# Petrie conjecture

## 1 Problem

If a compact Lie group $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ acts smoothly and non-trivially on a closed smooth manifold $M$$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)$?