# Petrie conjecture

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## 1 Problem

If a compact Lie group $G$${{Stub}} == Problem == ; 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}} == 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 {{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. == Further discussion == ; 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}} == 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$$M \simeq \CP^n$, Petrie [Petrie1972] restricted his attention to actions of the Lie group $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$$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 $M$$M$ are determined by the representations of $S^1$$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 $M$$M$ agrees with that of $\CP^n$$\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$$M \simeq \CP^4$.
• According to [Hattori1978], the Petrie conjecture holds if $M$$M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$$c_1(M) = (n+1)x$.
• Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978].
• By [Tsukada&Washiyama1979] and [Masuda1981], the Petrie conjecture is true if the fixed point set consists of three or four connected components.
• Masuda [Masuda1983] proved the Petrie conjecture in the case where $M$$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 $M$$M$ admits an appropriate smooth action of $Pin(2)$$Pin(2)$ and $\dim M \leq 22$$\dim M \leq 22$.
• It follows from [Dessai&Wilking2004] that the Petrie conjecture holds if $M$$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$$S^1$ acts in a Hamiltonian way on a compact symplectic manifold $M$$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 $M$$M$ determined by the cohomology ring $H^*(M;\Zz)$$H^*(M;\Zz)$?