Petrie conjecture

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1 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}} </wikitex> Results confirming that in some cases, the Petrie conjecture is true go back to {{cite|Dejter1976}}, {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2001}}, and {{cite|Dessai&Wilking2003}} == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts smoothly on a closed smooth manifold

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$M$, what constraints does this place on the topology of

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$M$ in general and on the Pontrjagin classes of

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$M$ 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

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$M$ which are homotopy equivalent to $\CP^n$$\CP^n$. He has formulated the following conjecture.

Conjecture 0.1 [Petrie1972].

Suppose that

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$M$ is a closed smooth manifold homotopy equivalent to $\CP^n$$\CP^n$ and that $S^1$$S^1$ acts smoothly and non-trivially on

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$M$. Then the total Pontrjagin class $p(M)$$p(M)$ of

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$M$ 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})$,

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

Results confirming that in some cases, the Petrie conjecture is true go back to [Dejter1976], [Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2001], and [Dessai&Wilking2003]

2 References

• [Dejter1976] I. J. Dejter, Smooth $S^1$$S^1$-manifolds in the homotopy type of $CP^3$$CP^3$, Michigan Math. J. 23 (1976), no.1, 83–95. MR0402789 (53 #6603) Zbl 0326.57009
• [Dessai&Wilking2003] Template:Dessai&Wilking2003
• [Dessai2001] Template:Dessai2001
• [Iberkleid1978] W. Iberkleid, Pseudolinear spheres, Michigan Math. J. 25 (1978), no.3, 359–370. MR512906 (80d:57023) Zbl 0377.57008
• [Petrie1972] T. Petrie, Smooth $S^1$$S^1$-actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010
• [Petrie1973] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR0322893 (48 #1254) Zbl 0262.57021
• [Tsukada&Washiyama1979] E. Tsukada and R. Washiyama, $S^1$$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