Continuous Lattices: Proceedings by B. Banaschewski, R.-E. Hoffmann

By B. Banaschewski, R.-E. Hoffmann

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Example text

6. SSIONS OF CERTAIN REPRESENTATIVE &CYCLES IN We will make the same assumptions and use the same notations about K as in the preceding section. For any c * r E K", let ( s * T > denote the cochain on integral coefficients of K" which takes the value 1 on the cell 6 * T and the value 0 on all other cells of K". The purpose of this section is to prove the following of Theorem 10. The (2m - 1)-dimensional imbedding class K has a representative cocycle vZm-'= PZ C {(ai,, aim-l>* (ajo aim>>9 Q2'"--?

T h e subcomplexes of K , L determined by those s;,T, for which aj, b, are # O will be denoted by 1x1 and l y l respectively. , a n y simplex c of 1x1 and any simplex T of I y I are in general position, then with respect to the oriented R"', the chains x, y have an intersection number ([2] Chap. 11) 0 ( x , y > = 2 ai b j O ( ~ i 9 TI), which is bilinear and possesses the following three properties (dim x=r, d im y=s and all intersection numbers are supposed to be defined): 0 ( x , y ) = ( -I)rJ O(y, x ) 0 ( x , a y ) = (-1 +s=m , r + s = m + 1, r , >' 0(c3x, y ) , (1) (2) and finally, change the orientation of R" and denote the intersection number w i t h respect to this otherwise oriented R" by @', then 0(x, y) = - O'(x, y ) , r +s = m .

R;. (3) (4) By 5", 7" and the construction we see easily Q),,,(hri,Toai)= 41. By conveniently choosing the orthogonal transformation T of R" to be orientation-preserving or orientation-reversing, we may always make'' ( c = f I as i n ( I ) ) 0 ( A r i , Tog;) = (5) (-l)di+'~. + Let rk C Sto T i , rI C KO- Sto ri-Sto T i , where dim r4 dim T, = m - 1. Then since dim r4 >, dim pi > 0 we have di m r , < m - 2 and T~ E I KOSt, Ti - St, ri I m-2. Hence, by 6", O ( h r k ,T or,) = o . ,= q l . I (a/, * 0 , ) -- Vo(6, * Case I.

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