# A New Proof of the Lefschetz Formula on Invariant Points by Hopf H. By Hopf H.

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P/~ commutes (~ d e n o t i n g the c a n o n i c a l If g : P ÷ P is a n o t h e r map, surjection). we have f o g = ~ 0 g and map 60 = ip/~. We will so is ~ a n d ~ I = ~-I write ~(~I,~2,k) mappings (2330) If f is b i j e c t i v e , = T(~1,~2,k) obtained in this Theorem: Proof: By : R ÷ R and that A < B implies immediately Finally we are to T is a s u b g r o u p from lift set of t r a n s p o r t of A u t ( R , < ) . TA ~ ~B for all follows by T the way. T is a s u b g r o u p (2329), denote of Bij(R).

Of k, (For t e c h n i c a l If we m a k e the p o s i t i o n of k constant. mentioned in m o s t bodies identical ambiguity frame ~ E F is k e p t fixed and w i l l not be e x p l i c i t l y cases. with the same since the sub-bodies however, (2211) in the a sub-body alterations the If, lie ordering to be some configurations and ~2 c N ~2 # ~)" a partial be E - d i s j o i n t in s e q u e l Two 71 is t h o u g h t Two at r e s t r-is = 1,2. <=> k I E k 2 and def equivalently, Clearly, Thus for i set of m i n i m a l can be the sub-bodies are should disappear.

3. 4. a, b < d ~ c(~) A this is follows: 6 ~ 6 (aUb)/co c 6 a v b. b < (avb) (avb). < d. 6 a v b, ~ 6 of a finite of (aUb)/Go U b such Assume a U b and definition (avb). Assume A consists (2239). 2. on any We define {c(~) I e 6 ( a U b ) / o g } . to disjoint V A equivalence (aUb)/co . 1. the (aUb)/co . e. 1. 2. : of H i + 1 6 b. Hi, D. 2. Now C I, C 2, we have A A B proves and B for 6 d and F, F = H I, Hi+ I cannot instance: such that regions G ~ in d a r e D. < d. spatial by assume regions of (2234) (iii).

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