Topology change and selection rules for high-dimensional 𝑆𝑝𝑖𝑛(1,𝑛)₀-Lorentzian cobordisms

Smirnov, Gleb; Torres, Rafael

doi:10.1090/tran/7939

Topology change and selection rules for high-dimensional $\mathrm {Spin}(1, n)_0$-Lorentzian cobordisms
HTML articles powered by AMS MathViewer

by Gleb Smirnov and Rafael Torres PDF

Trans. Amer. Math. Soc. 373 (2020), 1731-1747 Request permission

Abstract:

We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\mathrm {Spin}(1, n)_0$. This extends a result of Gibbons-Hawking on $\mathrm {Sl}(2,\mathbb {C})$-Lorentzian cobordisms between 3-manifolds and results of Reinhart and Sorkin on the existence of Lorentzian cobordisms. We compute the $\mathrm {Spin}(1, n)_0$-Lorentzian cobordism group for several dimensions. Restrictions on the gravitational kink numbers of $\mathrm {Spin}(1, n)_0$-weak Lorentzian cobordisms are obtained.

References

M. A. Aguilar, J. A. Seade, and A. Verjovsky, Indices of vector fields and topological invariants of real analytic singularities, J. Reine Angew. Math. 504 (1998), 159–176. MR 1656755
V. I. Arnol′d, Singularity theory, London Mathematical Society Lecture Note Series, vol. 53, Cambridge University Press, Cambridge-New York, 1981. Selected papers; Translated from the Russian; With an introduction by C. T. C. Wall. MR 631683
Andrew Chamblin, Some applications of differential topology in general relativity, J. Geom. Phys. 13 (1994), no. 4, 357–377. MR 1275275, DOI 10.1016/0393-0440(94)90015-9
Andrew Chamblin, Topology and causal structure, The Sixth Canadian Conference on General Relativity and Relativistic Astrophysics (Fredericton, NB, 1995) Fields Inst. Commun., vol. 15, Amer. Math. Soc., Providence, RI, 1997, pp. 185–188. MR 1463203
A. Chamblin and R. Penrose, Kinking and causality, Twistor Newsletter 34 (1992), 13 - 18.
A. Dold and H. Whitney, Classification of oriented sphere bundles over a $4$-complex, Ann. of Math. (2) 69 (1959), 667–677. MR 123331, DOI 10.2307/1970030
David Finkelstein and Charles W. Misner, Some new conservation laws, Ann. Physics 6 (1959), 230–243. MR 111506, DOI 10.1016/0003-4916(59)90080-6
Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
Robert P. Geroch, Topology in general relativity, J. Mathematical Phys. 8 (1967), 782–786. MR 213139, DOI 10.1063/1.1705276
Robert Geroch, Spinor structure of space-times in general relativity. I, J. Mathematical Phys. 9 (1968), 1739–1744. MR 234703, DOI 10.1063/1.1664507
Robert Geroch, Spinor structure of space-times in general relativity. II, J. Mathematical Phys. 11 (1970), 342–348. MR 255217, DOI 10.1063/1.1665067
G. W. Gibbons, Topology and topology change in general relativity, Classical Quantum Gravity 10 (1993), no. suppl., S75–S78. Les Journées Relativistes (Amsterdam, 1992). MR 1212801
G. W. Gibbons and S. W. Hawking, Selection rules for topology change, Comm. Math. Phys. 148 (1992), no. 2, 345–352. MR 1178148
G. W. Gibbons and S. W. Hawking, Kinks and topology change, Phys. Rev. Lett. 69 (1992), no. 12, 1719–1721. MR 1181660, DOI 10.1103/PhysRevLett.69.1719
S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), no. 3, 329–452. MR 2192936
Michel A. Kervaire, Relative characteristic classes, Amer. J. Math. 79 (1957), 517–558. MR 90051, DOI 10.2307/2372561
Michel Kervaire, Courbure intégrale généralisée et homotopie, Math. Ann. 131 (1956), 219–252 (French). MR 86302, DOI 10.1007/BF01342961
Michel A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72. MR 253347, DOI 10.1090/S0002-9947-1969-0253347-3
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
Robion C. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989. MR 1001966, DOI 10.1007/BFb0089031
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Robert J. Low, Topological changes in lower dimensions, Classical Quantum Gravity 9 (1992), no. 11, L161–L165. MR 1188985
Robert J. Low, Kinking number and the Hopf index theorem, J. Math. Phys. 34 (1993), no. 10, 4840–4842. MR 1235978, DOI 10.1063/1.530325
G. Lusztig, J. Milnor, and F. P. Peterson, Semi-characteristics and cobordism, Topology 8 (1969), 357–359. MR 246308, DOI 10.1016/0040-9383(69)90021-4
L. Markus, Line element fields and Lorentz structures on differentiable manifolds, Ann. of Math. (2) 62 (1955), 411–417. MR 73169, DOI 10.2307/1970071
J. Milnor, Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198–203. MR 157388
John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855–951 (Russian). MR 0221509
Peter Percell, Parallel vector fields on manifolds with boundary, J. Differential Geometry 16 (1981), no. 1, 101–104. MR 633628
V. V. Prasolov, Elements of homology theory, Graduate Studies in Mathematics, vol. 81, American Mathematical Society, Providence, RI, 2007. Translated from the 2005 Russian original by Olga Sipacheva. MR 2313004, DOI 10.1090/gsm/081
Daniel Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971). MR 290382, DOI 10.1016/0001-8708(71)90041-7
Bruce L. Reinhart, Cobordism and the Euler number, Topology 2 (1963), 173–177. MR 153021, DOI 10.1016/0040-9383(63)90031-4
R. D. Sorkin, Topology change and monopole creation, Phys. Rev. D (3) 33 (1986), no. 4, 978–982. MR 826200, DOI 10.1103/PhysRevD.33.978
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
P. Yodzis, Lorentz cobordism, Comm. Math. Phys. 26 (1972), 39–52. MR 310918
P. Yodzis, Lorentz cobordism. II, Gen. Relativity Gravitation 4 (1973), no. 4, 299–307. MR 401068, DOI 10.1007/bf00759849
C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 120654, DOI 10.2307/1970136