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A class of torus manifolds with nonconvex orbit space


Authors: Mainak Poddar and Soumen Sarkar
Journal: Proc. Amer. Math. Soc. 143 (2015), 1797-1811
MSC (2010): Primary 57R17, 57R91
DOI: https://doi.org/10.1090/S0002-9939-2014-12075-2
Published electronically: November 24, 2014
MathSciNet review: 3314091
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Abstract: We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch $ \chi _y$ genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring.


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Mainak Poddar
Affiliation: Departamento de Matemáticas, Universidad de los Andes, Bogota, Colombia
Email: mainakp@gmail.com

Soumen Sarkar
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
Email: soumensarkar20@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12075-2
Keywords: Almost complex, symplectic, Hirzebruch genus, moment angle complex, torus action
Received by editor(s): September 28, 2011
Received by editor(s) in revised form: July 11, 2012, and July 18, 2013
Published electronically: November 24, 2014
Additional Notes: The first author was partially supported by the Proyecto de investigaciones grant from the Universidad de los Andes
The second author was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (No. 2012-0000795)
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2014 American Mathematical Society