Petrie conjecture

Revision as of 16:08, 3 December 2010

1 Problem

If a compact Lie group ${{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 $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$. He proved that if $S^1$ acts smoothly on $M \simeq \CP^n$ with isolated fixed points, then the Pontrjagin classes of $M$ are determined by the representations of $S^1$ at the fixed points. Having this as well as other results in mind, Petrie 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 $x$ of $H^2(M; \mathbb{Z})$. {{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 manifold $M \simeq \mathbb{C}P^n$ 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 $\dim M \leq 6$. * According to {{cite|Hattori1978}}, the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$. * The results of {{cite|Musin1978}} and {{cite|Musin1980}} show in particular that the Petrie conjecture holds if $\dim M = 8$, i.e., $M \simeq \CP^4$. * 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 $k \geq 2$. * It follows from the work {{cite|James1985}} that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$. * The work of {{cite|Dessai2002}} confirms the Petrie conjecture if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$. * According to {{cite|Dessai&Wilking2004}}, the Petrie conjecture holds if $M$ admits to a smooth action of $T^k$ and $\dim M \leq 8k-4$. </wikitex> == Further discussion == <wikitex>; A related problem posed by Masuda and Suh {{cite|Masuda&Suh2008}} reads as follows. For two toric n$-manifolds with isomorphic cohomology rings, is there an isomorphism between the cohomology rings which preserves the Pontrjagin classes of the two manifolds? A symplectic version of the Petrie conjecture is discussed by Tolman {{cite|Tolman2010}}. In particular, in the case where $S^1$ acts in a Hamiltonian way on a compact symplectic manifold $M$, is the total Chern class of $M$ determined by the cohomology ring $H^*(M,\mathbb{Z})$? </wikitex> == References == {{#RefList:}} <!-- --> [[Category:Problems]] [[Category:Group actions on manifolds]]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 homotopy equivalent to $\CP^n$, $M \simeq \CP^n$, Petrie [Petrie1972] restricted his attention to actions of the Lie group $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$. He proved that if $S^1$ acts smoothly on $M \simeq \CP^n$ with isolated fixed points, then the Pontrjagin classes of $M$ are determined by the representations of $S^1$ at the fixed points. Having this as well as other results in mind, Petrie posed the following conjecture.

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 manifold $M \simeq \mathbb{C}P^n$ admits a smooth action of the torus $T^n$.
• By the work of [Dejter1976], the Petrie conjecture is true if $\dim M = 6$, i.e., $M \simeq \CP^3$ and hence, if $\dim M \leq 6$.
• According to [Hattori1978], the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$.
• Related results go back to [Musin1978] and [Musin1980], in particular, the late work proves the Petrie conjecture if $\dim M = 8$, i.e., $M \simeq \CP^4$.
• 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 $M$ admits a specific smooth action of $T^k$ for $k \geq 2$.
• It follows from the work [James1985] that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$.
• The work of [Dessai2002] confirms the Petrie conjecture if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$.
• According to [Dessai&Wilking2004], the Petrie conjecture holds if $M$ admits to a smooth action of $T^k$ and $\dim M \leq 8k-4$.

3 Further discussion

A related problem posed by Masuda and Suh [Masuda&Suh2008] reads as follows. For two toric $2n$-manifolds with isomorphic cohomology rings, is there an isomorphism between the cohomology rings which preserves the Pontrjagin classes of the two manifolds?

A symplectic version of the Petrie conjecture is discussed by Tolman [Tolman2010]. In particular, in the case where $S^1$ acts in a Hamiltonian way on a compact symplectic manifold $M$, is the total Chern class of $M$ determined by the cohomology ring $H^*(M,\mathbb{Z})$?

4 References

• [Dejter1976] I. J. Dejter, Smooth $S^1$-manifolds in the homotopy type of $CP^3$, 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 $Pin(2)$-action, Topology Appl. 122 (2002), no.3, 487–499. MR1911696 (2003f:58048) Zbl 0998.57048
• [Hattori1978] A. Hattori, $Spin^c$-structures and $S^1$-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 $S^1$ actions on homotopy $\mathbb{C}P^4$'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 $S^1$-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 $S^1$ 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 $S^1$-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, $S^1$-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 $S^1$ actions on cohomology complex projective spaces, Publ. Res. Inst. Math. Sci. 11 (1975/76), no.2, 483–496. MR0445528 (56 #3868) Zbl 0326.57008