Petrie conjecture
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The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. | The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. | ||
− | Petrie restricted his attention to smooth actions of the Lie group $S^1$ {{cite|Petrie1972}} | + | Petrie restricted his attention to smooth actions of the Lie group $S^1$, i.e., the circle {{cite|Petrie1972}} (or more generally, the torus $T^n$ for $n \geq 1$ {{cite|Petrie1973}}) |
+ | on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture. | ||
{{beginthm|Conjecture|{{cite|Petrie1972}}}} | {{beginthm|Conjecture|{{cite|Petrie1972}}}} |
Revision as of 01:31, 1 December 2010
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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 , i.e., the circle [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 ,The Petrie conjecture has not been confirmed in general, but if , the total Pontrjagin class of agrees with that of by the work of [Dejter1976] and [James1985]. According to [Hattori1978], the same is true if admits an invarint almost complex structure with the appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by [Petrie1973], [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], [Masuda1981], [Dessai2002] and [Dessai&Wilking2004].
References
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- [Dessai&Wilking2004] A. Dessai and B. Wilking, Torus actions on homotopy complex projective spaces, Math. Z. 247 (2004), no.3, 505–511. MR2114425 (2006c:57033) Zbl 1068.57034
- [Dessai2002] A. Dessai, Homotopy complex projective spaces with -action, Topology Appl. 122 (2002), no.3, 487–499. MR1911696 (2003f:58048) Zbl 0998.57048
- [Hattori1978] A. Hattori, -structures and -actions, Invent. Math. 48 (1978), no.1, 7–31. MR508087 (80e:57051) Zbl 0395.57020
- [Iberkleid1978] W. Iberkleid, Pseudolinear spheres, Michigan Math. J. 25 (1978), no.3, 359–370. MR512906 (80d:57023) Zbl 0377.57008
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- [Masuda1981] M. Masuda, On smooth -actions on cohomology complex projective spaces. The case where the fixed point set consists of four connected components, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no.1, 127–167. MR617869 (82i:57031) Zbl 0462.57019
- [Muslin1978] Template:Muslin1978
- [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
- [Wang1975] K. Wang, Differentiable circle group actions on homotopy complex projective spaces, Math. Ann. 214 (1975), 73–80. MR0372895 (51 #9099) Zbl 0285.57025
- [Yoshida1976] Template:Yoshida1976