Petrie conjecture
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What about the case where $M$ is [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]? | What about the case where $M$ is [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]? | ||
− | In such a context, Petrie restricted his attention to smooth actions of the Lie group $S^1$, the circle {{cite|Petrie1972}} ( | + | In such a context, Petrie restricted his attention to smooth actions of the Lie group $S^1$, the circle {{cite|Petrie1972}} (more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}}), and he posed |
+ | the following conjecture. | ||
{{beginthm|Conjecture|{{cite|Petrie1972}}}} | {{beginthm|Conjecture|{{cite|Petrie1972}}}} | ||
Suppose that $M$ is a closed smooth $2n$-manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ | Suppose that $M$ is a closed smooth $2n$-manifold homotopy equivalent to $\CP^n$ and that $S^1$ acts smoothly and non-trivially on $M$. Then the total Pontrjagin class $p(M)$ of $M$ agrees with that of $\CP^n$, i.e., for a generator $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ | ||
{{endthm}} | {{endthm}} | ||
− | Petrie {{cite|Petrie1973}} has proven that his conjecture holds true if the action of $S^1$ on $M$ extends to a smooth action of $T^n$. | + | Petrie {{cite|Petrie1973}} has proven that his conjecture holds true if the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$. |
The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. | The Petrie conjecture has not been confirmed in general, but if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}. | ||
According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Another special cases | According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with the first Chern class $c_1(M) = (n+1)x$. Another special cases |
Revision as of 04:18, 1 December 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Introduction
Tex syntax error, what constraints does this place on the topology of
Tex syntax errorin general and on the Pontrjagin classes of
Tex syntax errorin particular? What about the case where
Tex syntax erroris homotopy equivalent to ?
In such a context, Petrie restricted his attention to smooth actions of the Lie group , the circle [Petrie1972] (more generally, the torus for [Petrie1973]), and he posed the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose thatTex syntax erroris a closed smooth -manifold homotopy equivalent to and that acts smoothly and non-trivially on
Tex syntax error. Then the total Pontrjagin class of
Tex syntax erroragrees with that of , i.e., for a generator ,
Tex syntax errorextends to a smooth action of the torus .
The Petrie conjecture has not been confirmed in general, but if , the statement is true by the work of [Dejter1976] and [James1985].
According to [Hattori1978], the conjecture is also true ifTex syntax erroradmits an invarint almost complex structure with the first Chern class . Another special cases
where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and
[Masuda1981]. By the work of [Dessai2002], the total Pontrjagin class ofTex syntax erroragrees with that of if and the action of on
Tex syntax errorextends
to an appropriate action of . Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that and the action
of onTex syntax errorextends to a smooth action of for .
A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].
References
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- [Dessai2002] A. Dessai, Homotopy complex projective spaces with -action, Topology Appl. 122 (2002), no.3, 487–499. MR1911696 (2003f:58048) Zbl 0998.57048
- [Hattori1978] A. Hattori, -structures and -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
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- [Muslin1978] Template:Muslin1978
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- [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, -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
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