Petrie conjecture

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1 Problem

If a compact Lie group $G$${{Stub}} == Problem == <wikitex>; If a compact Lie group $G$ acts smoothly and non-trivially 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? In the case where $M$ is [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]], $M \simeq \CP^n$, Petrie {{cite|Petrie1972}} restricted his attention to smooth actions of the Lie group $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$, and posed the following conjecture. {{beginthm|Conjecture|{{cite|Petrie1972}}}} Suppose that $S^1$ acts smoothly and non-trivially on a closed smooth n$-manifold $M \simeq \CP^n$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., $p(M) = (1+x^2)^{n+1}$ for a generator $x \in H^2(M; \mathbb{Z})$. {{endthm}} </wikitex> == Progress to date == <wikitex>; As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases. * Petrie {{cite|Petrie1973}} has verified his conjecture under the assumption that the action of $S^1$ on $M \simeq \mathbb{C}P^n$ extends to a smooth action of the torus $T^n$. * By the work of {{cite|Dejter1976}}, the Petrie conjecture is true if $\dim M = 6$, i.e., $M \simeq \CP^3$, and more generaly, if $\dim M \leq 6$. * According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Remember $\dim M = 2n$. * Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}, and {{cite|Muslin1978}}. * By {{cite|Tsukada&Washiyama1979}}, {{cite|Masuda1981}}, and {{cite|Masuda1983}}, the Petrie conjecture is true if the fixed point set consists of three or four connected components. * It follows from {{cite|James1985}} that the Petrie conjecture is also true if $\dim M = 8$, i.e., $M \simeq \CP^4$. * The work of {{cite|Dessai2002}} confirms the Petrie conjecture under the assumption that the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$ and $\dim M \leq 22$. * According to {{cite|Dessai&Wilking2004}}, the Petrie conjecture holds if the action of $S^1$ on $M$ extends to a smooth action of the torus $T^k$ and $\dim M \leq 8k-4$. </wikitex> == Further discussion == <wikitex>; A symplectic version of the Petrie conjecture is discussed in the article of {{cite|Tolman2010}}. </wikitex> == References == {{#RefList:}} <!-- --> [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts smoothly and non-trivially 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?

In the case where $M$$M$ is homotopy equivalent to $\CP^n$$\CP^n$, $M \simeq \CP^n$$M \simeq \CP^n$, Petrie [Petrie1972] restricted his attention to smooth actions of the Lie group $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$$S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$, and posed the following conjecture.

Conjecture 1.1 [Petrie1972]. Suppose that $S^1$$S^1$ acts smoothly and non-trivially on a closed smooth $2n$$2n$-manifold $M \simeq \CP^n$$M \simeq \CP^n$. Then the total Pontrjagin class $p(M)$$p(M)$ of $M$$M$ agrees with that of $\CP^n$$\CP^n$, i.e.,

$p(M) = (1+x^2)^{n+1}$$p(M) = (1+x^2)^{n+1}$ for a generator $x$$x$ of $H^2(M; \mathbb{Z})$$H^2(M; \mathbb{Z})$.

2 Progress to date

As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However, it has been proven in the following special cases.

• Petrie [Petrie1973] has verified his conjecture under the assumption that the action of $S^1$$S^1$ on $M \simeq \mathbb{C}P^n$$M \simeq \mathbb{C}P^n$ extends to a smooth action of the torus $T^n$$T^n$.
• By the work of [Dejter1976], the Petrie conjecture is true if $\dim M = 6$$\dim M = 6$, i.e., $M \simeq \CP^3$$M \simeq \CP^3$, and more generaly, if $\dim M \leq 6$$\dim M \leq 6$.
• According to [Hattori1978], the Petrie conjecture holds if $M$$M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$$c_1(M) = (n+1)x$. Remember $\dim M = 2n$$\dim M = 2n$.
• Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978], and [Muslin1978].
• By [Tsukada&Washiyama1979], [Masuda1981], and [Masuda1983], the Petrie conjecture is true if the fixed point set consists of three or four connected components.
• It follows from [James1985] that the Petrie conjecture is also true if $\dim M = 8$$\dim M = 8$, i.e., $M \simeq \CP^4$$M \simeq \CP^4$.
• The work of [Dessai2002] confirms the Petrie conjecture under the assumption that the action of $S^1$$S^1$ on $M$$M$ extends to an appropriate action of $Pin(2)$$Pin(2)$ and $\dim M \leq 22$$\dim M \leq 22$.
• According to [Dessai&Wilking2004], the Petrie conjecture holds if the action of $S^1$$S^1$ on $M$$M$ extends to a smooth action of the torus $T^k$$T^k$ and $\dim M \leq 8k-4$$\dim M \leq 8k-4$.

3 Further discussion

A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].

4 References

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• [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 $Pin(2)$$Pin(2)$-action, Topology Appl. 122 (2002), no.3, 487–499. MR1911696 (2003f:58048) Zbl 0998.57048
• [Hattori1978] A. Hattori, $Spin^c$$Spin^c$-structures and $S^1$$S^1$-actions, Invent. Math. 48 (1978), no.1, 7–31. MR508087 (80e:57051) Zbl 0395.57020
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• [Masuda1981] M. Masuda, On smooth $S^1$$S^1$-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
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• [Muslin1978] Template:Muslin1978
• [Petrie1972] T. Petrie, Smooth $S^1$$S^1$-actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010
• [Petrie1973] T. Petrie, Torus actions on homotopy complex projective spaces, Invent. Math. 20 (1973), 139–146. MR0322893 (48 #1254) Zbl 0262.57021
• [Tolman2010] S. Tolman, On a symplectic generalization of Petrie's conjecture, Trans. Amer. Math. Soc. 362 (2010), no.8, 3963–3996. MR2638879 Zbl 1216.53074
• [Tsukada&Washiyama1979] E. Tsukada and R. Washiyama, $S^1$$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
• [Wang1975] K. Wang, Differentiable circle group actions on homotopy complex projective spaces, Math. Ann. 214 (1975), 73–80. MR0372895 (51 #9099) Zbl 0285.57025
• [Yoshida1975/76] T. Yoshida, On smooth semifree $S^1$$S^1$ actions on cohomology complex projective spaces, Publ. Res. Inst. Math. Sci. 11 (1975/76), no.2, 483–496. MR0445528 (56 #3868) Zbl 0326.57008