Petrie conjecture

(Difference between revisions)
Jump to: navigation, search
m
Line 1: Line 1:
−
{{Stub}}== Introduction ==
+
{{Stub}}== Problem ==
<wikitex>;
<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?
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?
Line 9: Line 9:
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 verified his conjecture under the assumtion that the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$.
+
== Progress to date ==
−
As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However, if $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
+
As of November 30, 2010, the Petrie conjecture has not been confirmed in general. However it has been proven in the following 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
+
* Petrie {{cite|Petrie1973}} has verified his conjecture under the assumption that the action of $S^1$ on $M$ extends to a smooth action of the torus $T^n$.
−
where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and
+
* If $\dim M \leq 8$, the statement is true by the work of {{cite|Dejter1976}} and {{cite|James1985}}.
−
{{cite|Masuda1981}}. By the work of {{cite|Dessai2002}}, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ if $\dim M \leq 22$ and the action of $S^1$ on $M$ extends
+
* According to {{cite|Hattori1978}}, the conjecture is also true if $M$ admits an invariant almost complex structure with the first Chern class $c_1(M) = (n+1)x$.
−
to an appropriate action of $Pin(2)$. Similarly, the work of {{cite|Dessai&Wilking2004}} confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$ and the action
+
* Other special cases where the Petrie conjecture holds are described by {{cite|Wang1975}}, {{cite|Yoshida1976}}, {{cite|Iberkleid1978}}, {{cite|Muslin1978}}, {{cite|Tsukada&Washiyama1979}}, and {{cite|Masuda1981}}.
−
of $S^1$ on $M$ extends to a smooth action of $T^k$ for $k \geq 1$.
+
* By the work of {{cite|Dessai2002}}, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ 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 {{cite|Dessai&Wilking2004}} confirms the Petrie conjecture under the assumption that $\dim M \leq 8k-4$ and the action of $S^1$ on $M$ extends to a smooth action of $T^k$ for $k \geq 1$.
+
== Further discussion ==
A symplectic version of the Petrie conjecture is discussed in the article of {{cite|Tolman2010}}.
A symplectic version of the Petrie conjecture is discussed in the article of {{cite|Tolman2010}}.

Revision as of 13:07, 1 December 2010

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

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? What about the case where M is homotopy equivalent to \CP^n?

In such a context, Petrie restricted his attention to smooth actions of the Lie group S^1, the circle [Petrie1972] (more generally, the torus T^k for k \geq 1 [Petrie1973]), and he posed the following conjecture.

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}.

1 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 action of S^1 on M extends to a smooth action of the torus T^n.
  • 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 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], [Yoshida1976], [Iberkleid1978], [Muslin1978], [Tsukada&Washiyama1979], and [Masuda1981].
  • By the work of [Dessai2002], the total Pontrjagin class of M agrees with that of \CP^n 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 \dim M \leq 8k-4 and the action of S^1 on M extends to a smooth action of T^k for k \geq 1.

2 Further discussion

A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].

3 References

Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=Petrie_conjecture&oldid=5887"
Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox