Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends

Author:
Tatsuhiko Yagasaki

Journal:
Trans. Amer. Math. Soc. **362** (2010), 5745-5770

MSC (2010):
Primary 57S05, 58D05

DOI:
https://doi.org/10.1090/S0002-9947-2010-05101-3

Published electronically:
June 17, 2010

MathSciNet review:
2661495

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose is a noncompact connected oriented -manifold and is a positive volume form on . Let denote the group of orientation-preserving diffeomorphisms of endowed with the compact-open topology and let denote the subgroup of -preserving diffeomorphisms of . In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of . This argument, together with Moser's theorem, enables us to deduce two selection theorems for the groups and . The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of on the space of volume forms. This implies that is a strong deformation retract of the group consisting of , which preserves the set of -finite ends of .

The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of . Let denote the subgroup consisting of all which fix the ends of . S. R. Alpern and V. S. Prasad introduced the topological vector space of end charges of and the end charge homomorphism , which measures the mass flow toward ends induced by each . We show that the homomorphism has a continuous section. This induces the factorization , and it implies that is a strong deformation retract of .

**1.**S. R. Alpern and V. S. Prasad, Typical Dynamics of Volume-Preserving Homeomorphisms,*Cambridge Tracts in Mathematics*, Cambridge University Press, 2001.**2.**A. Banyaga, Formes-volume sur les variétés à bord,*Enseignement Math.*(2) 20 (1974) 127 - 131. MR**0358649 (50:11108)****3.**A. Banyaga, The Structure of Classical Diffeomorphism Groups,*Mathematics and Its Applications*400, Kluwer Academic Publishers Group, Dordrecht, 1997. MR**1445290 (98h:22024)****4.**R. Berlanga, Groups of measure-preserving homeomorphisms as deformation retracts,*J. London Math. Soc. (2)*68 (2003) 241 - 254. MR**1980255 (2004d:57040)****5.**R. Berlanga and D. B. A. Epstein, Measures on sigma-compact manifolds and their equivalence under homeomorphism,*J. London Math. Soc. (2)*27 (1983) 63 - 74. MR**686504 (84m:28023)****6.**R. Bott and L. W. Tu, Differential Forms in Algebraic Topology,*Graduate Texts in Mathematics*82, Springer-Verlag, New York-Berlin, 1982. MR**658304 (83i:57016)****7.**A. Fathi, Structures of the group of homeomorphisms preserving a good measure on a compact manifold,*Ann. Scient. Ec. Norm. Sup. (4)*13 (1980) 45 - 93. MR**584082 (81k:58042)****8.**R. E. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds,*Trans. Amer. Math. Soc.*255 (1979) 403 - 414. MR**542888 (80k:58031)****9.**J. Moser, On the volume elements on a manifold,*Trans. Amer. Math. Soc.*120 (1965) 286 - 294. MR**0182927 (32:409)****10.**D. McDuff, On groups of volume-preserving diffeomorphisms and foliations with transverse volume form,*Proc. London Math. Soc.*(3) 43 (1981) 295 - 320. MR**628279 (83g:58007)****11.**J. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity,*Ann. of Math.*42 (1941) 874 - 920. MR**0005803 (3:211b)****12.**T. Yagasaki, Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends,*Fund. Math.*197 (2007) 271 - 287. MR**2365892 (2009f:57056)**

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Additional Information

**Tatsuhiko Yagasaki**

Affiliation:
Division of Mathematics, Graduate School of Science and Technology, Kyoto Institute of Technology, Kyoto, 606-8585, Japan

Email:
yagasaki@kit.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-2010-05101-3

Keywords:
Group of volume-preserving diffeomorphisms,
mass flow,
end charge homomorphism,
$\sigma$-compact manifold

Received by editor(s):
June 9, 2008

Published electronically:
June 17, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

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