# A Comprehensive Introduction to Differential Geometry by Michael Spivak PDF

By Michael Spivak

Publication through Michael Spivak, Spivak, Michael

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5) O ~ Der(F V,A) h*> Der (F ,A) ~ HOmFv((VF) ab,A) Since A is in derivation tion d' : G V vanishes on VG , thus giving rise to a deriva- since A is in ~ Der(G,A) VMod__Fv , we have h* : Der(F V,A) -T Der(F,A) This implies that both homomorphisms Since f' Corollary that every ~ A . 5) are monomorphic. 5) is monomorphic. 5) we conclude that ker f~ = V(GI,A) Ji' : Fi ~ FV @9 V(G2,A) have left inverses (see 55 both split. 6). 7. In this s e c t i o n we considering the with . Thus V S i n c e the c W ~ H 2 (F,A)) e V ( G 2,A) isomorphism this c o m p l e t e s , is c l e a r l y Exact shall d e d u c e sequence let an e x a c t V(-,-) V ¢ W and and induced by the the proof.

4). 2) can be c o n t i n u e d to the right) We o b t a i n , Ext-sequence.

To prove this, let that that proof d' (x) = d' ( x ) - I f ( x ) - d ' (y)" ( f ( x ) ) - l ] - f ( x ) - f ( y ) fl(x)"fl(Y) fl two homomorphisms shows The : ~ G fl . Then = It , x then induces inducing ~ and ~ = f l ( x ) " (f(x)) -I of. 3 and ~ . Conversely, e induces is t h u s then the f,fi:G calculation a derivation complete. 4). 4. 10) [e] (see with then construct exact the e x a c t (sx) a function , set. 216-217). the c o h o m o l o g y [57], a function by . e. lll). 11) A = A[E] A ) ~ > ZQ ® G I G associated .