Tatsuhiko Yagasaki
Faculty of Engineering and Design, Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto, Japan
Measure-preserving homeomorphisms of noncompact manifolds
and mass flow toward ends
In this talk we discuss topological properties of
groups of measure-preserving homeomorphisms of noncompact manifolds (with the compact-open topology). Suppose $M$ is a connected $n$-manifold and $\mu$ is a good Radon measure on $M$. In the case that $M$ is compact, A. Fathi showed that
|
${\cal H}(M; \mu)$ |
$\subset$ |
${\cal H}(M, \mu\mbox{-reg})$ |
$\subset$ |
${\cal H}(M)$ |
| $n \geq 3$ |
|
SDR |
|
weak HD |
|
| $n = 1, 2$ |
($\ell_2$-mfd) |
SDR |
|
HD |
|
We are concerned with extension of these results to the noncompact case.
Suppose $M$ is noncompact. R. Berlanga showed that
${\cal H}(M; \mu) \subset {\cal H}(M, \mu\mbox{-end-reg})$ : SDR
and in [1] we have shown that
| Theorem. |
| |
(SDR) | | HD | |
|
$n = 2$ | ${\cal H}(M;\mu)_0$ | $\subset$ | ${\cal H}(M, \mu\mbox{-end-reg})_0$ | $\subset$ | ${\cal H}(M)_0$ |
|
| $\ell_2$-mfd | | ANR | | $\ell_2$-mfd |
Our next goal is to study a relation between the group ${\cal H}(M; \mu)$ and
the subgroup ${\cal H}^c(M; \mu)$ of $\mu$-preserving homeomorphisms of $M$ with compact support.
For this purpose we introduce a sort of mass flow homomorphism.
Let ${\cal B}_c(M)$ denote the set of Borel subsets of $M$ with compact frontier.
The mass flow toward ends induced by $h \in {\cal H}(M, \mu)_0$ is measured by
the function
$J_h^\mu : {\cal B}_c(M) \longrightarrow {\R} : J_h^\mu(C) = \mu(C - h(C)) - \mu(h(C) - C)$.
These functions $J_h^\mu$ form a topological vector space $V_\mu$ and we obtain a continuous group homomorphism $J^\mu : {\cal H}(M; \mu)_0 \to V_\mu$. In [2] we have shown
| Theorem. |
(1) |
$J^\mu$ has a continuous (non-homomorphic) section. |
|
(2) |
${\cal H}(M; \mu)_0 \cong {\rm Ker}\,J^\mu \times V_\mu$ ${\rm Ker}\,J^\mu \subset {\cal H}_E(M; \mu)$ : SDR |
The study of relation ${\cal H}^c(M; \mu)_0 \subset {\rm Ker}\,J^\mu$ is in progress.
REFERENCES
[1] T. Yagasaki. Groups of measure-preserving homeomorphisms of noncompact 2-manifolds, Topology and its Appli., 154 (2007) 1521–1531.
[2] T. Yagasaki. Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends, arXiv math.GT/0512231.
[3] T. Yagasaki. Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends, (preprint)
