Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.

By Hatcher A.

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Extra resources for Algebraic topology. Errata (web draft, Nov. 2004)

Example text

We e a s i l y g e t from (2) and a l s o Assuming a(X) < w e w i l l t h e n have + m , which i m p l i e s Z a(Xil\+l-Ah) h>j + 0 , as j + m t h i s I together with ( 1) I implies which, t o g e t h e r w i t h ( 4 ) , g i v e s ( 3 ) . d. From t h e . v e r y d e f i n i t i o n of a-measurable is completely a d d i t i v e , t h a t i s d i s j o i n t sequence of a ( U h Mh) sets and ( 1 ) = ChCi(\) , we g e t t h a t if {\I is a a-measurable s e t s . The f o l l o w i n g p r o p e r t y i s q u i t e r e l e v a n t and w e l l known: PROPOSITION 2.

V ~ i ,7 PROOF. c 1 1 From t h e d e f i n i t i o n o f Substituting 6,6, with GiSh+ and c2 and A A = LAhAh. h , we have E ( v ~ ~ v ~) 6 kv, ~ - v . ~ i h k we obtain k ,6ivj6h6h6ivj= = h,i,i 6ivj6h6i6hvj = h,i,j -L 6hvi6hvk6ivj6kvj h,k,i,j I MINIMAL CONES x6hVh f o r which we have used t h e i d e n t i t i e s h EVh6h = 0 h i n p l a c e of (vh6ivk-vi6hvk)~k Writing again 23 = 0 , 6,6, . w e obtain k 6 v . ( v 6 v -v 6 v )6 6 v . , i ] h i k Sivj6ivk6kvs6svj = -c4-) i h k k h j , k , i , I, s 6 6 v =-c f o r which we have used t h e i d e n t i t y EShv,=O .

W e obtain k 6 v . ( v 6 v -v 6 v )6 6 v . , i ] h i k Sivj6ivk6kvs6svj = -c4-) i h k k h j , k , i , I, s 6 6 v =-c f o r which we have used t h e i d e n t i t y EShv,=O . cl(x) for Moreover, f o r a l l i . e ( 6 , 6 , V j ) 2 )- 2 and x w e have vjvk6i6kvs6i6jvs= k,i,j,s I f w e choose for a l l 6n+lc1(x) = 0 : For t h e s e r e a s o n s , a t t h e p o i n t h,i,j w e have CY . v (x) = 0 , , h thus, for i

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