Petrie conjecture
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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 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 $x \in H^2(M; \mathbb{Z})$, $$p(M) = (1+x^2)^{n+1}.$$ | ||
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The Petrie conjecture has not been confirmed in general (no counterexample is know at the moment). However, if $\dim M \leq 8$, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ | The Petrie conjecture has not been confirmed in general (no counterexample is know at the moment). However, if $\dim M \leq 8$, the total Pontrjagin class of $M$ agrees with that of $\CP^n$ | ||
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[[Category:Problems]] | [[Category:Problems]] | ||
[[Category:Group actions on manifolds]] | [[Category:Group actions on manifolds]] | ||
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Revision as of 23:43, 30 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Introduction
Tex syntax error, what constraints does this place on the topology of
Tex syntax errorin general and on the Pontrjagin classes of
Tex syntax errorin particular. Petrie restricted his attention to smooth actions of the Lie group [Petrie1972] (or more generally, the torus for [Petrie1973]) on closed smooth manifolds
Tex syntax errorwhich are homotopy equivalent to . He has formulated the following conjecture.
Conjecture 0.1 [Petrie1972].
Suppose thatTex syntax erroris a closed smooth manifold homotopy equivalent to and that acts smoothly and non-trivially on
Tex syntax error. Then the total Pontrjagin class of
Tex syntax erroragrees with that of , i.e., for a generator ,
Tex syntax erroragrees with that of
by the work of [Dejter1976] and [James1978].
[Iberkleid1978], [Tsukada&Washiyama1979], [Dessai2002],[Dessai&Wilking2004], and [Tolman2010].
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
- [James1978] Template:James1978
- [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