Petrie conjecture
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1 Problem
If a compact Lie group acts smoothly and non-trivially on a closed smooth manifold
, what constraints does this place on the topology of
in general and on the Pontrjagin classes of
in particular?
In the case where is homotopy equivalent to
,
, Petrie [Petrie1972] restricted his attention to smooth actions of the Lie group
, and posed the following conjecture.
Conjecture 1.1 [Petrie1972].
Suppose that acts smoothly and non-trivially on a closed smooth
-manifold
. Then the total Pontrjagin class
of
agrees with that of
, i.e.,
for a generator
of
.
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 action of
on
extends to a smooth action of the torus
.
- By the work of [Dejter1976], the Petrie conjecture is true if
, i.e.,
, and more generaly, if
.
- According to [Hattori1978], the Petrie conjecture holds if
admits an invariant almost complex structure with the first Chern class
.
- Other special cases where the Petrie conjecture holds are described by [Wang1975], [Yoshida1975/76], [Iberkleid1978], and [Muslin1978].
- 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 by assuming that the action of
on
extends to a specific action of
for
.
- It follows from [James1985] that the Petrie conjecture is also true if
, i.e.,
.
- The work of [Dessai2002] confirms the Petrie conjecture if the action of
on
extends to an appropriate smooth action of
and
.
- According to [Dessai&Wilking2004], the Petrie conjecture holds if the action of
on
extends to a smooth action of the torus
and
.
3 Further discussion
A symplectic version of the Petrie conjecture is discussed in the article of [Tolman2010].
4 References
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-manifolds in the homotopy type of
, Michigan Math. J. 23 (1976), no.1, 83–95. MR0402789 (53 #6603) Zbl 0326.57009
- [Dessai&Wilking2004] A. Dessai and B. Wilking, Torus actions on homotopy complex projective spaces, Math. Z. 247 (2004), no.3, 505–511. MR2114425 (2006c:57033) Zbl 1068.57034
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- [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
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