Petrie conjecture

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Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$${{Stub}}== Introduction == <wikitex>; The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. Petrie restricted his attention to smooth actions of the Lie group $S^1$, i.e., the circle {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}}) on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} As Petrie {{cite|Petrie1973}} has shown, the total Pontrjagin class of $M$ agrees with that of $\CP^n$, if the action of $S^1$ on $M$ extends to a smooth action of $T^n$. The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and {{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds, if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$. has sho and {{cite|Dessai&Wilking2004}}. == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]] </wikitex>G$ acts smoothly on a closed smooth manifold $M$$M$, what constraints does this place on the topology of $M$$M$ in general and on the Pontrjagin classes of $M$$M$ in particular.

Petrie restricted his attention to smooth actions of the Lie group $S^1$$S^1$, i.e., the circle [Petrie1972] (or more generally, the torus $T^k$$T^k$ for $k \geq 1$$k \geq 1$ [Petrie1973]) on closed smooth manifolds $M$$M$ which are homotopy equivalent to $\CP^n$$\CP^n$. He has formulated the following conjecture.

Conjecture 0.1 [Petrie1972].

Suppose that $M$$M$ is a closed smooth manifold homotopy equivalent to $\CP^n$$\CP^n$ and that $S^1$$S^1$ acts smoothly and non-trivially on $M$$M$. Then the total Pontrjagin class $p(M)$$p(M)$ of $M$$M$ agrees with that of $\CP^n$$\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$$x \in H^2(M; \mathbb{Z})$,

$\displaystyle p(M) = (1+x^2)^{n+1}.$ $$\displaystyle p(M) = (1+x^2)^{n+1}.$$

As Petrie [Petrie1973] has shown, the total Pontrjagin class of $M$$M$ agrees with that of $\CP^n$$\CP^n$, if the action of $S^1$$S^1$ on $M$$M$ extends to a smooth action of $T^n$$T^n$. The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$$\dim M \leq 8$, the statement is true by the work of [Dejter1976] and [James1985]. According to [Hattori1978], the conjecture is also true if $M$$M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981]. Moreover, by the work of [Dessai2002], the Petrie conjecture holds, if $\dim M \leq 22$$\dim M \leq 22$ and the action of $S^1$$S^1$ on $M$$M$ extends to an appropriate action of $Pin(2)$$Pin(2)$.

has sho and [Dessai&Wilking2004].

References

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