Petrie conjecture

Revision as of 02:07, 22 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 $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$ 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., $p(M) = (1+x^2)^{n+1}$ 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 $\dim M \leq 6$. * 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>; A related problem posed by Masuda and Suh {{cite|Masuda&Suh2008}} reads as follows. For two toric n$-manifolds with isomorphic cohomology rings, is there an isomorphism between the cohomology rings which preserves the Pontrjagin classes of the two manifolds? A stroger version of the question reads as follows. {{beginthm|Question}} For two toric n$-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 $H^{2i}(M; \mathbb{R}) \cong H^{2i}(\mathbb{C}P^n; \mathbb{R})$ for all $i \geq 0$, is it true that $H^{j}(M; \mathbb{Z}) \cong H^{j}(\mathbb{C}P^n; \mathbb{Z})$ for all $j \geq 0$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\mathbb{Z})$? {{endthm}} </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 $S^1 = \{z \in \mathbb{C} \ \colon \ |z| = 1\}$ 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 [Petrie1972] 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 $M \simeq \CP^n$ 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 $\dim M \leq 6$.
• Related results go back to [Musin1978] and [Musin1980], in particular, the latter work shows that the Petrie conjecture holds if $\dim M = 8$, i.e., $M \simeq \CP^4$.
• According to [Hattori1978], the Petrie conjecture holds if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$.
• 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 $k \geq 2$.
• The work of [James1985] confirms the result of [Musin1980] that the Petrie conjecture is true if $\dim M = 8$, i.e., $M \simeq \CP^4$.
• According to [Dessai2002], the Petrie conjecture holds if $M$ admits an appropriate smooth action of $Pin(2)$ and $\dim M \leq 22$.
• It follows from [Dessai&Wilking2004] that the Petrie conjecture holds if $M$ admits 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 stroger version of the question reads as follows.

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

4 References

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