Petrie conjecture
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1 Introduction
The Petrie conjecture was formulated in the following context: suppose that a Lie group acts smoothly on a closed smooth manifold
, what constraints does this place on the topology of
in general and on the Pontrjagin classes of
in particular.
Petrie restricted his attention to smooth actions of the Lie group [Petrie1972] (or more generally, the torus
for
[Petrie1973]) on closed smooth manifolds
which are homotopy equivalent to
. He has formulated the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose that![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\CP^n](/images/math/3/6/6/366bb12893fb823ceace7c638591875b.png)
![S^1](/images/math/7/a/6/7a6956585935bf7ca7c8ece6cab79334.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![p(M)](/images/math/0/2/c/02c42206226d935c7bb982720e0a668d.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\CP^n](/images/math/3/6/6/366bb12893fb823ceace7c638591875b.png)
![x \in H^2(M; \mathbb{Z})](/images/math/d/4/e/d4ed508d99aeb5016587b9dadc11c62a.png)
![\displaystyle p(M) = (1+x^2)^{n+1}.](/images/math/1/5/2/152f7c8d80ee709a970ee7e352b1240f.png)
The Petrie conjecture has not been confirmed in general, and no counterexample is know at the moment. However, in some special cases, the conjecture has beed proven by [Dejter1976], [Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2002],[Dessai&Wilking2004], and [Tolman2010].
2 References
- [Dejter1976] I. J. Dejter, Smooth
-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
- [Dessai2002] A. Dessai, Homotopy complex projective spaces with
-action, Topology Appl. 122 (2002), no.3, 487–499. MR1911696 (2003f:58048) Zbl 0998.57048
- [Iberkleid1978] W. Iberkleid, Pseudolinear spheres, Michigan Math. J. 25 (1978), no.3, 359–370. MR512906 (80d:57023) Zbl 0377.57008
- [Petrie1972] T. Petrie, Smooth
-actions on homotopy complex projective spaces and related topics, Bull. Amer. Math. Soc. 78 (1972), 105–153. MR0296970 Zbl 0247.57010
- [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