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Local homology of groups of volume preserving diffeomorphisms. I. (English) Zbl 0577.58005

Given a topological group G, consider the homotopy fibre \(\bar BG\) of \(BG^ d\to BG\), where \(G^ d\) is G with discrete topology, and B indicates the classifying space. This space depends only on the germ of G at the identity, and its homology \(H_*(\bar BG)\) is called the local homology of G by J. Mather [Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 35-37 (1975; Zbl 0333.57015)]. Now let W be a non-compact oriented smooth manifold with a volume form \(\omega\), connected, without boundary, all ends trivial, and of infinite \(\omega\)-volume. Consider the group \(Diff_{\omega}(W)\) of \(\omega\)-preserving diffeomorphisms with the compact \(C^{\infty}\)-topology and its identity component \(Diff_{\omega 0}(W)\). There is a homeomorphism \(\Phi\) : Diff\({}_{\omega 0}(W)\to H^{n-1}(W,{\mathbb{R}})\), called flux and given by \(\Phi (f)z=\int_{c}\omega\), where z is an (n-1)-cycle in W and c is an n-chain in W with boundary \(f_*(z)-z\). Let \(Diff^{\Phi}_{\omega 0}(W)\) be the kernel of \(\Phi\).
Let \(\Gamma^ n_{sl}\) be the groupoid of germs of volume preserving diffeomorphisms of \({\mathbb{R}}^ n\). Let \(\tau\) : TW\(\to BSL(n,{\mathbb{R}})\) classify the tangent bundle TW. Consider the pullback \(\tau^*(B \Gamma^ n_{s\ell})\) of the fibration B \(\Gamma\) \({}^ n_{s\ell}\to BSL(n,{\mathbb{R}})\) and let \(S_{\omega}(W)\) be the space of all sections of \(\tau^*(B\Gamma^ n_{s\ell})\to TW\) with the compact open topology; the \(\pi_ 1(S_{\omega}(W))=H^{n-1}(W,{\mathbb{R}})\). Let \(\tilde S_{\omega}(W)\) be the universal cover of \(S_{\omega}(W).\)
Then the first main result of the paper is that there is a mapping (for dim \(W\neq 2)\) \(\tilde f_ W: \bar B Diff^{\Phi}_{\omega 0}(W)\to \tilde S_{\omega}(W)\) which is an isomorphism in integer homology and covers a homology equivalence \(f_ W: \bar B Diff_{\omega 0}(W)\to S_{\omega}(W)\). For \(n=1\) both \(\tilde f_ W\) and \(f_ W\) are homotopy equivalences.
The second main result concerns the group \(Diff^ c_ 0(W)\) of compactly supported \(\omega\)-preserving diffeomorphisms of W with the usual direct limit topology over all compact subsets. Let \(S^ c_{\omega}(W)\) be the space of sections of \(\tau^*(B\Gamma^ n_{s\ell})\to TW\) which equal a certain base section \(s_ 0\) off some compact subset, again with the direct limit topology, and let \(S^ c_{\omega 0}(W)\) be the connected component containing \(s_ 0\). Then the second main theorem says that there is a homology equivalence \(f^ c_ W: \bar B Df^ c_{\omega 0}(W)\to S^ c_{\omega 0}(W)\). As a corollary this yields \(\pi_{n+1}(\bar B\Gamma^ n_{s\ell})=0\) for \(n\geq 3\) and \(\pi_ 3(\bar B\Gamma^ 2_{s\ell})={\mathbb{R}}\).
Reviewer: P.Michor

MSC:

58D05 Groups of diffeomorphisms and homeomorphisms as manifolds

Citations:

Zbl 0333.57015
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References:

[1] A. BANYAGA , Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique (Comm. Math. Helv., Vol. 53, 1978 , pp. 174-227). MR 80c:58005 | Zbl 0393.58007 · Zbl 0393.58007
[2] R. E. GREENE and K. SHIOHAMA , Diffeomorphisms and Volume Preserving Embeddings of Non-Compact Manifolds (Trans. A.M.S., Vol. 255, 1979 , pp. 403-414). MR 80k:58031 | Zbl 0418.58002 · Zbl 0418.58002
[3] A. HAEFLIGER , Homotopy and Integrability , in : Manifolds, Amsterdam 1970 (Springer Lect. Notes, No. 197, 1971 , pp. 133-163). MR 44 #2251 | Zbl 0215.52403 · Zbl 0215.52403
[4] J. C. HAUSMANN and D. HUSEMOLLER , Acyclic Maps (Enseign. Math., 1979 ). MR 80k:55044 | Zbl 0412.55008 · Zbl 0412.55008
[5] A. KRYGIN , Continuation of Diffeomorphisms Preserving Volume (Funct. Anal. and Appl., Vol. 5, 1971 , pp. 147-150). MR 51 #4309 | Zbl 0236.57016 · Zbl 0236.57016
[6] J. MATHER , The Vanishing of the Homology of Certain Groups of Homeomorphisms (Topology, Vol. 10, 1971 , pp. 297-298). MR 44 #5973 | Zbl 0207.21903 · Zbl 0207.21903
[7] J. MATHER , Integrability in Codimension 1 (Comm. Math. Helv., Vol. 48, 1973 , pp. 195-233). MR 50 #8556 | Zbl 0284.57016 · Zbl 0284.57016
[8] D. MCDUFF , Foliations and Monoids of Embeddings , in : Geometric Topology, Cantrell, Academic Press, 1979 , pp. 429-444. MR 82m:57014 | Zbl 0473.57016 · Zbl 0473.57016
[9] D. MCDUFF , The Homology of Some Groups of Diffeomorphisms (Comm. Math. Helv., Vol. 55, 1980 , pp. 97-129). MR 81j:57018 | Zbl 0448.57015 · Zbl 0448.57015
[10] D. MCDUFF , On groups of Volume Preserving Diffeomorphisms and Foliations with Transverse Volume Form (Proc. London Math. Soc., (3), Vol. 43, 1981 , pp. 295-320). MR 83g:58007 | Zbl 0411.57028 · Zbl 0411.57028
[11] D. MCDUFF , On Tangle Complexes and Volume Preserving Diffeomorphisms of Open 3-Manifolds (Proc. London Math. Soc., (3), Vol. 43, 1981 , pp. 321-333). MR 83g:58008 | Zbl 0411.57029 · Zbl 0411.57029
[12] G. ROUSSEAU , Difféomorphisms d’une variété symplectique non-compacte (Comm. Math. Helv., Vol. 53, 1978 , pp. 622-633). MR 80a:58010 | Zbl 0393.53017 · Zbl 0393.53017
[13] G. B. SEGAL , Classifying Spaces Related to Foliations (Topology, Vol. 17, 1978 , pp. 367-382). MR 80h:57036 | Zbl 0398.57018 · Zbl 0398.57018
[14] S. SMALE , Diffeomorphisms of the 2-sphere (Proc. A.M.S., Vol. 10, 1959 , pp. 621-626). MR 22 #3004 | Zbl 0118.39103 · Zbl 0118.39103
[15] W. THURSTON , Foliations and Groups of Diffeomorphisms (Bull. A.M.S., 80, 1974 , pp. 304-307). Article | MR 49 #4027 | Zbl 0295.57014 · Zbl 0295.57014
[16] W. THURSTON , On the Structure of the Group of Volume Preserving Diffeomorphisms , Preprint c. 1973 .
[17] J. MATHER , Foliations and Local Homology of Groups of diffeomorphisms (Proc. Int. Congr. Math., Vancouver, 1974 , Vol. 2, pp. 35-37). MR 55 #4205 | Zbl 0333.57015 · Zbl 0333.57015
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