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

By Edoardo Ballico, Ciro Ciliberto

Best geometry and topology books

Extra info for Algebraic Curves and Projective Geometry. Proc. conf Trento, 1988

Example text

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 .