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Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral


Authors: Toshiyuki Mano and Humihiko Watanabe
Journal: Proc. Amer. Math. Soc. 140 (2012), 3867-3881
MSC (2010): Primary 33C05; Secondary 14K25, 55N25, 14F40, 32C35
DOI: https://doi.org/10.1090/S0002-9939-2012-11221-3
Published electronically: March 8, 2012
MathSciNet review: 2944728
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Abstract: We study twisted cohomology and homology groups on a one-dimensional complex torus minus $ n$ distinct points with coefficients in a certain local system of rank one. This local system comes from the integrand of the Riemann-Wirtinger integral introduced by Mano. We construct bases of non-vanishing cohomology and homology groups, give an interpretation as a pairing of a cohomology class and a homology class to the Riemann-Wirtinger integral, and finally describe briefly the Gauss-Manin connection on the cohomology groups.


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Additional Information

Toshiyuki Mano
Affiliation: Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, Japan
Email: tmano@math.u-ryukyu.ac.jp

Humihiko Watanabe
Affiliation: Kitami Institute of Technology, 165, Koencho, Kitami 090-8507, Hokkaido, Japan
Email: hwatanab@cs.kitami-it.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2012-11221-3
Keywords: Theta function, integral representation
Received by editor(s): December 7, 2009
Received by editor(s) in revised form: August 21, 2010, January 31, 2011, April 19, 2011, and April 28, 2011
Published electronically: March 8, 2012
Additional Notes: The first author was supported in part by GCOE, Kyoto University and MEXT Grant-in-Aid for Young Scientists (B) (No. 21740118).
The second author was supported in part by Grant-in-Aid for Scientific Research (C) (No. 19540158), JSPS
Dedicated: Dedicated to Professor Keizo Yamaguchi on his sixtieth birthday
Communicated by: Ted Chinburg
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.