Abstract
In this article, we show the existence of conjugations on many smooth simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X 6 to admit a conjugation with fixed point set M 3 is the obvious one: the existence of a degree-halving ring isomorphism between the \({\mathbb Z_2}\)-cohomologies of X and M. As a consequence certain 6-manifolds, for which Puppe (J Fixed Point Theory Appl 2(1):85–96, 2007) proved the non-existence of non-trivial orientation-preserving finite group actions, do admit many involutions.
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References
Bak, A.: K-theory of forms. In: Annals of Mathematics Studies, vol. 98. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1981)
Curtis, C., Reiner, I.: Representation theory of finite groups and associative algebras. In: Pure and Applied Mathematics XI. Wiley-Interscience Publishers (1962)
Davis M., Januskiewicz T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62, 417–451 (1991)
Dovermann K.H., Masuda M., Schultz R.: Conjugation involutions on homotopy complex projective spaces. Jpn. J. Math. 12, 1–35 (1986)
Franz M., Puppe V.: Steenrod squares on conjugation spaces. C. R. Math. Acad. Sci. Paris 342(3), 187–190 (2006)
Hambleton, I., Hausmann, J.-C. Conjugation spaces and 4-manifolds. arXiv:0906.5057 (2009)
Hausmann J.-C., Holm T., Puppe V.: Conjugation spaces. Algebraic Geom. Topol. 5, 923–964 (2005)
Hausmann, J.-C., Knutson, A.: The cohomology ring of polygon spaces. Annales de l’Institut Fourier, pp. 281–321 (1998)
Kreck M.: Surgery and duality. Ann. Math. 149, 707–754 (1999)
Kreck M.: Simply connected asymmetric manifolds. J. Topol. 2(2), 249–261 (2009)
Kreck M.: Corrigendum to “Simply connected asymmetric manifolds”. J. Topol. 4(1), 254–255 (2011)
Olbermann, M.: Conjugations on 6-Manifolds. PhD Thesis, University of Heidelberg (2007). http://www.ub.uni-heidelberg.de/archiv/7450/
Olbermann M.: Conjugations on 6-manifolds. Math. Ann. 342(2), 255–271 (2008)
Olbermann M.: Involutions on S 6 with 3-dimensional fixed point set. Algebraic Geom. Topol. 10(4), 1905–1932 (2010)
Postnikov, M.M.: The structure of the ring of intersections of three-dimensional manifolds. Doklady Akad. Nauk. SSSR (N.S.) 61, 795–797 (1948, in Russian)
Puppe V.: Do manifolds have little symmetry?. J. Fixed Point Theory Appl. 2(1), 85–96 (2007)
Ranicki, A.: Algebraic and geometric surgery. In: Oxford Mathematical Monograph. Oxford University Press (2002)
Stong R.E.: Notes on cobordism theory. Princeton University Press, Princeton (1968)
Sullivan D.: On the intersection ring of compact three manifolds. Topology 14(3), 275–277 (1975)
Wall C.T.C.: Classification problems in differential topology. V. On certain 6-manifolds. Invent. Math. 1, 355–374 (1966)
Wall, C.T.C.: Surgery on Compact Manifolds, 2nd edn. Edited and with a foreword by A.A. Ranicki. Mathematical Surveys and Monographs, vol. 69. American Mathematical Society, Providence (1999)
Zubr A.V.: Classification of simply connected six-dimensional spin manifolds. Math. USSR Izvestija 9(4), 793–812 (1975)
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Olbermann, M. Conjugations on 6-manifolds with free integral cohomology. Math. Ann. 353, 65–93 (2012). https://doi.org/10.1007/s00208-011-0673-0
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DOI: https://doi.org/10.1007/s00208-011-0673-0