Groups of Automorphisms of Manifolds by Prof. Dan Burghelea, Prof. Richard Lashof, Prof. Melvin

By Prof. Dan Burghelea, Prof. Richard Lashof, Prof. Melvin Rothenberg (auth.)

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F Then of 4. )~ . (APt(F)) Irm i ('~pL(F) i m p l y the a b o v e b y ( 3 . 2 a n d ( 3 . 3 ) . 53 P r o o f of T h e o r e m 4. 3'. category. We apply the a r g u m e n t of 4. 3 in the smooth In Step 2 we replace ~ P l ( F , 8F) by Ad(F, 8F), the corresponding smooth complex (see Chapter 2). Let D d+ C F be a smooth embedding with a F C ~ D d + = D d - i , 8 D d _ intDd-i transverse to 8F. complex whose k-simplices are diffeomorphisms ~(diA k X D+d) C diA k X D +d Let ~d(D d+ ,D d-l) be the 9: A k X D d+ -* A k X D d+ ' and ~IA k X S d+ = identity.

E . E(D p,V~ g) has the s a m e homotoP7 type as its s u b s p a c e e m b e d d i n g s w h i c h a g r e e with g o n D p as well as Evidently we have a m a p ~: E(D p, V, g; D_P) -* F(D p, V; g). 8D p. N p S i m i l a r l y for E(D ,V; g). rel p ~ : ~. el(E(DP, V; g)) -* ~ri (F(D , V; g), such that Thus we get a h o m o m o r p h i s m wrel(E(D p, V; g)) ~ i > xrel(F(DP, V; g)) I ~rel(E(DP, V; g; DP)) I commutes. b) In c o m p o s i n g ot and ~ we get a h o m o m o r p h i s m r eIl(. D p-i , V ; g ] D P - t ) ) 6:lr;el(E(DP, V;g)) -* ~ri+ C o n s i d e r the s u b s p a c e F0(DP, v ; g)C F(DP, v ; g) of e m b e d d i n g s which also coincide with g in a neighborhood of D P - t ; and s i m i l a r l y , c) .

If f-1 (p) r (L, a L ) aFV = ~ we assume (L, OL) w h i c h m a k e s we assume on we can and have selected an s-structure it a f i n i t e P D p a i r in t h e s e n s e of [ 18 ]. the structure agrees w i t h t h e o n e o n DL a l r e a d y on If w e a r e in C a s e I determined by ~+V L-~ OL. 0. Let ~r:E -* K b e a S e r r e c o n n e c t e d a n d a f i n i t e CW c o m p l e x . L e t pE K a n d Then there exists a unique s-structure subcomplex K' C K, is a subcomplex of h K , : C K , - ' w - l ( K ') C, h K , = h I C K , Cp = F.

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