On crossed homomorphisms on symplectic mapping class groups

Author:
Ryoji Kasagawa

Journal:
Trans. Amer. Math. Soc. **360** (2008), 4815-4839

MSC (2000):
Primary 53C15, 57S05; Secondary 55R40, 57R20

DOI:
https://doi.org/10.1090/S0002-9947-08-04618-7

Published electronically:
April 11, 2008

MathSciNet review:
2403705

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a symplectic manifold with a relation between Chern classes of it and the cohomology class of the symplectic form , we construct a crossed homomorphism on the symplectomorphism group of with values in the cohomology group of and show an application to the symplectic flux group. Moreover we see that descends to a crossed homomorphism on the symplectic mapping class group of and show a nontrivial example of it.

**1.**K. S. Brown, Cohomology of groups, Springer, Berlin, Heidelberg, New York, Springer 1982. MR**672956 (83k:20002)****2.**J. Kedra, D. Kotschick and S. Morita,*Crossed flux homomorphisms and vanishing theorems for flux groups*, Geom. Funct. Anal. 16(2006), no. 6 1246-1273. MR**2276539****3.**D. Kotschick and S. Morita,*Signatures of foliated surface bundles and the symplectomorphism groups of surfaces*, Topology 44(2005), no. 1, 131-149. MR**2104005 (2005k:57050)****4.**D. Kotschick and S. Morita,*Characteristic classes of foliated surface bundles with area-preserving holonomy*, J. Differential Geom. 75(2007), no. 2, 273-302. MR**2286823****5.**D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford Univ. Press, 1995. MR**1373431 (97b:58062)****6.**J.K. Moser,*On the volume elements on a manifold*, Trans. Amer. Math. Soc., 120(1965), 286-294. MR**0182927 (32:409)****7.**S. Morita,*Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles*, J. Differential Geometry 47(1997), 560-599. MR**1617632 (99h:57029)****8.**A. Reznikov,*Characteristic classes in symplectic topology*, Selecta Math. 3(1997), 601-642. MR**1613528 (2000f:53116)****9.**E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966. MR**0210112 (35:1007)****10.**W. Thurston,*Some simple examples of symplectic manifolds*, Proceedings of the American Mathematical Society, 55(1976), 467-468. MR**0402764 (53:6578)****11.**R. Trapp,*A linear representation of the mapping class group and the theory of winding numbers*, Topology and its Appl. 43(1992), 47-64. MR**1141372 (92k:57027)****12.**I. Vaisman, Symplectic Geometry and Secondary Characteristic Classes, Progress in Math. 72, Birkhäuser, Boston, Basel, 1987. MR**932470 (89f:58062)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
53C15,
57S05,
55R40,
57R20

Retrieve articles in all journals with MSC (2000): 53C15, 57S05, 55R40, 57R20

Additional Information

**Ryoji Kasagawa**

Affiliation:
Department of Mathematics, College of Science and Technology, Nihon University, 1-8 Kanda, Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan

Email:
kasagawa@math.cst.nihon-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-08-04618-7

Received by editor(s):
July 26, 2006

Published electronically:
April 11, 2008

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.