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Equivariant smoothing of piecewise linear manifolds

Published online by Cambridge University Press:  13 March 2017

CHRISTIAN LANGE
CHRISTIAN LANGE*
Affiliation:
Mathematisches Institut der Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany e-mail: [email protected]
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Abstract

We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 369 - 380
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Bourguignon, J. P. and Ezin, J. P. Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc. 301 (1987), 723–736.Google Scholar
[2] Cerf, J. Sur les diffomorphismes de la sphère de dimension trois (Γ4 = 0). Lecture Notes in Math. no. 53 (Springer-Verlag, Berlin-New York, 1968), xii+133 pp.Google Scholar
[3] Chern, S. An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc. 6 (1955), 771–782.Google Scholar
[4] Davis, M. W. Lectures on orbifolds and reflection groups. Adv. Lect. Math. 16 (2011), 63–93.Google Scholar
[5] Dinkelbach, J. and Leeb, B. Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom. Topol. 13 (2009), 1129–1173.Google Scholar
[6] Forster, O. Lectures on Riemann surfaces. Graduate Texts in Math. (Springer–Verlag, New York–Berlin, 1981).Google Scholar
[7] Hatcher, A. Algebraic topology (Cambridge University Press, Cambridge, 2002).Google Scholar
[8] Hirsch, M. W. and Mazur, B. Smoothings of piecewise linear manifolds. Annals of Math. Stud. no. 80 (Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974).Google Scholar
[9] Hudson, J. F. P. Piecewise linear topology. University of Chicago Lecture Notes (W. A. Benjamin, Inc., New York-Amsterdam, 1969).Google Scholar
[10] Illman, S. Smooth equivariant triangulations of G-manifolds for G a finite group. Math. Ann. 233 (1978), 199–220.Google Scholar
[11] Kervaire, M. A. and Milnor, B. J. W. Groups of homotopy spheres. I. Ann. of Math. (2) 77 (1963), 504–537.Google Scholar
[12] Kirby, R. C. and Siebenmann, L. C. Foundational essays on topological manifolds, smoothings and triangulations. Ann. Math. Stud. no. 88 (Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977).Google Scholar
[13] Kwasik, S. and Lee, L. B. Locally linear actions on 3-manifolds. Math. Proc. Camb. Phil. Soc. 104 (1988), 253–260.Google Scholar
[14] Lange, Ch. and Mikhaîlova, M. A. Classification of finite groups generated by reflections and rotations. Transform. Groups 21 (2016), 1155–1201.Google Scholar
[15] Lange, Ch. Characterization of finite groups generated by reflections and rotations. J. Topol. doi:10.1112/jtopol/jtw020.Google Scholar
[16] Lange, Ch. Some results on orbifold quotients and related objects. KUPS - Kölner Universitäts Publikations Server (2016).Google Scholar
[17] Mani, P. Automorphismen von polyedrischen Graphen. Math. Ann. 192 (1971), 279–303.Google Scholar
[18] Milnor, J. Lectures on the h-cobordism theorem (Princeton University Press, Princeton, N.J., 1965).Google Scholar
[19] Moise, E. E. Geometric topology in dimensions 2 and 3. Graduate Texts in Math. vol. 47 (Springer–Verlag, New York–Heidelberg, 1977).Google Scholar
[20] Munkres, J. R. Elementary differential topology. Lectures given at Massachusetts Institute of Technology, Fall (Princeton University Press, Princeton, N.J., 1966).Google Scholar
[21] Rolfsen, D. Knots and links. Mathematics Lecture Series, no. 7 (Publish or Perish, Inc., Berkeley, California, 1976).Google Scholar
[22] Rourke, C. P. and Sanderson, B. J. Introduction to piecewise-linear topology. Ergeb. Math. Grenzgeb. Band 69, (1972).Google Scholar
[23] Thurston, W. P. Three-dimensional geometry and topology. vol. 1 (Princeton Mathematical Series, 1997).Google Scholar
[24] Whitehead, J. On C 1 complexes. Annals of Math. 41 (1940), 809–832.Google Scholar
[25] Zimmermann, B. P. On finite groups acting on spheres and finite subgroups of orthogonal groups. Sib. Èlektron. Mat. Izv. 9 (2012), 1–12.Google Scholar