MATH CLASS 27. Contents

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1 MATH CLASS 27 Contents 1. Reduced and relatve homology and cohomology 1 2. Elenberg-Steenrod Axoms Axoms for unreduced homology Axoms for reduced homology Axoms for cohomology 5 These notes are based on Algebrac Topology from a Homotopcal Vewpont, M. Agular, S. Gtler, C. Preto A Concse Course n Algebrac Topology, J. Peter May More Concse Algebrac Topology, J. Peter May and Kate Ponto Algebrac Topology, A. Hatcher 1. Reduced and relatve homology and cohomology Defnton 1.1. For A X a sub-complex, let C n (A) C n (X) as the cells of A are a subset of the cells of X. Let C (X, A) = C (X)/C (A) and H (X, A) = H (C (X, A)). Let C (X, A) = ker(c (X) C (A)) = Hom(C (X, A), Z) and H (X) = H (C (X, A)). and We can gve smlar defntons for H (X, A; M) H (X, A; M). Defnton 1.2. For X a based CW-complex wth based pont a zero cell, 1

2 2 MATH CLASS 27 Let C (X) = C (X)/C ( ) and H (X) = H ( C (X)). Let C (X) = ker(c (X) C ( )) = Hom( C (X), Z) and H (X) = H ( C (X)). Remark 1.3. Note that C (X, A) = C (X)/C (A) = C (X/A). Indeed, X/A has a CW-structure all cells n A dentfed to the base pont and one cell for each cell not n A. Therefore, we have a natural somorphsm H (X, A) = H (X/A). Smlarly, H (X, A) = H (X/A) Unreduced cohomology can be though of as a functor from the homotopy category of pars of topologcal spaces to abelan groups: H (, ; M) : hcwpars Ab where H (X; M) = H (X, ; M) Axoms for unreduced homology. 2. Elenberg-Steenrod Axoms Defnton 2.1. A famly of functors E q ( ) : hcwpars Ab ndexed over the ntegers and natural transformatons : E q (X, A) E q 1 (A) whch satsfes the followng axoms s called an unreduced homology theory. (1) (Exactness) There s a long exact sequence... E q (A) E q (X) E q (X, A) E q 1 (A)... (2) (Excson) If (X; A, B) s a CW trad, then E q (A, A B) E q (X, B)

3 (3) (Addtvty) If (X, A) = (X, A ), then E q (X, A ) = Eq (X, A) MATH CLASS 27 3 Let M be an abelan group. The followng addtonal property, whch may or may not be satsfed, s called the dmenson axom: (Dmenson for M) E ( ) = M concentrated n degree zero. For (X, A) a par of spaces, let (ΓX, ΓA) be a CW-approxmaton. Defnng E (X, A) := E (ΓX, ΓA) we have that the above axoms are equvalent to the followng. Defnton 2.2. A famly of functors E q ( ; ) : htop-pars Ab ndexed over the ntegers and natural transformatons : E q (X, A) E q 1 (A) whch satsfes the followng axoms s called an unreduced homology theory. (1) (Exactness) There s a long exact sequence... E q (A) E q (X) E q (X, A) E q 1 (A)... (2) (Excson) If (X; A, B) s an excsve trad, then E q (A, A B) E q (X, B) (3) (Addtvty) If (X, A) = (X, A ), then E q (X, A; M) = E q (X, A ) (4) (Weak equvalence) If f : (X, A) (Y, B) s a weak equvalence, then E (f) s an somorphsm. Let M be an abelan group. The followng addtonal property, whch may or may not be satsfed, s called the dmenson axom: (Dmenson) E ( ) = M concentrated n degree zero Axoms for reduced homology.

4 4 MATH CLASS 27 Defnton 2.3. A famly of functors Ẽ q ( ) : h CWTop Ab ndexed over the ntegers (1) (Exactness) If A s a subcomplex of X Ẽ q (A) Ẽq(X) Ẽq(X/A) s exact. (2) (Suspenson) There are natural somorphsms Σ : Ẽq(X) Ẽq+1(ΣX) (3) (Addtvty) If X = X, then Ẽ q (X ) = Ẽ q (X). Let M be an abelan group. The followng addtonal property, whch may or may not be satsfed, s called the dmenson axom: (Dmenson for M) Ẽ (S 0 ) = M concentrated n degree zero. Agan, for well based space X, let (ΓX, Γ ) (X, ) be a CW approxmaton wth Γ a pont. Then we can defne Ẽ q (X) := Ẽq(ΓX). The axoms translate to: Defnton 2.4. A famly of functors Ẽ q ( ) : htop Ab ndexed over the ntegers (where Top s the category of well-based spaces) whch satsfes the followng axoms s called a reduced homology theory. (1) (Exactness) If A X s a cofbraton, then Ẽ q (A) Ẽq(X) Ẽq(X/A) s exact. (2) (Suspenson) There are natural somorphsms Σ : Ẽq(X) Ẽq+1(ΣX)

5 MATH CLASS 27 5 (3) (Addtvty) If X = X, then Ẽ q (X ) = Ẽ q (X). (4) (Weak equvalence) If f : X Y s a weak equvalence, then Ẽ (f) s an somorphsm. Let M be an abelan group. The followng addtonal property, whch may or may not be satsfed, s called the dmenson axom: (Dmenson for M) Ẽ (S 0 ) = M concentrated n degree zero. Remark 2.5. Note that Exactness mples the followng: If A f X C f s a cofber sequence, then Ẽ q (A) Ẽq(X) Ẽq(C f ) s exact. Indeed, consder the commuatve dagram A f r X M f then A M f s a cofbraton so there s a commutatve dagram, where Ẽq(r) s an somorphsm and the top row s exact: Ẽ q (A) Ẽ q (M f ) Ẽ q (M f /A) = Ẽ q(r) Ẽ q (A) Ẽ q (X) Ẽ q (C f ) Proposton 2.6. Gven a reduced cohomology theory Ẽ and a cofber sequence A X C f, there s a long exact sequence... Ẽ n+1 (C f ) Ẽ n (A) Ẽ n (X) Ẽ n (C f ) Ẽ n 1 (A) Axoms for cohomology. All of the above axoms generalze to contravarant functors E q ( ), Ẽ q ( ) wth all arrows reversed n the obvous way. However, the Addtvty axom has a dfferent flavor:

6 6 MATH CLASS 27 (Unreduced Addtvty) If (X, A) = (X, A ), then E q (X, A) = E q (X, A ). (Reduced Addtvty) If X = X, then Ẽ q (X) = Ẽ q (X ). Remark 2.7 (Passage from reduced to unreduced). Suppose that we are gven an unreduced homology theory E. Let X ponted CW -complex. Note that X s the dentty. Ths mples that the long exact sequence... E q+1 (X, ) E q ( ) E q (X) E q (X, ) E q 1 ( )... splts nto splt short exact sequences 0 E q ( ) E q (X) E q (X, ) 0 so that Then, the functor E (X) = E (X, ) E ( ). Ẽ (X) = E (X, ) gves a reduced homology theory. Conversely, gven a reduced homology theory Ẽ and a CW-par (X, A). Then lettng E (X, A) = Ẽ (X + /A + ). In partcular, E (X) = E (X, ) = Ẽ (X + / + ) = Ẽ (X + ). Ths wll gve an unreduced homology theory. Note that nothng n ths remark appealed to the dmenson axom.