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

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

ISBN-10: 3540160612

ISBN-13: 9783540160618

ISBN-10: 3540397450

ISBN-13: 9783540397458

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

Example text

3) simply groups abelian and finitely generated, otherwise U: H 2 x H 2 intersection > H4 : ~ form. 2). i) : b4(1) Now is h i g h l y , : free Hi(x,~) sequence spaces forms. 6) SBF-space X, that is (H*X,U) ~ (V, U). Moreoper, for SBF-spacen X,Y each map ~: (H*Y,U) ----~(H*X,U) F: X We d e r i v e in ,gI~F'is r e a ~ z a b l e by a ma N ~ Y ~aith H*I' = ~. from this I-i c o r r e s p o n d e d result that the to the h o m o t o p y equivalence types classes of S B F - s p a c e s . 7) if X is an S B F - s p a c e the h o m o t o p y Remark: closed type which of M(V,U) realizes topological manifold result symmetric bilinear form (V,U) .

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) !

2) induces the connecting homomorphism ~ in the following diagram: Hom(H3,FTH ~) << 2 , A Ext(g3,1"2i 3) > H 3 (X, FTH~ ) > H4 2 , (X, F2i 3 ) ~] ,2. 3) 42 Here ~ is the Bockstein h o m o m o r p h i s m versal coefficient have im(A~) theorem. 2 (X,Tr~). 6). 6). 2) Thus we see that in the definition = im ~. Next we define d(~2) d(<02)(~) = ( ( I Q ) ) . 4). 8), ~ 6 Hom(H2,Z~) . Here the compositions b4 H4 . 6) diagram [x',x"] × [x,x'] o > Ix,x"] o Mi(X',X" ) x Mi(X,X' ) .... ~ Mi(X,X" ) Therefore the sets Mi(X,X') are quotient functors.

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


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