Hn+2 Hn®Z/2 A H n i > ~n+l exact sequence *z/2 The column is a short exact sequence, Hn+ 3 is free abelian and i ® 1 the h o m o m o r p h i s m Fn+2(i) i in the row of the diagram. and the map 4.

For a map ~: A ~ Hn+ 2 -*: Fbn+l(Hn+2'v) be the map between = Hom(~,~/4), @: G(A) • G(Hn+ 2) be a map as in (3) and let F~+I(A,v) push outs, see For the inclusion Therefore ; let see (6) (5) (b), induced by Hn • Ext(~,cok bn+ 3) with (4). j : ~: ker bn+ 2 c Hn+ 2 we get pj ~n+3 = J*P~n+3 : ]*bn+2 = 0. the element {~n+2} = A is welldefined. J ~n+3 6 Ext(ker bn+2,cok An extension Zn+2' bn+3) which represents (7) this element, fits into the row of (5) (a) such that this row is an exact sequence. Since £-13" is surjective on -i (bn+ 2) we see that each exact row as in (a) is obtained via (7) by an appropriate 8n+3 " Next we define proper maps.

C) Each A3-system S is realizable, that is, for S there exists an n A~-polyhedron X such t h a t S ~ SX are p r o p e r l y isomorphic. (D) Each proper map 4: S(X) map X Theorem -- ~ S(X') is realizable by a continuous ~ X'. 3 (14). A c t u a l l y A n - s y s t e m s 3 S: A 3 ) An-systems, (here (i) is an easy corollary of this result and of and p r o p e r maps form a c a t e g o r y and S is a functor, n A3n denotes the full h o m o t o p y category of A ~ - p o l y h e d r a ) . My student M.

### Algebraic Topology Göttingen 1984: Proceedings of a Conference held in Göttingen, Nov. 9–15, 1984 by C. Allday, V. Puppe (auth.), Larry Smith (eds.)

by George

4.1