By Conforto F.
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Proof - 1) + 2): Given an arrow-map ( g , h ) : q + p and a homotopy H : A x I + B of h, form the homotopy H ( q x 1) : D x I + B ; because p g ( z ) = H ( q x l ~ ) ( z , Ofor ) every z E D, it follows that there exists a map G : D x I + E such that G(-,0) = g and p G = H ( q x 1). 2) + 3): Define the maps g : B' n E + E and h : B' n E -----) B by g(X, e ) = e and h(X,e) = p ( e ) , for every (A, e) E B' n E ; notice E , n E ) into that (9,h ) is an arrow-map of the arrow (B' fl E , ~ B I ~ B' p .
Now from condition 2) we obtain an arrow-map (G, H ) : ~ B I x~ 11 E -+ p ; then define I ' : B' n E + E' to be the adjoint of G under the exponential law. 3) + 1): Let H' : A + B' be the adjoint of H : A x I -+ B. Since ( H ' ( z ) , g ( z ) )E B' n E for every z E A , define the map G' : A + E' by G'(z) = r ( H ' ( z ) , g ( z ) )and set G : A x I + E to be the adjoint of G'. 1 is a fibration. 1 is the so-called covering homotopy property for the fibration ( E , p , B ) . 2 is a pictorial representation of this property.
12 Prove that if X ; , i = 1 , 2 are two CoH-groups, then X 1 A X 2 is a commutative CoH-group. 13 Let X ; , i = 1 , 2 be two CoH-groups with CoH-multiplications p;, respectively. 11, the space X 1 A X2 has two different CoH-group multiplications vi inherited from p ; , respectively, i = 1,2. Then, for every based space (Y,yo), the set [XI A X z , Y ] ,has two seemingly different group structures induced by v1 and vz; furthermore, prove that these two group structures on [XI A X 2 ,Y ] . coincide.
Abelsche Funktionen und algebraische Geometrie MAg by Conforto F.