Abstract
Let (M m, T) be a smooth involution on a closed smooth m-dimensional manifold and \({F = \bigcup_{j=0}^{n} F^j}\) (n ≤ m) its fixed point set, where F j denotes the union of those components of F having dimension j. In this paper we show that, if the top dimensional component F n is indecomposable, then m ≤ 2n + 1. We also give examples to show that this bound is best possible. This gives an improvement for the famous Five Halves Theorem of J. Boardman when the top dimensional component of the fixed point set is indecomposable.
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References
Borel A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts. Ann. Math. 57(2), 115–207 (1953)
Borel A., Hirzebruch F.: Characteristic classes and homogeneous spaces I. Am. J. Math. 80, 458–538 (1958)
Boardman, J.M.: Cobordism of involutions revisited. In Proceedings of the second conference on compact transformation groups, University of Massachusetts, Amherst, Mass., Part I, (1971), pp. 131–151. Lecture Notes in Mathematics, vol. 298, Springer, Berlin (1972)
Boardman J.M.: On manifolds with involution. Bull. Am. Math. Soc. 73, 136–138 (1967)
Brown R.L.W.: Immersions and embeddings up to cobordism. Can. J. Math. 23(6), 1102–1115 (1971)
Conner P.E., Floyd E.E.: Differentiable Periodic Maps. Springer, Berlin (1964)
Conner P.E., Floyd E.E.: Fibring within a cobordism class. Michigan Math. J. 12, 33–47 (1965)
Kelton S.M.: Involutions fixing \({ {\textnormal{I} {\hskip 0.75pt}\! \textnormal{R}\textnormal{P}}^j \cup F^n}\). Topol. Appl. 142, 197–203 (2004)
Kelton S.M.: Involutions fixing \({ {\textnormal{I} {\hskip 0.75pt}\! \textnormal{R}\textnormal{P}}^j \cup F^n}\), II. Topol. Appl. 149(1-3), 217–226 (2005)
Kosniowski C., Stong R.E.: Involutions and characteristic numbers. Topology 17, 309–330 (1978)
Morava, J.: Cobordism of involutions revisited, revisited. Homotopy invariant algebraic structures, Baltimore, MD (1998), pp. 15–18. Contemp. Math., 239, American Mathematical Society, Providence, RI (1999)
Pergher P.L.Q.: Bounds on the dimension of manifolds with certain Z 2 fixed sets. Mat. Contemp. 13, 269–275 (1996)
Pergher P.L.Q., Figueira F.G.: Dimensions of fixed point sets of involutions. Arch. Math. (Basel) 87(3), 280–288 (2006)
Pergher P.L.Q., Figueira F.G.: Bounds on the dimension of manifolds with involution fixing \({F^n \cup F^2}\). Glasg. Math. J. 50(3), 595–604 (2008)
Pergher P.L.Q., Figueira F.G.: Involutions fixing \({F^n\cup F^2}\). Topol. Appl. 153(14), 2499–2507 (2006)
Pergher P.L.Q., Figueira F.G.: Two commuting involutions fixing \({F^n\cup F^{n-1}}\). Geom. Dedicata 117, 181–193 (2006)
Pergher P.L.Q., Ramos A., de Oliveira R.: \({Z_2^k}\)-actions fixing \({RP^2 \cup RP^{even}}\). Algebr. Geom. Topol. 7, 29–45 (2007)
Pergher P.L.Q., Stong R.E.: Involutions fixing \({\{point\} \cup F^n}\). Transform. Groups 6, 78–85 (2001)
Royster D.C.: Involutions fixing the disjoint union of two projective spaces. Indiana Univ. Math. J. 29(2), 267–276 (1980)
Thom R.: Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28, 17–86 (1954)
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The author was partially supported by CNPq and FAPESP.
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Pergher, P.L.Q. Involutions whose top dimensional component of the fixed point set is indecomposable. Geom Dedicata 146, 1–7 (2010). https://doi.org/10.1007/s10711-009-9419-5
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DOI: https://doi.org/10.1007/s10711-009-9419-5
Keywords
- Involution
- Projective space bundle
- Indecomposable manifold
- Splitting principle
- Stiefel–Whitney class
- Characteristic number