of Math. (2) 40 (1939), 149204. Unitary representations of the inhomogeneous Lorentz group including reflections, Group Theoretical Concepts and Methods in Elementary Particle Physics (Lectures Istanbul Summer School Theoret. Phys., 1962), Gordon and Breach, New York, 1964, pp. 3780. MR 30#1210. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/7906 