Petrie conjecture

(Difference between revisions)
Jump to: navigation, search
Line 3: Line 3:
The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly 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.
The Petrie conjecture was formulated in the following context: suppose that a Lie group $G$ acts smoothly 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.
−
Petrie restricted his attention to smooth actions of the Lie group $S^1$, i.e., the circle {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}})
+
Petrie restricted his attention to smooth actions of the Lie group $S^1$, the circle {{cite|Petrie1972}} (or more generally, the torus $T^k$ for $k \geq 1$ {{cite|Petrie1973}})
−
on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]]. He has formulated the following conjecture.
+
on closed smooth manifolds $M$ which are [[Fake complex projective spaces|homotopy equivalent to $\CP^n$]].
{{beginthm|Conjecture|{{cite|Petrie1972}}}}
{{beginthm|Conjecture|{{cite|Petrie1972}}}}
−
Suppose that $M$ is a closed smooth 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}}
−
As Petrie {{cite|Petrie1973}} has shown, the total Pontrjagin class of $M$ agrees with that of $\CP^n$, if the action of $S^1$ on $M$ extends to a smooth action of $T^n$.
+
Petrie {{cite|Petrie1973}} has shown that the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if the action of $S^1$ on $M$ extends to a smooth action of $T^n$.
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 appropriate first Chern class. Another special cases
According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invarint almost complex structure with appropriate first Chern class. Another special cases
where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and
where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and
−
{{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds, if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate
+
{{cite|Masuda1981}}. Moreover, by the work of {{cite|Dessai2002}}, the Petrie conjecture holds if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends to an appropriate action of $Pin(2)$.
−
action of $Pin(2)$.
+
Similarly, the work of {{cite|Dessai&Wilking2004}} confirms the Petrie conjecture under the assumption that the action of $S^1$ on $M$ extends to an effective action of $T^k$
−
+
for $n \leq 4k - 2$.
−
+
−
has sho and {{cite|Dessai&Wilking2004}}.
+
== References ==
== References ==

Revision as of 02:28, 1 December 2010

This page has not been refereed. The information given here might be incomplete or provisional.

Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group G acts smoothly 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.

Petrie restricted his attention to smooth actions of the Lie group S^1, the circle [Petrie1972] (or more generally, the torus T^k for k \geq 1 [Petrie1973]) on closed smooth manifolds M which are homotopy equivalent to \CP^n.

Conjecture 0.1 [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}),
\displaystyle p(M) = (1+x^2)^{n+1}.

Petrie [Petrie1973] has shown that the total Pontrjagin class of M agrees with that of \CP^n if the action of S^1 on M extends to a smooth action of 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 [Dejter1976] and [James1985]. According to [Hattori1978], the conjecture is also true if M admits an invarint almost complex structure with appropriate first Chern class. Another special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981]. Moreover, by the work of [Dessai2002], the Petrie conjecture holds if \dim M \leq 22 and the action of S^1 on M extends to an appropriate action of Pin(2). Similarly, the work of [Dessai&Wilking2004] confirms the Petrie conjecture under the assumption that the action of S^1 on M extends to an effective action of T^k for n \leq 4k - 2.

References

Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=Petrie_conjecture&oldid=5848"
Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox