Algebraic Topology, Gottingen 1984 by L. Smith

By L. Smith

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Example text

It is possible to show that in any category C, if you can construct arbitrary coproducts and coequalizers, then you can construct all colimits. fn+1 / X2 f2 / · · · fn−1 / Xn fn / / · · · be Xn+1 a telescope diagram, and let X∞ its colimit. If k ≤ l, write fk,l for the unique map in this diagram from Xk to Xl . 35 Let X1 ··· fn−1 f1 / Xn fn,r(n) ··· fr(n−1),r(n)  / Xr(n) fn fr(n),r(n+1) / Xn+1  fn+1 / ··· fn+1,r(n+1) / Xr(n+1) fr(n+1),r(n+2) / ···. Show that X∞ is also the colimit of the bottom row, and that the induced map X∞ → X∞ is the identity map.

And maps jn : Xn → Xn+1 between them. Then we may form the diagram X0 j0 / X1 j1 / ··· jn−1 / Xn jn / Xn+1 jn+1 / ···. , the union of the Xn ); then X, or anything homeomorphic to X, is called a CW complex. The space Xn ⊆ X is called the n-skeleton of X A subcomplex of a CW complex X is a subspace K ⊆ X which is a CW complex constructed by using some, but not necessarily all, of the cells used to construct X. For example, each skeleton Xn ⊆ X is a subcomplex of X. 1 Note that the 0-dimensional disk D0 , which is a single point, is its own interior!

X × {0}) ∪ (X × {1}) ∪ ({∗} × I) (b) If X is a pointed space, then there is a copy of I ⊆ Σ0 X, namely ∗ × I. Show that ΣX ∼ = Σ0 X/I. (c) Suppose X is a CW complex. Describe a CW decomposition of ΣX in terms of the given one for X. We will often use this identification, because it gives us a handy notation for points in ΣX; a typical point can be written as [x, t], the equivalence class of (x, t) ∈ X ×I. 31 For this problem, we use the spheres S n ⊆ Rn+1 centered at ( 21 , 0, . . , 0), radius 1 2 and basepoint at the origin 0.

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