Algebraic Curves and Projective Geometry. Proc. conf Trento, by Edoardo Ballico, Ciro Ciliberto

By Edoardo Ballico, Ciro Ciliberto

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Then ( E ' , p ' , B ) is dominated by ( E , p , B ) if there exist arrow-maps (u,1 ~: p) + p' and ( u , 1 B ) : p' -, p such that uu is fibre homotopic to 1p over B. Prove that if ( E ' , p ' , B ) is dominated by a weak fibration ( E , p ,B ) , then (E',p',B ) is also a weak fibration. 3 Cofibrations - Let X E Top be given; a subspace A c X is a strong deformation retract of X if there exists a homotopy H : X x I X such that x H ( z , O ) = 2, 2 E H(z,l ) E A , 2 E X H(a,t) = a, (a,t) E A x I .

3 that a’ P’ is the lifting of LY Q p; thus, + and therefore, deg is a homomorphism. For every n E Z, let cr’(n) : I + R be defined by a’(n)(t)= tn. Now take ~ ( n=)pa’(n);clearly, deg(a(n)) = n and thus, deg is onto. 3. HOMOTOPY GROUPS 33 Finally, let [a]E 7rl(S1,eo)be such that deg([a]) = 0; this implies that the unique lifting a’of a is a loop of R. Since R is contractible, a’ cg (the constant map at 0), implying that a ce,. Hence, deg is one-to-one. 0 - - EXERCISES 1 3 . 1 Let (x,~~),(Y,y~) E Top, be given; let pr1: xx Y -----t x , pr2 :x x Y _t Y be the projection maps.

0 - - EXERCISES 1 3 . 1 Let (x,~~),(Y,y~) E Top, be given; let pr1: xx Y -----t x , pr2 :x x Y _t Y be the projection maps. Prove that the function given by = ((Pd*b)(a), (pr2)*(n)(a)) is an isomorphism for every n 2 1 . 2 Prove that the maps s1-+ S’ and x S1 , z H (z,eo) s1+ s1x s1, z H (e,,z) are not homotopic. 3 Show that if (X,Z,) is an H-space, 7rl(X,zo)is commutative. 4 Prove that S1 is not a retract of B2. 5 Prove that if p : E _t B and p’ : E’ --+B’ are covering maps, then p x p ’ : E x El- B x B’ is a covering map; hence, prove that the usual 2-dimensional torus can be covered by R2 .

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