Abstract
Conjugation spaces are spaces with an involution such that the fixed point set of the involution has \({\mathbb{Z} _2}\)-cohomology ring isomorphic to the \({\mathbb{Z} _2}\)-cohomology of the space itself, with the difference that all degrees are divided by two (e.g. \({\mathbb{C} {\rm P}^n}\) with the complex conjugation has \({\mathbb{R} {\rm P}^n}\) as fixed point set). One also requires that a certain conjugation equation is fulfilled. We give a new characterisation of conjugation spaces and apply it to the following realization problem: given M, a closed orientable 3-manifold, does there exist a simply connected 6-manifold X and a conjugation on X with fixed point set M? We give an affirmative answer.
Similar content being viewed by others
References
Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge University Press, Cambridge (1993)
Baues, H.J.: Obstruction Theory on Homotopy Classification of Maps. Lecture Notes in Mathematics, vol. 628. Springer, Heidelberg (1977)
Borel, A.: On periodic maps of certain K(π, 1). Collect. Pap. III, 57–60 (1983)
Conner P.E., Raymond F., Weinberger P. (1972) Manifolds with no periodic maps. In: Proceedings 2nd Conference Compact Transformation Groups, Part II, Springer Lecture Notes, vol. 299
Davis, M., Januskiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62, 417–451 (1991)
Franz, M., Puppe, V.: Steenrod squares on conjugation spaces. arxiv: math.AT/050157 (2005)
Hausmann, J.-C., Holm, T., Puppe, V.: Conjugation spaces. Algebraic Geometric Topology 5, 923–964 (2005)
Hausmann, J.-C., Knutson, A.: The cohomology ring of polygon spaces. Ann. de l’Institut Fourier 281–321 (1998)
Hambleton, I., Kreck, M., Teichner, P.: Nonorientable 4-manifolds with fundamental group of order 2. Transactions AMS, vol 344, Number 2 (1994)
Kreck, M.: Surgery and duality. Ann. Math. 149, 707–754 (1999)
Kreck, M.: Simply connected asymmetric manifolds. Preprint (2007)
Nash, J.: Real algebraic manifolds. Ann. Math. 56, 405–421 (1952)
Olbermann, M.: Conjugations on 6-Manifolds. Ph.D. Thesis, University of Heidelberg (2007). Available at http://www.ub.uni-heidelberg.de/archiv/7450/
Puppe, V.: Do manifolds have little symmetry? J. Fixed Point Theory Appl. 2(1), 85–96 (2007)
Ranicki, A.: Algebraic and Geometric Surgery. Oxford University Press, Oxford Mathematical Monograph, Oxford (2002)
Stong, R.E.: Notes on cobordism theory. Princeton University Press, Mathematical notes, Princeton (1968)
Tognoli, A.: Su una congettura di Nash. Ann. Sci. Norm. Sup. Pisa 27, 167–185 (1973)
Wall, C.T.C.: Surgery on compact manifolds, 2nd edn. In: Ranicki, A.A. (eds) Mathematical Surveys and Monographs, vol. 69, American Mathematical Society, Providence, RI (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partially supported by the DFG grant KR 814.