Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 527 - Homotopy Theory Spring 2013 Homework 1 Solutions Problem 1. Show that the following conditions on a topological space X are equivalent. 1. X is contractible. 2. The identity map idX : X → X is null-homotopic. 3. For any space Y , every continuous map f : X → Y is null-homotopic. 4. For any space W , every continuous map f : W → X is null-homotopic. 5. For any space W , the projection map pW : W × X → W is a homotopy equivalence. Solution. Throughout this problem, let cz denote any constant map with value z (with appropriate source and target). (1 ⇔ 2) Consider the following equivalent statements. X is contractible. ⇔ The unique map f : X → ∗ is a homotopy equivalence, i.e. has a homotopy inverse g : ∗ → X. ⇔ There is a point g(∗) = x0 ∈ X such that the constant map cx0 = g ◦f : X → X is homotopic to the identity map idX . ⇔ The map idX is null-homotopic. (3 ⇒ 2) Particular case Y = X and f = idX . (2 ⇒ 3) Assume idX ' cx0 for some point x0 ∈ X. Writing f as the composite f = f ◦ idX , we obtain f = f ◦ idX ' f ◦ cx0 = cf (x0 ) so that f is null-homotopic. (4 ⇒ 2) Particular case W = X and f = idX . (2 ⇒ 4) Assume idX ' cx0 for some point x0 ∈ X. Writing f as the composite f = idX ◦ f , we obtain f = idX ◦ f ' cx0 ◦ f = cx0 so that f is null-homotopic. (5 ⇒ 1) Particular case W = ∗. (1 ⇒ 5) Assume idX ' cx0 for some point x0 ∈ X. Then the map (idW , cx0 ) : W → W × X is homotopy inverse to the projection pW : W × X → W . Indeed, one composite is already the identity pW ◦ (idW , cx0 ) = idW while the other composite satisfies (idW , cx0 ) ◦ pW = idW × cx0 ' idW × idX = idW ×X . 1 Problem 2. Let C be a locally small category with finite products, including a terminal object. Let G be a group object in C. Show that for any object X of C, the hom-set HomC (X, G) is naturally a group. In other words, the structure maps of G induce a group structure on HomC (X, G), and this assignment HomC (−, G) : C op → Gp is a functor. Solution. Since the functor HomC (X, −) : C → Set preserves limits (in particular finite products), the set HomC (X, G) inherits a group object structure in Set from G, so that HomC (X, G) is a group. Explicitly, its multiplication map is µ∗ HomC (X, G) × HomC (X, G) ∼ = HomC (X, G × G) −→ HomC (X, G) and likewise for the unit and inverse structure maps. To prove naturality, it suffices to prove that for any morphism f : X → Y in C, the induced map of sets f ∗ : HomC (Y, G) → HomC (X, G) is a group homomorphism. This follows from commutativity of the diagram HomC (X, G) × HomC (X, G) o ∼ = µ∗ O O f ∗ ×f ∗ HomC (Y, G) × HomC (Y, G) o / HomC (X, G × G) O f∗ ∼ = f∗ µ∗ HomC (Y, G × G) 2 HomC (X, G) / HomC (X, G). Problem 3. Consider S 1 as the unit circle in R2 with basepoint (1, 0), and consider the “pinch” map p : S 1 → S 1 /S 0 ∼ = S1 ∨ S1 which collapses the equator S 0 ⊂ S 1 , i.e. identifies the points (1, 0) and (−1, 0). a. Show that the pinch map is (pointed) homotopy coassociative. More precisely, the diagram p / S1 p S1 ∨ S1 S1 ∨ S1 / p∨id S1 ∨ S1 ∨ S1 id∨p commutes up to pointed homotopy. Solution. Denote the two summand inclusions by ιi : S 1 → S 1 ∨ S 1 for i = 1, 2. Using the model S 1 ∼ = I/∂I, the pinch map can be explicitly written as ( ι1 (2s) if 0 ≤ s ≤ 12 p(s) = ι2 (2s − 1) if 12 ≤ s ≤ 1. The composite via the top right is if 0 ≤ s ≤ 14 ι1 (4s) (p ∨ id) ◦ p(s) = ι2 (4s − 1) if 41 ≤ s ≤ 12 ι3 (2s − 1) if 21 ≤ s ≤ 1 whereas composite via the bottom left is if 0 ≤ s ≤ 12 ι1 (2s) (id ∨ p) ◦ p(s) = ι2 (4s − 2) if 12 ≤ s ≤ 34 ι3 (4s − 3) if 34 ≤ s ≤ 1. The formula 4s if 0 ≤ s ≤ 1+t ι1 ( 1+t ) 4 2+t H(s, t) = ι2 (4s − 1 − t) if 1+t ≤ s ≤ 4 4 4s−2−t ι3 ( 2−t ) if 2+t ≤ s ≤ 1 4 defines a pointed homotopy H : S 1 × I → S 1 ∨ S 1 ∨ S 1 from (p ∨ id) ◦ p to (id ∨ p) ◦ p. In fact, a very similar argument shows that S 1 is a homotopy cogroup object in Top∗ . (Do not show this.) Comultiplication is the pinch map S 1 → S 1 ∨ S 1 , the counit is the constant map S 1 → ∗, and the coinverse S 1 → S 1 reverses the last component (viewed in R2 ). 3 b. Conclude that for any pointed space (X, x0 ), the set π1 (X, x0 ) is naturally a group. More precisely, the structure maps of S 1 as homotopy cogroup object induce a group structure on π1 (X, x0 ), and moreover this assignment defines a (covariant) functor π1 : Top∗ → Gp. Solution. Since S 1 is a homotopy cogroup object in Top∗ , it becomes a cogroup object in the homotopy category Ho(Top∗ ), and therefore a group object in the opposite category Ho(Top∗ )op . We have π1 (X, x0 ) = [S 1 , (X, x0 )]∗ = HomHo(Top∗ ) S 1 , (X, x0 ) = HomHo(Top∗ )op (X, x0 ), S 1 which is naturally a group, by Problem 2. 4 Problem 4. For pointed spaces, show that the smash product distributes over the wedge. More precisely, there is a natural isomorphism X ∧ (Y ∨ Z) ∼ = (X ∧ Y ) ∨ (X ∧ Z). of pointed spaces. Don’t forget to argue that the isomorphism is natural. Solution. Recall that the wedge is the coproduct in Top∗ . Applying the functor X ∧ − to the natural summand inclusions Y → Y ∨ Z and Z → Y ∨ Z produces maps X ∧ Y → X ∧ (Y ∨ Z) X ∧ Z → X ∧ (Y ∨ Z) which together define a natural map ϕ : (X ∧ Y ) ∨ (X ∧ Z) → X ∧ (Y ∨ Z) of pointed spaces. It remains to show that ϕ is an homeomorphism, thus an isomorphism in Top∗ . To construct the inverse of ϕ, consider the map ψ in the commutative diagram e ψ X ∧ (Y ∨ Z) / (X ∧ Y ) ∨ (X ∧ Z) O < O O ψ X × (Y ∨ Z) (X ∧ Y ) q (X ∧ Z) O O O X × (Y q Z) o ∼ = (X × Y ) q (X q Z) in Top. The bottom is an isomorphism because the functor X × − : Top → Top preserves (arbitrary) coproducts. Now the map X × (Y q Z) → X × (Y ∨ Z) is a quotient map, hence so is the composite X × (Y q Z) → X × (Y ∨ Z) → X ∧ (Y ∨ Z). on the left-hand side of the diagram. Since ψ is constant on the equivalence classes defined by this quotient map, it induces a unique continuous map ψe : X ∧ (Y ∨ Z) → (X ∧ Y ) ∨ (X ∧ Z) making the diagram commute. By construction, ψe is clearly the set-theoretic inverse of ϕ. 5