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Almost complex torus manifolds - a problem of Petrie type.
- Source :
-
Proceedings of the American Mathematical Society . Jul2024, Vol. 152 Issue 7, p3153-3164. 12p. - Publication Year :
- 2024
-
Abstract
- The Petrie conjecture asserts that if a homotopy \mathbb {CP}^n admits a non-trivial circle action, its Pontryagin class agrees with that of \mathbb {CP}^n. Petrie proved this conjecture in the case where the manifold admits a T^n-action. An almost complex torus manifold is a 2n-dimensional compact connected almost complex manifold equipped with an effective T^n-action that has fixed points. For an almost complex torus manifold, there exists a graph that encodes information about the weights at the fixed points. We prove that if a 2n-dimensional almost complex torus manifold M only shares the Euler number with the complex projective space \mathbb {CP}^n, the graph of M agrees with the graph of a linear T^n-action on \mathbb {CP}^n. Consequently, M has the same weights at the fixed points, Chern numbers, cobordism class, Hirzebruch \chi _y-genus, Todd genus, and signature as \mathbb {CP}^n, endowed with the standard linear action. Furthermore, if M is equivariantly formal, the equivariant cohomology and the Chern classes of M and \mathbb {CP}^n also agree. [ABSTRACT FROM AUTHOR]
- Subjects :
- *COMPLEX manifolds
*EULER number
*CHERN classes
*PROJECTIVE spaces
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 152
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 177610126
- Full Text :
- https://doi.org/10.1090/proc/16768