By Strom J.
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Extra resources for Modern homotopy theories
Example text
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. 46 2. 35 to other diagram shapes. 5 Formal Properties of Pushout and Pullback Squares We conclude with some formal properties of pushout and pullback squares. These rules, and their homotopy-theoretical analogs, will be crucial throughout our study of homotopy theory.
20 Prove Corollary 27 by showing that map((X, A), (Y, B)) / map(X, Y ) map(A, B) / map(A, Y ) is a pullback square. 60 3. 7 Thus, we consider space of maps map((X, A), (Y, B)) as the pair map((X, A), (Y, B)), map(X, B) . big space subspace The product of two pairs (X, A) and (Y, B) in T is also a pair in T : (X, A) × (Y, B) = X × Y , A × Y ∪ X × B . big space subspace With these preliminaries, you can generalize some of the results of Theorem 25 to maps of pairs. 21 (a) Show that the exponential law holds for maps of pairs.
Products of CW Complexes. If the open cells Ci (X) of X are indexed by I and the open cells Cj (Y ) of Y are indexed by J , we have for each i ∈ I and j ∈ J the composition Ci (X) × Cj (Y ) χi ×χj / X × Y. 56 3. 11 (a) Suppose Ci (X) ∼ = int(Dn ) and Cj (Y ) ∼ = int(Dm ). Show that Ci (X) × n+m Cj (Y ) ∼ int(D ). = (b) Show that the inclusion Ci (X) × Cj (X) → X × Y is continuous. (c) Show that X × Y is the union of the product cells Ci (X) × Cj (Y ). This problem suggests that the set equation X ×Y = Ci (X) × Cj (Y ) I×J might actually a representation of X × Y as a CW complex.