By Alexander Zabrodsky (Eds.)
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Extra info for Hopf Spaces
Assume i n d u c t i v e l y t h a t connected and H n ( v ( n ) , z ) = Hn,,(c(f),Z). men Y 45 Postnikov systems n n ( v ( n ) ) = Hn+l(c(f),z). isomorphism of i s then k(n) Hn. M ( ~ )= Let ,z) ,n) * (C(f) M(n) V(n) -+ * v ( ~ )y i e l d an and Y(n) y(n+l) = c(k(n))* *= If f("): Y ( n ) = H e n ( Y ) then X Moore) approAmation of n. i s a f a i r l y n a t u r a l construction and H t ( f ) i s n Htn While i n dim: Y i s c a l l e d t h e homoZqy (or Y -+ uniquely defined f o r any map f HE n does not s h a r e t h e s e conveniences: There i s no w a y one can choose u n i v e r s a l l y m a p s f: x -+ Y m n ( f ) : mn(X) mn(Y).
X' , p ' Suppose Y. h: X,p -> [X v Y , Y] I-I: X x Y holds. 1 -+ -+ i s an isomorphism F ( h v 1) can and as Y Hence for any map [X x h: X X, Y] + Y + [X v X , Y] fiber h i s an H-space. 4. Definitio :, A map h: X equivalent t o a f i b e r of a map i s commutative where N X -+Y -+ Y g: Y i s a principal f i b r a t i o n i f -+ B. : i s t h e f i b r a t i o n induced by g from Em. is Homotopy p r o p e r t i e s of H-spaces 50 Let F: Y be a commutative diagram and l e t Then fl,fo = flyy f(Y,Y) by g: Y + B , is a pair and LfOY h: X Let F: Y F and Y + PB' f = f PB' fo fl be a homotopy ,F ' fog - B'fl.
X OX On x R X = (RX x x Add c o r e l Ei LX) The s t r u c t u r e of ( R X , Add) w1 = 1, on L X LX n (LX [ , H-space 1. A s observed i n 1 . 1 . 2 ( a ) , [M,X] hence Add- AddT, RX - F$ LX Ppa2 = Addw2. and and on L X x RX i = 0,m. c o r e l Eiy x i = 0,- 1, 1 x Ei x x RX) if Add X,p RX x - & is HC. w 2 = T. AddT. i s an H-space f o r any space has a m u l t i p l i c a t i o n w i t h u n i t p*: [M,X] x [M,X] = [M,X I t can be e a s i l y seen t h a t i f (commutative). X,p is x HA XI -+ [M,X].