Petrie conjecture
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− | As Petrie {{cite|Petrie1973}} | + | As Petrie {{cite|Petrie1973}} has shown, the total Pontrjagin class of $M$ agrees with that of $\CP^n$, if the action of $S^1$ on $M$ extends to a smooth action of $T^n$. |
− | The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, | + | The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. |
According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases | According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases | ||
where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and | where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and | ||
− | {{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, | + | {{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds, if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate |
action of $Pin(2)$. | action of $Pin(2)$. | ||
Revision as of 02:08, 1 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Introduction
Tex syntax error, what constraints does this place on the topology of
Tex syntax errorin general and on the Pontrjagin classes of
Tex syntax errorin 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 manifoldsTex syntax errorwhich are homotopy equivalent to . He has formulated the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose thatTex syntax erroris a closed smooth manifold homotopy equivalent to and that acts smoothly and non-trivially on
Tex syntax error. Then the total Pontrjagin class of
Tex syntax erroragrees with that of , i.e., for a generator ,
Tex syntax erroragrees with that of , if the action of on
Tex syntax errorextends to a smooth action of .
The Petrie conjecture has not been confirmed in general, but if , the statement is true by the work of [Dejter1976] and [James1985].
According to [Hattori1978], the conjecture is also true ifTex syntax erroradmits an invarint almost complex structure with appropriate first Chern class. Another special cases
where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and
[Masuda1981]. Moreover, by the work of [Dessai2002], the Petrie conjecture holds, if and the action of onTex syntax errorextends to an appropriate
action of .
has sho and [Dessai&Wilking2004].
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&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
- [James1985] D. M. James, Smooth actions on homotopy 's, Michigan Math. J. 32 (1985), no.3, 259–266. MR803831 (87c:57031) Zbl 0602.57026
- [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