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

Example text

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. 4) above is an a b b r e v i a t i o n of our result in [3], we d e s c r i b e many p r o p e r t i e s of a Remark: [4] since F-system a s s o c i a t e d to H. 5) the isomorphism ~: F b n_i(Hn) ~ H o m ( H n , F n _ i) Thus @n+l is d e t e r m i n e d by P@n+l- = bn in this case.

M o r e o v e r , the g r o u p s groups Tn+2(X [Xi,Xj] ), ~ (X) were also obtained by Hilton n+l j were computed by Brown-Copeland [5] in c a s e X i a n d X. are M o o r e spaces, t h a t is i,j 6 { 4 , 5 , 9 , 1 0 } . T h e m o r e d i f f i c u l t p a r t s of the 3 t a b l e s e e m to b e n e w results. W e p o i n t o u t that we h a v e S p a n i e r W h i t e h e a d d u a l i t y [Xi,Xj] ~ [DXj,DXi] where DD = identity DX I - and where X3 DX 2 - X 2 DX4(r ) - X8(r ) DX 5(p,r) DX 6 : mx7(t) - X9(P,r) X6 : X10(t) DXll(r,t) : Xl1(t,r) In the f o l l o w i n g t a b l e we use the n o t a t i o n : g := (p,p') j :: m a x ( t , r ' ) i := m i n ( r , r ' ) i':: m i n ( r + 1 , r ' ) m :: m i n ( r , t ' ) m':: min(r+l,t') n :: m i n ( t , t ' ) k :: min(t,r') m" := m i n ( r , t ' + l ) n":: min(t,t'+l) !

T=: {s C T I s < t}, the co-boundary since ordered j (sl,s2)" sets we have ITII * IT21 = IT 1 * T21 denoting (t,i) ordered with ITII ~ X 1 T two partially and I~I = IT1 a and T1 (ti,s i E Ti; i = 1,2). e. (t) for all < ~ Flags: To If T t E T. e. if F = t < t I < o-o < t. < "° , then F(i) = t. eo if there exist F' = (F N F') ~ { t ' } , t,t' C T I s' < t'} and {s C F I s > t} = {s' C F' I s' > t'}o If to be k - w a l l - n e i g h b o u r s we denote this by with t # t' and F = in w h i c h case we have n e c e s s a r i l y I s < t} = {s' C F' F' and starting with {s e F and if T is locally finite-dimensional, we define t = F(k) or - equivalently - t' = F'(k) Pure and locally pure tessellations: We define a t e s s e l l a t i o n and if all flags in F(T) T to be pure if have the same cardinality.

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