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 actions of the Lie group $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$ and proved that if the fixed point set of the action consists only of isolated fixed points, then the Pontrjagin classes of $M$ are determined by the representations of $S^1$ at the fixed points. Motivated by this result, Petrie {{cite|Petrie1972}} posed the following conjecture. {{beginthm|Conjecture|(Petrie conjecture)}} 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., $p(M) = (1+x^2)^{n}$ for a generator $x$ of $H^2(M; \mathbb{Z})$. {{endthm}} </wikitex> == Progress to date == <wikitex>; As of December 21, 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 manifold $M \simeq \CP^n$ admits 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 hence, if $\dim M \leq 6$. * Related results go back to {{cite|Musin1978}} and {{cite|Musin1980}}, in particular, the latter work shows that the Petrie conjecture holds if $\dim M = 8$, i.e., $M \simeq \CP^4$. * According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$. * Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1975/76}}, {{cite|Iberkleid1978}}. * By {{cite|Tsukada&Washiyama1979}} and {{cite|Masuda1981}}, the Petrie conjecture is true if the fixed point set consists of three or four connected components. * Masuda {{cite|Masuda1983}} proved the Petrie conjecture in the case where $M$ admits a specific smooth action of $T^k$ for $k \geq 2$. * The work of {{cite|James1985}} confirms the result of {{cite|Musin1980}} that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$. * According to {{cite|Dessai2002}}, the Petrie conjecture holds if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$. * It follows from {{cite|Dessai&Wilking2004}} that the Petrie conjecture holds if $M$ admits a smooth action of $T^k$ and $\dim M \leq 8k-4$. </wikitex> == Further discussion == <wikitex>; Masuda and Suh {{cite|Masuda&Suh2008}} posed the following question about the invariance of Pontrjagin classes for toric n$-manifolds. {{beginthm|Question}} For two toric n$-manifolds with isomorphic cohomology rings, is it true that any isomorphism between the cohomology rings preserves the Pontrjagin classes of the two manifolds? {{endthm}} A symplectic version of the Petrie conjecture is discussed by Tolman {{cite|Tolman2010}}. In particular, the following question has been posed. {{beginthm|Question}} If the circle $S^1$ acts in a Hamiltonian way on a compact symplectic manifold $M$ with $H^{2i}(M;\Rr) \cong H^{2i}(\CP^n; \Rr)$ for all $i \geq 0$, is it true that $H^{j}(M;\Zz) \cong H^{j}(\CP^n; \Zz)$ for all $j \geq 0$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\Zz)$? {{endthm}} </wikitex> == References == {{#RefList:}} <!-- --> [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts smoothly and non-trivially 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? In the case where

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$M$ is homotopy equivalent to $\CP^n$$\CP^n$, $M \simeq \CP^n$$M \simeq \CP^n$, Petrie [Petrie1972] restricted his attention to actions of the Lie group $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$$S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$ and proved that if the fixed point set of the action consists only of isolated fixed points, then the Pontrjagin classes of

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$M$ are determined by the representations of $S^1$$S^1$ at the fixed points. Motivated by this result, Petrie [Petrie1972] posed the following conjecture.

Conjecture 1.1 (Petrie conjecture).

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

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$M$ agrees with that of $\CP^n$$\CP^n$, i.e.,

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

2 Progress to date

As of December 21, 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 manifold $M \simeq \CP^n$$M \simeq \CP^n$ admits a smooth action of the torus $T^n$$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 hence, if $\dim M \leq 6$$\dim M \leq 6$.
• Related results go back to [Musin1978] and [Musin1980], in particular, the latter work shows that the Petrie conjecture holds if $\dim M = 8$$\dim M = 8$, i.e., $M \simeq \CP^4$$M \simeq \CP^4$.
• According to [Hattori1978], the Petrie conjecture holds if

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  $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$$c_1(M) = (n+1)x$.
• Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978].
• By [Tsukada&Washiyama1979] and [Masuda1981], the Petrie conjecture is true if the fixed point set consists of three or four connected components.
• Masuda [Masuda1983] proved the Petrie conjecture in the case where

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  $M$ admits a specific smooth action of $T^k$$T^k$ for $k \geq 2$$k \geq 2$.
• The work of [James1985] confirms the result of [Musin1980] that the Petrie conjecture is true if $\dim M = 8$$\dim M = 8$, i.e., $M \simeq \CP^4$$M \simeq \CP^4$.
• According to [Dessai2002], the Petrie conjecture holds if

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  $M$ admits an appropriate smooth action of $Pin(2)$$Pin(2)$ and $\dim M \leq 22$$\dim M \leq 22$.
• It follows from [Dessai&Wilking2004] that the Petrie conjecture holds if

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  $M$ admits a smooth action of $T^k$$T^k$ and $\dim M \leq 8k-4$$\dim M \leq 8k-4$.

3 Further discussion

Masuda and Suh [Masuda&Suh2008] posed the following question about the invariance of Pontrjagin classes for toric $2n$$2n$-manifolds.

Question 3.1. For two toric $2n$$2n$-manifolds with isomorphic cohomology rings, is it true that any isomorphism between the cohomology rings preserves the Pontrjagin classes of the two manifolds?

A symplectic version of the Petrie conjecture is discussed by Tolman [Tolman2010]. In particular, the following question has been posed.

Question 3.2.

If the circle $S^1$$S^1$ acts in a Hamiltonian way on a compact symplectic manifold

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$M$ with $H^{2i}(M;\Rr) \cong H^{2i}(\CP^n; \Rr)$$H^{2i}(M;\Rr) \cong H^{2i}(\CP^n; \Rr)$ for all $i \geq 0$$i \geq 0$, is it true that $H^{j}(M;\Zz) \cong H^{j}(\CP^n; \Zz)$$H^{j}(M;\Zz) \cong H^{j}(\CP^n; \Zz)$ for all $j \geq 0$$j \geq 0$? Is the total Chern class of

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$M$ determined by the cohomology ring $H^*(M;\Zz)$$H^*(M;\Zz)$?

4 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&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$$S^1$-actions, Invent. Math. 48 (1978), no.1, 7–31. MR508087 (80e:57051) Zbl 0395.57020
• [Iberkleid1978] W. Iberkleid, Pseudolinear spheres, Michigan Math. J. 25 (1978), no.3, 359–370. MR512906 (80d:57023) Zbl 0377.57008
• [James1985] D. M. James, Smooth $S^1$$S^1$ actions on homotopy $\mathbb{C}P^4$$\mathbb{C}P^4$'s, Michigan Math. J. 32 (1985), no.3, 259–266. MR803831 (87c:57031) Zbl 0602.57026
• [Masuda&Suh2008] M. Masuda and D. Y. Suh, Classification problems of toric manifolds via topology, Toric topology, Amer. Math. Soc. (2008), 273–286. MR2428362 (2010d:14071) Zbl 1160.57032
• [Masuda1981] M. Masuda, On smooth $S^1$$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
• [Masuda1983] M. Masuda, Integral weight system of torus actions on cohomology complex projective spaces, Japan. J. Math. (N.S.) 9 (1983), no.1, 55–86. MR722536 (84m:57027) Zbl 0531.57037
• [Musin1978] O. R. Musin, Unitary actions of $S^1$$S^1$ on complex projective spaces, Uspekhi Mat. Nauk 33 (1978), no.6(204), 225–226. MR526030 (81a:57037)
• [Musin1980] O. R. Musin, Actions of the circle on homotopy complex projective spaces, Mat. Zametki 28 (1980), no.1, 139–152, 170. MR585071 (81m:57027)
• [Petrie1972] T. Petrie, Smooth $S^1$$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