Petrie conjecture
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− | + | As Petrie {{cite|Petrie1973}} shows himself, 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 $T^n$. | |
− | by the work of {{cite|Dejter1976}} and {{cite|James1985}}. | + | The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, it is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. |
− | According to {{cite|Hattori1978}}, the | + | 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 |
− | complex structure with appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, | + | where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and |
− | {{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 |
− | 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 02:04, 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 ,As Petrie [Petrie1973] shows himself, the total Pontrjagin class of agrees with that of , if the action of on extends to a smooth action of . The Petrie conjecture has not been confirmed in general, but if , it is true by the work of [Dejter1976] and [James1985]. According to [Hattori1978], the conjecture is also true if admits 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 on extends to an appropriate action of .
has sho and [Dessai&Wilking2004].
References
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