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

**Read or Download Algebraic Topology Göttingen 1984: Proceedings of a Conference held in Göttingen, Nov. 9–15, 1984 PDF**

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

**Example text**

3). of F 4 for which we thus have the natural sequence t > ) F4 )) FT(H2) V i a the s e c o n d a r y b o u n d a r y tb5: ~ r4 H5 The f o l l o w i n g b 5 in . 9) H3 (X'~Z/2) Here £ integral A is the surjection Steenrod and by the trivial square. >> H 2 *Z/2 in the u n i v e r s a l coefficient The map ~ is given on r(C) t h e o r e m and Sq2 is the @ C ~ D map on C ® D, we use H 2 * ~ / 2 c C ~ ~ / 2 . 1) w h i c h start with b 5. C. W h i t e h e a d [23]. 10) for i > 5. 0). 1) which are realizable by a l-connected 5-dimensional polyhedron.

Are well known are available We w a n t the o t h e r h a n d differentiable Since our results we do not r e s t r i c t the s e t of a l l m a p s homomorphism in t e r m s On triangulable. for all S B F - s p a c e s to c o m p u t e homology below to b e between manifolds in t h i s section to m a n i f o l d s . SBF-spaces which induce (~. 9) let } V* be the d u a l of ~. G(~) (V,U) T h e n we s e t (F(v*) ~ m/2 the s u b g r o u p • F(V*) ~ v*)/U U is g e n e r a t e d b y the following elements (where i C ~/2 b v C F (v*) and b W is a g e n e r a t o r ) : (i) [x,y] (ii) yx ~ x (iii) [x,y] ~ (iv) bv ® (v) bv ® y (vi) (G ® ipz/2)ob W + where C x,y,z ~(W*) For each C V* are = G(~) I + [z,x] ® y + (yx) ~ y + [y,z] ® x [y,x] ~ x i (C~ ®

4). 4) equivalence on D N of p r o b l e m (**). We do this w o u l d be a never ending task. 4) we use the following results. We show that for a simply c o n n e c t e d complex X the P-groups Fn(X) termined by the h o m o l o g y H,(X) (b4X,~4X .... ,bnX,~nX) and Fn_ [(A;X) . Here we use inductively the is i n t r o d u c e d in [3], in §2 are e s s e n t i a l l y de- and by the sequences of b o u n d a r y invariants [4]. 3) are r e a l i z a b l e by an a p p r o p r i a t e X. This, together with conditions on the r e a l i z a b i l i t y of h o m o l o g y homomorphisms, yields the result.