# Download Algebraic Topology (Colloquium Publications, Volume 27) by Solomon Lefschetz PDF

. ~ jP ....... Ho I has a solution Ho: I x Xo x Xo -----4. l: Xl = (s, P1(xd), -> (s E I, Xl E X d. l(X1) = H 1(0, xd· Then ilP1A(XO) = pflA(Xo) = pHo(l, xo) = (1, Po(xo» = i1po(xo), and therefore Pl A = Po . l = Pl' 0 0 40 I Introductory Notions Define maps g: 1 x I x Xo u I x i x Xo ~ X, K: I x I x Xo ~ I x B by g(s, t, xo) K(s, t, xo) J = , Ho(s, xo) (t = 0), fl A(Xo) (s = 1), H 1(s, A(XO)) (t = 1); I = (s, Po(xo)). The reader may verify that g is well defined and that K is an extension of po g.

O is a nondegenerate simplicial map. For example, if g is a simplicial map of L into K, then cjJ g is standard. However, not every standard map has this form. In fact, iff is a standard map, then f maps the (n - I)-skeleton r;-1 of L into the point eo; and it follows that, if Ci is any n-simplex of K, one can alter f inside Ci without changing it elsewhere in such a way that the resulting map is still standard. Thus standard maps are more flexible than simplicial maps; and this flexibility will be most useful in proving the main theorem of this section.