Petrie conjecture
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− | Results confirming that in some cases, the Petrie conjecture is true go back to {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2001}}, and {{cite|Dessai&Wilking2003}} | + | Results confirming that in some cases, the Petrie conjecture is true go back to {{cite|Dejter1976}}, {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2001}}, and {{cite|Dessai&Wilking2003}} |
== References == | == References == |
Revision as of 04:19, 30 November 2010
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1 Introduction
The Petrie conjecture was formulated in the following context: suppose that a Lie group acts smoothly on a closed smooth manifold , what constraints does this place on the topology of in general and on the Pontrjagin classes of in particular.
Petrie restricted his attention to smooth actions of the Lie group [Petrie1972] (or more generally, the torus for [Petrie1973]) on closed smooth manifolds which are homotopy equivalent to . He has formulated the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose that is a closed smooth manifold homotopy equivalent to and that acts smoothly and non-trivially on . Then the total Pontrjagin class of agrees with that of , i.e., for a generator ,Results confirming that in some cases, the Petrie conjecture is true go back to [Dejter1976], [Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2001], and [Dessai&Wilking2003]
2 References
- [Dejter1976] I. J. Dejter, Smooth -manifolds in the homotopy type of , Michigan Math. J. 23 (1976), no.1, 83–95. MR0402789 (53 #6603) Zbl 0326.57009
- [Dessai&Wilking2003] Template:Dessai&Wilking2003
- [Dessai2001] Template:Dessai2001
- [Iberkleid1978] W. Iberkleid, Pseudolinear spheres, Michigan Math. J. 25 (1978), no.3, 359–370. MR512906 (80d:57023) Zbl 0377.57008
- [Petrie1972] T. Petrie, Smooth -actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010
- [Petrie1973] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR0322893 (48 #1254) Zbl 0262.57021
- [Tsukada&Washiyama1979] E. Tsukada and R. Washiyama, -actions on cohomology complex projective spaces with three components of the fixed point sets, Hiroshima Math. J. 9 (1979), no.1, 41–46. MR529325 (80j:57043) Zbl 0411.57037