By Hatcher A.

**Read Online or Download Algebraic topology. Errata (web draft, Nov. 2004) PDF**

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Let f : M → G be a monoid homomorphism from a monoid M to a group G. Write mg = f (m) · g. Let Q = B(∗, M, G) be the category with objects g ∈ G and morphisms (m, g) ∈ M × G from mg to g (m,g) mg −−−−→ g . Then there is a fiber sequence up to homotopy Bf |Q| −−−−→ B M −−−−→ BG . 3. 4 to the monoids Mn = G L n (B) and groups G n = G L n (A) we obtain categories Q n for each natural number n. There are stabilization maps i : Q n → Q n+1 , Mn → Mn+1 and G n → G n+1 , with (homotopy) colimits Q ∞ , M∞ and G ∞ , and a quasifibration |Q ∞ | −−−−→ BG L ∞ (B) −−−−→ BG L ∞ (A) .

With d ∈ N interpreted as the complex vector space Cd , and morphism spaces V(d, e) = U (d) if d = e, ∅ otherwise from d to e. The sum functor ⊕ takes (d, e) to d + e and embeds U (d) × U (e) into U (d + e) by the block sum of matrices. The tensor functor ⊗ takes (d, e) to de and maps U (d) × U (e) to U (de) by means of the left lexicographic ordering, which identifies {1, . . , d}×{1, . . , e} with {1, . . , de}. Both of these functors are continuous. The zero and unit objects are 0 and 1, respectively.

Let A be a connective S-algebra. The commutative square K (A) G K (A∧p ) K (π0 (A)) G K (π0 (A∧p )) becomes homotopy Cartesian after p-completion. We apply this with A = ku or A = K (C). Also in the second case ku ∧p , by Suslin’s theorem on the algebraic K -theory of algebraically A∧p closed fields [30]. Then π0 (A) = Z and π0 (A∧p ) = Z p . It is known that K (Z p ) has telescopic complexity 1, by B¨okstedt–Madsen [6], [7] for p odd and by the third author [26] for p = 2. It is also known that K (Z) has telescopic complexity 1 for p = 2, by Voevodsky’s proof of the Milnor conjecture and Rognes–Weibel [25].