Petrie conjecture

Revision as of 21:02, 30 November 2010 (view source) Krzysztof Pawałowski (Talk | contribs) ← Older edit		Revision as of 22:28, 30 November 2010 (view source) Krzysztof Pawałowski (Talk | contribs) Newer edit →
Line 10:		Line 10:
	</wikitex>		</wikitex>
−	Results confirming that in some cases the Petrie conjecture is true go back to {{cite|Dejter1976}}, {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2002}},{{cite|Dessai&Wilking2004}}, and {{cite|Tolman2010}}.	+	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 {{cite|Dejter1976}}, {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2002}},{{cite|Dessai&Wilking2004}}, and {{cite|Tolman2010}}.
	== References ==		== References ==

---

Revision as of 22:28, 30 November 2010

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

1 Introduction

The Petrie conjecture was formulated in the following context: suppose that a Lie group ${{Stub}}== Introduction == <wikitex>; 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 $S^1$ {{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. {{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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ {{endthm}} </wikitex> Results confirming that in some cases the Petrie conjecture is true go back to {{cite|Dejter1976}}, {{cite|Iberkleid1978}}, {{cite|Tsukada&Washiyama1979}}, {{cite|Dessai2002}},{{cite|Dessai&Wilking2004}}, and {{cite|Tolman2010}}. == References == {{#RefList:}} [[Category:Problems]] [[Category:Group actions on manifolds]]G$ acts smoothly on a closed smooth manifold

Tex syntax error

$M$, what constraints does this place on the topology of

Tex syntax error

$M$ in general and on the Pontrjagin classes of

Tex syntax error

$M$ in particular. Petrie restricted his attention to smooth actions of the Lie group $S^1$ [Petrie1972] (or more generally, the torus $T^k$ for $k \geq 1$ [Petrie1973]) on closed smooth manifolds

Tex syntax error

$M$ which are homotopy equivalent to $\CP^n$. He has formulated the following conjecture.

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 $S^1$-manifolds in the homotopy type of $CP^3$, 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 $Pin(2)$-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 $S^1$-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, $S^1$-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