By Georges de Rham
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Extra info for Lectures on introduction to algebraic topology, (Tata Institute of Fundamental Research. Lectures on mathematics and physics. Mathematics, 44)
Example text
Of each of a, the I. sets . L e t j b e a n i n t e g e r w i t h 11 j c n I n o r d e r t o show t h a t ai a n d bi h a v e a p o i n t o f B i , we may a s s u m e t h a t j + i. I f j > i , t h e n a i c o n t a i n s t h e p o i n t a i fl a , 5 Bit , a n d bi i s i n c i d e n t , On t h e . o t h e r h a n d , i f j < i , t h e n bj fl a i l i e s in w i t h bi fl bi EE,' g/ n a i , a n d b i p a s s e s t h r o u g h a i n bi E B_! I . Now, we e m b e d d _B: i n a b l o c k i n g s e t _ B i .
3 . 6 THEOREM. b a ( n , t , q ) < _ b p ( n , t , q ) . PROOF. A s s u m e t o t h e c o n t r a r y t h a t b, > _ b p t l . F i x i n P = P G ( b p t 1 , q ) a h y p e r p l a n e H , a n d d e n o t e by B a n n - f o l d t - b l o c k i n g s e t of H . By o u r a s s u m p t i o n , t h e r e e x i s t s a n n - f o l d t - b l o c k i n g s e t B' i n t h e a f f i n e s p a c e P-H. C o n s e q u e n t l y BuB' would b e a n n - f o l d t - b l o c k i n g set On n-Fold Blocking Sets of 35 P =PG ( bp + 1 , q ) , a c o n t r a d i c t i o n .
An , b, s b z t - . t b n i-1 I n o r d e r t o c h o o s e a i a n d bi , a t m o s t of i -1 4 ( j - 1 ) = 2 ( i - l ) ( i - Z ) t h e a l i n e s d i f f e r e n t from 1 t h r o u a h P; are forbidden. Bv t h e h y p o i h e s i s 2 ( n - l ) ( n - 2 ) 5 q-2, h e n c e f o r e v e r y i s n o n e c a n f i n d two l i n e s a i , bi s a t i s f y i n g t h e a s s u m p t i o n . ,? , Now, we d e f i n e f o r a n y i w i t h 1 B; S t e p 2. a; 2; Namely: 1 . *bi sis n ,. ,a i n b, b , , a i fl b,-l or b i c o n t a i n s a , b i n ai-, , .