Petrie conjecture

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Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$${{Stub}}== Introduction == <wikitex>; The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly on a closed smooth manifold $M$, what constraints does this place on the topology of $M$ in general and on the Pontrjagin classes of $M$ in particular. Petrie restricted his attention to smooth actions of the Lie group $S^1$ {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}}) on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $M$ is a closed smooth manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} The Petrie conjecture has not been confirmed in general (no counterexample is know at the moment). However, if $\dim M \leq 8$, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ by the work of {{cite|Dejter1976}} and {{cite|James1978}}. {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2002}},{{cite|Dessai&Wilking2004}}, and {{cite|Tolman2010}}. == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]] </wikitex>G$ acts smoothly on a closed smooth manifold $M$$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$$S^1$ [Petrie1972] (or more generally, the torus $T^k$$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})$,

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

The Petrie conjecture has not been confirmed in general (no counterexample is know at the moment). However, if $\dim M \leq 8$$\dim M \leq 8$, the total Pontrjagin class of $M$$M$ agrees with that of $\CP^n$$\CP^n$ by the work of [Dejter1976] and [James1978].

[Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2002],[Dessai&Wilking2004], and [Tolman2010].

References

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• [James1978] Template:James1978
• [Petrie1972] T. Petrie, Smooth $S^1$$S^1$-actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010
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• [Tolman2010] S. Tolman, On a symplectic generalization of Petrie's conjecture, Trans. Amer. Math. Soc. 362 (2010), no.8, 3963–3996. MR2638879 Zbl 1216.53074
• [Tsukada&Washiyama1979] E. Tsukada and R. Washiyama, $S^1$$S^1$-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