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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
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The Structure of Classical Diffeomorphism Groups (Hardcover, 1997 ed.)
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The Structure of Classical Diffeomorphism Groups (Hardcover, 1997 ed.)
Series: Mathematics and Its Applications, 400
Expected to ship within 10 - 15 working days
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In the 60's, the work of Anderson, Chernavski, Kirby and Edwards
showed that the group of homeomorphisms of a smooth manifold which
are isotopic to the identity is a simple group. This led Smale to
conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1,
of a smooth manifold M, with compact supports, and isotopic to the
identity through compactly supported isotopies, is a simple group
as well. In this monograph, we give a fairly detailed proof that
DifF(M)o is a simple group. This theorem was proved by Herman in
the case M is the torus rn in 1971, as a consequence of the
Nash-Moser-Sergeraert implicit function theorem. Thurston showed in
1974 how Herman's result on rn implies the general theorem for any
smooth manifold M. The key idea was to vision an isotopy in
Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a
deep connection between the local homology of the group of
diffeomorphisms and the homology of the Haefliger classifying space
for foliations. Thurston's paper [180] contains just a brief sketch
of the proof. The details have been worked out by Mather [120],
[124], [125], and the author [12]. This circle of ideas that we
call the "Thurston tricks" is discussed in chapter 2. It explains
how in certain groups of diffeomorphisms, perfectness leads to
simplicity. In connection with these ideas, we discuss Epstein's
theory [52], which we apply to contact diffeomorphisms in chapter
6.
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