Algebraic Topology Göttingen 1984: Proceedings of a by C. Allday, V. Puppe (auth.), Larry Smith (eds.)

By C. Allday, V. Puppe (auth.), Larry Smith (eds.)

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Additional info for Algebraic Topology Göttingen 1984: Proceedings of a Conference held in Göttingen, Nov. 9–15, 1984

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3) ~ is explicitly described by certain natural structure of the functor F. We now obtain the group F4(b4,~ 4) as follows. 4) with $4 : {Z3 } . By use of the composition i: F(H 2) P ) cok b 4 > > ~3 we obtain the push out d i a g r a m p F(H 2) ~ Z 2 @ F(H 2) ~gH 2 2 )> F2H 2 > r~3 ~ ~"2 @ n3'l~H2 p This yields F4(b4,~ 4) as an abelian group. 2). 6) ~3(A;M(H2,2)~ > F3(A) (b4,~ 4) Hom(A,g3M(H2,2)) > Hom(A,F(H2)) Again the extension problem diagram determines F3(A) (b4,~ 4) as an abelian D 5 can be computed, polyhedra for the left hand column see is complete is solved.

The e x t e n s i o n ~/2 > ) Z/4 Let )) 2/2 yields the exact sequence A'2/2 > > A'Z/4 > A'2/2 ~ B y H o m ( A * Z / 2 , A @ ~/2 : E x t ( A * Z / 2 , A ® 2/2) p r e s e n t s the c o n n e c t i n g h o m o m o r p h i s m A ®Z/2 > > A®2/4 >> A ~ 2 / 2 we choose an e x t e n s i o n G(A) w h i c h reabove: A>> A * Z/2 > G(A) For each ~ 6 Hom(A,B) {G(A)} A~/2 (2) there is a h o m o m o r p h i s m ~ such that the following d i a g r a m commutes: A~2/2 > ~ > G(A) A >> A ' Z / 2 (3) B®Z/2 > > G(B) >> B ' Z / 2 31 Moreover, we define G(A) b y the c o m m u t a t i v e d i a g r a m (4) Ext (A,ZZ/2) > ~ ) G(A) II H o m ( A ~Z/2,Z/4) > ~ >> Horn(A,~/2) tf i[ > Hom(G(A),Z/4) ~ Hom(A@ZZ/2,Z/4) Remark: For the Moore space of A in degree G(A) = ~n+2M(A,n) n we have isomorphisms (n ~ 4) , G(A) = ~nM(A,n+l) = [M(A,n+I),S n] C o n s i d e r the d i a g r a m with n ~ 4 is a d i a g r a m of u n b r o k e n arrows as in as in (b); groups, as usual > (a) and 3 (b) below.

9) is not true in dimension 5. J. 10); see (I. § 5). 2 [13] computed the h o m o t o p y group Zn+2 of A n - p o l y h e d r o n for n ~ 3. Our c o m p u t a t i o n of S4 solves this p r o b l e m for n = 2. §3 The classification of maps between simply connected 4-dimensional polyhedra w i t h the n o t a t i o n in section §2 we can state our result on the set of h o m o t o p y classes [X,X'] where X and X' are simply connected 4-dimensional polyhedra. 9). I0) a good c h a r a c t e r i z a t i o n of the subset H,[X,X'] c Hom(H,H').

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