By Routh E.J.
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Extra info for A treatise on the dynamics of a system of rigid bodies
2 . 2. 7 Observe now t h a t t h e following homotopies a l l coincides w i t h t h e a c t i o n . remain i n i s given by X,(x,~) +. x,X + - fl pl fl: 5: admits an H-structure H-map) so that fo X1,u1 * N p -+ f f 1 0' over lifts to a Given X2,u2. 3. Homotopy p r o p e r t i e s of H-spaces Postnikov systems Given a space IXnshnshn ,n-lY (PI) X i s a system k 1 of spaces and maps so t h a t n hn: X -+ Xn nm(xn) = o hn,n-lhn N m 5 n. i s an isomorphism f o r m > n. for xn-1 i s + K(n ( X ) , n + l ) n Xn-l nm(h ) n satisfies: xn hn,n-l: ('3) A Postnikov system f o r X.
3. p: i s a l s o a multiplication. p hence, A and IJ (G) i s a multiplication i: X v X c X If Obviously The s e t of H-structures f o r i s fixed then t h e assignment a : given by w + pw. Theorem ( [ C o ~ e l a n d ] ~ If ): is in = wA, i s t h e set X (F) * [X A XYX] i s an inverse t o t h e functions = HD(l,;,p) H-structures f o r X PWY P i*-l 11 i*-l(F) D x X X,p Hence: i s an H-space then the s e t of 1-1 correspondence w i t h [x A X,X]. 4. Let Lemma: be an H-map. f : X,p + X 1 , p l equivaZence with a hmotopy inverse Proof: (f A + = HD(l,p,p) then g i s an H-map.
1. of Definition: X,u be an H-space and i s a map 0 : X x Y + Y on Y h: X + Y X,p Let i s given by so t h a t Y a space. 4 s t a t e s t h a t if X,p a c t s on then t h e f i b e r of Y s o t h a t t h e p r o j e c t i o n on If Y,Py Y h admits a m u l t i p l i c a t i o n , i s multiplicative. i s an H-space, an a c t i o n rl of X,px on Y,py i s said t o be an H-action i f i n a d d i t i o n i s commutative. 2. Lemma: If X,p H-acts on Y,py then h: X + Y, h ( x ) = rl(x,+) is an H-map. *,h(x2).
A treatise on the dynamics of a system of rigid bodies by Routh E.J.