By Liang-shin Hahn
The aim of this booklet is to illustrate that advanced numbers and geometry will be combined jointly fantastically. This ends up in effortless proofs and traditional generalizations of many theorems in airplane geometry, equivalent to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem. The booklet is self-contained - no history in advanced numbers is believed - and will be coated at a leisurely velocity in a one-semester direction. the various chapters could be learn independently. Over a hundred workouts are incorporated. The ebook will be compatible as a textual content for a geometry direction, or for an issue fixing seminar, or as enrichment for the coed who desires to recognize extra.
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Extra info for Complex numbers and geometry
Therefore the integer j∗ (α) can be any multiple of 2. August 26, 2009 16:21 9in x 6in 32 b789-ch02 M. Kervaire and J. Milnor Let us study the eﬀect of replacing ε by εα = ε+j(α)ε on the homology of the modiﬁed manifold. 6, where i carries ε into an element λ of order l > 1. Evidently lε must be a multiple of ε , say: lε + l ε = 0. Since ε is not a torsion element, these two elements can satisfy no other relation. Since εα = ε + j∗ (α)ε it follows that lεα + (l − lj(α))ε = 0. Now using the sequence ε i α α Hk M 0 → Hk Mα → 0, Z→ we see that the inclusion homomorphism iα carries ε into an element λα ∈ Hk Mα of order |l − lj(α)|.
Adams , . Proof. Let Σ be a homotopy n-sphere. Then the only obstruction to the triviality of τ ⊕ ε1 is a well-deﬁned cohomology class on (Σ) ∈ H n (Σ; πn−1 (SOn+1 )) = πn−1 (SOn+1 ). The coeﬃcient group may be identiﬁed with the stable group πn−1 (SO). But these stable groups have been computed by Bott , as follows, for n ≥ 2: The mod 8 residue class: 0 πn−1 (SO) 1 2 3 4 5 6 7 Z Z2 Z2 0 Z 0 0 0. ) Case 1. n ≡ 3, 5, 6 or 7 (mod 8). Then πn−1 (SO) = 0, so that on (Σ) is trivially zero. Case 2.
Definition. The linking number L(λ, µ) is the rational number modulo 1 deﬁned by L(λ, µ) = ν · µ. This linking number is well deﬁned, and satisﬁes the symmetry relation L(µ, λ) + (−1)pq L(λ, µ) = 0 (compare Seifert and Threlfall ). 4. The ration l /l modulo 1 is, up to sign, equal to the selflinking number L(λ, λ). Proof. Since lε + l ε = 0 in Hk M0 , we see that the cycle lε + l ε on bM0 bounds a chain c on M0 . Let c1 = ϕ(x0 ×Dk+1 ) denote the cycle in ϕ(S k ×Dk+1 ) ⊂ M with boundary ε . Then the chain c − l c1 , has boundary lε; hence (c − l c1 )/l has boundary ε, representing the homology class λ in Hk M .
Complex numbers and geometry by Liang-shin Hahn