By Francesca Aicardi (auth.), V. I. Arnold, I. M. Gelfand, V. S. Retakh, M. Smirnov (eds.)
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Sample text
1a cannot be transformed into the other without crossing of the discriminant of the parallel coorientations self-tangencies K+ if the cusps births are forbidden. They are indeed distinguished only by the values of the invariants f+ and f-. Different paths connecting the two curves in fh,o are shown in fig. 25. Discriminants and local invariants of planar fronts 47 Figure 25. Examples of the F+ and F- discriminants. The values of f+ and f- are shown in the connected components of the complement In table 8 a list of oriented and cooriented fronts in the oriented plane with the values of their 10 local additive invariants is presented.
F3, 'Y run on the set of signs (+, - ), and TJ runs Discriminants and local invariants of planar fronts 41 Every reduced elementary jump defines a l-cochain whose value at the I-chain positively crossing transversally the corresponding stratum of the discriminant is equal to 2, being zero if the crossing takes place at any other stratum. 2. 1. L11TC and L11TD are the elementary RO-independent jumps defined at the strata of the cusp crossings. L11T~+ is the jump at the cusp crossing strata which is nonzero only if the cusp is of type up and the branch going away from the cusp point is positive.
2. -I -3 -2 -3 0 -1 -6 -7- -:? 1.. (2) 0 @ @ @) @I(QI~ -~ -1. 0 2. 2. r' c- 4 0 :t 0 4 0 o 0 -1 1" 1" 0 0 0 0 0 0 -2, 0 1 4- 0 :l. 10 1 -2 ·1 -2 -4- -2 -1. -J. =1 JI'='- ~ Q (B) @ @ 8J ®I® (ill S)--4 -1 L Q 6 ~ w0 0 0 0 0 0 -1 CO 1 o (5 @ -2 -; -2 -3 -4- -1 -J. 22. b § fl ~ -3, ® 0 0 0 0 ; -20 tg ~ 0 o 0 I~: :L c¢xJ 0 0 1 3 1 1 0 :, 1 1-4 4 0 oj 1 ; -t -8 14 -4 0 J 3 -i -I 8 ®~® -8 -11 \l -lG. 2.. 2. 3 -1 0 0 '5 1 1 ~ 50 F. 2 The RO-dependent l-cocyc1es We solve the coherence system at the strata of co dimension 2 for the RO-dependent local strata in two steps.