Algebraic Topology (topics course) Spring 2010 John E. Harper. Series 6

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1 Algebraic Toology (toics course) Sring 2010 John E. Harer Series 6 Let R be a ring and denote by Ch + R (res. Mod R) the category of non-negative chain comlexes over R (res. the category of left R-modules). Define a ma f : MN in Ch + R to be (i) a weak equivalence if it is a homology isomorhism, (ii) a fibration if the ma : M k N k is an eimorhism for each k 1, (iii) a cofibration if it has the left lifting roerty with resect to all acyclic fibrations. The urose of this series is to give a roof of the following roosition. Proosition 1. These three classes of mas give Ch + R the structure of a model category. Exercise 1. Prove Proosition 2. Proosition 2. The left-hand solid commutative diagram A i f g X B h A k i k g k h k X k k k 1 1 X k 1 X k (k 0) in Ch + R has a lift if and only if the right-hand sequence of lifting roblems in Mod R has a solution, if and only if the sequence of lifting roblems A k g k X k i k Cy k 1 (X) Cyk 1 ( ) k (k 0) in Mod R has a solution. Exercise 2. Prove Proosition 3. Proosition 3. Let : X be a ma in Ch + R. (a) The ma is an acyclic fibration if and only if the induced ma X k Cy k 1 (X) Cyk 1 ( ) k is an eimorhism for each k 0. (b) If is an acyclic fibration, then the induced ma ( k ) : Cy k (X)Cy k ( ) is an eimorhism for each k 0. (c) If the induced ma X k Cy k 1 (X) Cyk 1 ( ) k is an eimorhism for each k 0, then the induced ma ( k ) : Cy k (X)Cy k ( ) is an eimorhism for each k 0. Exercise 3. Prove Proosition 4. 1

2 2 Proosition 4. Let i: AB be a ma in Ch + R. If the ma i k : A k is a monomorhism with coker(i k ) a rojective R-module for each k 0, then i is a cofibration. Definition 5. Let A be a left R-module and n 1. The chain comlex (A) in Ch + R has the form Dn(A) : 0 0 A id A 0 0 and is defined degreewise by (A) k := { A, for k = n, n 1, 0, otherwise. The n-disk chain comlex in Ch + R is defined by Dn := (R). Note that the ma 0 is a weak equivalence for each n 1; i.e., the n-disk chain comlex is acyclic. Exercise 4. Prove Proosition 6. Proosition 6. Let n 1. There is an adjunction Mod R Ch + R Ev n with left adjoint on to and Ev n the evaluation functor defined objectwise by Ev n (B) := B n ; i.e., there are isomorhisms natural in A, B. Exercise 5. Prove Proosition 7. hom Ch + ( (A), B) = hom ModR (A, Ev n (B)) R Proosition 7. Let n 1. A solid commutative diagram of the form 0 X in Ch + R is equivalent to an element y n. A lift in such a solid commutative diagram is equivalent to an element x X n such that n x = y. Exercise 6. Prove Proosition 8. Proosition 8. A ma : X in Ch + R is a fibration if and only if it has the right lifting roerty with resect to the set of mas j n : 0, n 1. Definition 9. Let A be a left R-module and n 0. The chain comlex S n (A) in Ch + R has the form S n (A) : 0 0 A 0 0 and is defined degreewise by S n (A) k := { A, for k = n, 0, otherwise.

3 3 The n-shere chain comlex S n in Ch + R is defined by Sn := S n (R). For notational convenience, define the chain comlexes S 1 := 0, D 0 := S 0 (R), and denote by j n : S n 1 the natural inclusion ma in Ch + R. Exercise 7. Prove Proosition 10. Proosition 10. Let n 0. There is an adjunction S n Mod R Ch + R Cy n with left adjoint on to and Cy n the n-dimensional cycles functor defined objectwise by Cy n (B) := ker( : B n B n 1 ); i.e., there are isomorhisms natural in A, B. Exercise 8. Prove Proosition 11. hom Ch + (S n (A), B) = hom ModR (A, Cy R n (B)) Proosition 11. Let n 1. A solid commutative diagram of the form S n 1 j n in Ch + R is equivalent to an element (y, z) n Cy n 1 (X) such that n 1 z = y. A lift in such a solid commutative diagram is equivalent to an element x X n such that n x = y and x = z. Exercise 9. Prove Proosition 12. Proosition 12. A ma : X in Ch + R is an acyclic fibration if and only if it has the right lifting roerty with resect to the set of mas X j n : S n 1, n 0. Recall the following roosition which is a secial case of the roerty that homology commutes with filtered colimits. Proosition 13. Let n 0 and consider any diagram of the form G 0 i 1 G 1 i 2 G 2 i 3 G 3 in Ch + R. Then the natural ma colim k H n (G k = ) H n (colim k G k ) in Mod R is an isomorhism. Exercise 10. Use Proosition 8 together with a small object argument to rove Proosition 14. Proosition 14. Let : X be a ma in Ch + R. Then has a factorization X j X q in Ch + R as an acyclic cofibration j followed by a fibration q; i.e., MC5(ii) is satisfied. Exercise 11. Prove Proosition 15 using the factorizations constructed in the roof of Proosition 14.

4 4 Proosition 15. Consider any solid commutative diagram of the form A i g X B h in Ch + R. If i is an acyclic cofibration and is a fibration, then the diagram has a lift; i.e., MC4(ii) is satisfied. Exercise 12. Use Proosition 12 together with a small object argument to rove Proosition 16. Proosition 16. Let : X be a ma in Ch + R. Then has a factorization X j q in Ch + R as a cofibration j followed by an acyclic fibration q; i.e., MC5(i) is satisfied. Exercise 13. Prove Proosition 17. Proosition 17. Every identity ma in Ch + R is a fibration, cofibration, and weak equivalence. Exercise 14. Prove Proosition 18. Proosition 18. The three classes of mas in Ch + R weak equivalences, fibrations, and cofibrations are each closed under comosition. Exercise 15. Prove Proosition 19. Proosition 19. The category Ch + R has all small limits and colimits, and they are calculated degreewise. In articular, MC1 is satisfied. Exercise 16. Prove Proosition 20. Proosition 20. The class of weak equivalences in Ch + R satisfies the two out of three axiom MC2. Exercise 17. Prove Proosition 21. Proosition 21. The three classes of mas in Ch + R weak equivalences, fibrations, and cofibrations are each closed under retracts; i.e., MC3 is satisfied. Exercise 18. Prove Proosition 1. Exercise 19. Use the factorizations constructed in the roof of Proosition 16 together with Proosition 4 to rove Proosition 22. Proosition 22. (a) A ma i: AB in Ch + R is a cofibration if and only if the ma i k : A k is a monomorhism with coker(i k ) a rojective R-module for each k 0. (b) A chain comlex B Ch + R is cofibrant if and only if is a rojective R-module for each k 0. (c) Every chain comlex B Ch + R is fibrant. Exercise 20. Please read [1, Sections 7-8] and [2, ]; see also [3, 2.3].

5 5 References [1] W. G. Dwyer and J. Saliński. Homotoy theories and model categories. In Handbook of algebraic toology, ages North-Holland, Amsterdam, [2] P. G. Goerss and K. Schemmerhorn. Model categories and simlicial methods. In Interactions between homotoy theory and algebra, volume 436 of Contem. Math., ages Amer. Math. Soc., Providence, RI, [3] M. Hovey. Model categories, volume 63 of Mathematical Surveys and Monograhs. American Mathematical Society, Providence, RI, 1999.