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By Prof. Dan Burghelea, Prof. Richard Lashof, Prof. Melvin Rothenberg (auth.)
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Additional info for Groups of Automorphisms of Manifolds
Place of F , and Hence a s i m U a r d i a g r a m (with (F, aF) (Dd+, D d-~) in place o~ D d) to that ix* the proo~ o~ 4. Z' shows that ~rm_~(,~d(F , aF)) -- ~rm_~(~Pt (F, a~')) li~ts to ~ r m . i ( A P i (~', ~)~')), and hence f is h o m o t o p i c r a PL-buncUe map i~ one o~ the conaitions a) o r b) holds. 5. Fibrations over Manifolds S e c t i o n 0. In the l a s t c h a p t e r , bundle projection. trary manifold. we e x a m i n e d the p r o b l e m of d e f o r m i n g We now c o n s i d e r t h i s p r o b l e m w h e n We w o r k s i m u l t a n e o u s l y in t h e Sm f: V -," S m is r e p l a c e d by an a r b i - pL a n d s m o o t h c a t e g o r y .
3 from 3. rel(A(V)), C(N) in place of ~(N), etc. J F r o m Corollaries 3. Z',3,3' we get: 36 Let Theorem A'. V n be a c o m p a c t k - c o n n e c t e d m a n i f o l d , n >_ 5, a n d D n C Int V n. a) Diff Cate$or~r. i. j < 2k-4 Z. j < ~k-g or (k W X I satisfy the conditions do, w e g e t f r o m ( 3 . 1 1 ) t h e r e s u l t : 35 T h e o r e m 3. i'. n Given V , n >__5, and w n c i. ~ri(SW ) -~ iri(W), i = 0, i , Z. Tri(V , W ) = 0 f o r i~ r, r < n-4 Int V n. Suppose ( o r r < _ n - 3 if n > 5 a n d ~i(sv) _~ ~ i ( v ) , i = 0, i), 3. W is k - c o n n e c t e d , 0<_k<_ r. Then ~j(y) = 0 for a) j <__inf(Zr-3, r+k-Z), Diff C a t e g o r y b) j <__inf(Zr-3,r+k-g,r+Z), P L Category. Corollary 3. g'. Given V n, n > 5, and D n c k-connected ( k < n - 4 if 8V is not 1-connected).
Groups of Automorphisms of Manifolds by Prof. Dan Burghelea, Prof. Richard Lashof, Prof. Melvin Rothenberg (auth.)
W X I satisfy the conditions do, w e g e t f r o m ( 3 . 1 1 ) t h e r e s u l t : 35 T h e o r e m 3. i'. n Given V , n >__5, and w n c i. ~ri(SW ) -~ iri(W), i = 0, i , Z. Tri(V , W ) = 0 f o r i~ r, r < n-4 Int V n. Suppose ( o r r < _ n - 3 if n > 5 a n d ~i(sv) _~ ~ i ( v ) , i = 0, i), 3. W is k - c o n n e c t e d , 0<_k<_ r. Then ~j(y) = 0 for a) j <__inf(Zr-3, r+k-Z), Diff C a t e g o r y b) j <__inf(Zr-3,r+k-g,r+Z), P L Category. Corollary 3. g'. Given V n, n > 5, and D n c k-connected ( k < n - 4 if 8V is not 1-connected).