The homotopy type of the space of diffeomorphisms. I

Abstract

A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D i f f left-parenthesis upper D Superscript n Baseline comma partial-differential right-parenthesis asymptotically-equals normal upper Omega Superscript n plus 1 Baseline left-parenthesis Top Subscript n Baseline slash upper O Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Diff</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≃<!-- ≃ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mtext>Top</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\operatorname {Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text {Top}_n}/{O_n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained. The main technique is fibrewise smoothing.