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This can be the court cases of a global workshop on knot thought held in July 1996 at Waseda collage convention Centre. It was once geared up via the foreign study Institute of Mathematical Society of Japan. The workshop used to be attended via approximately a hundred and eighty mathematicians from Japan and 14 different nations.

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But lim(. . ← −Z← −Z← − Z) = ← p=2 If we consider mixing these operations in groups, the following remarks are appropriate. i) localizing and then profinitely completing is simple and often gives zero,   Gp if ∩ = ∅ (G ) = p∈ ∩ 0 if ∩ = ∅ . g. (G, l, ) = (Z , ∅, p) gives Qp = 0. g. ¯ p = Q ⊗ Zp , the “field of p-adic numbers”, usually a) (Z0 )¯p = Q denoted by Qp . Qp is the field of quotients of Zp (although it 27 Algebraic Constructions is not much larger because only 1/p has to be added to Zp to make it a field).

Is just the localization S1 → S1 studied above. ii) π = Z/pn for π = 0 if p ∈ / π = Z/pn if p ∈ . For general π, take finite direct sums and then direct limits of the first two cases. Step 2. The case (X − →X) = K(π, n) → − K(π , n) . If localizes homology, then it localizes homotopy as in Step 1 because π = Hn X, π = Hn X . If localizes homotopy, then we use induction, Step 1, diagram III in remark b) and remark c) to see that localizes homology. 48 Step 3. The general case X − →X. If localizes homology, apply the Hurewicz theorem for n = 1 to see that localizes π1 .

Proof: We have to check exactness for ( )⊕(0) i−j 0 → Z(p) −−−−→ Zp ⊕ Q −−→ Qp → 0 where i and j are the natural inclusions Zp −−→ Qp , Q −−→ Qp . ⊗Q ⊗Zp Take n ∈ Z and q ∈ Zp then (n/pa ) + q = (n + pa q)/pa can be an arbitrary p-adic number. Thus i − j is onto. It is clear that ( ) ⊕ (0) has zero kernel. To complete the proof only note that a rational number n/m is also a p-adic integer when m is not divisible by p. Thus n/m is in Z localized at p. Corollary The ring of integers localized at p is the fibre product of the rational numbers and the ring of p-adic integers over the p-adic numbers.