Autumn Semester 2016{2017 MATH41071/MATH61071 Algebraic topology 7 Cell complexes and cellular homology Cell complexes In this section we will consider topological spaces constructed inductively out of open and closed cells (instead of simplices), i.e. topological (sub-)spaces being homeomorphic to closed or open unit ball. 7.1 Definition. A finite cell complex (or finite CW complex) is a Haus- ` dorff space X which is a finite disjoint union X = α eα of subspaces that are open cells, together with a a characteristic map for every cell eα: a con- k k tinuous map fα : D ! X that maps B homeomorphically to eα and maps k−1 the boundary @B = S continuously into a union of cells eβ that all have k smaller dimension than k. The image fα(D ) is denotede ¯α. The union of cells of dimension less or equal then k is called the k-skeleton of X and will be denoted by Xk. A a pair of cell complexes is a pair (X; A) that consists of a finite cell complex X and a subcomplex (i.e. the union of a subset of cells) A. In the following we will usually drop the word finite . Cellular complexes are more flexible and easier to handle than simplicial complexes in many situations: If X and Y are cellular complexes, then the product X ×Y is again a cellular complex. If (X; A) is a cellular pair then X=A is again a cellular complex. It has one new vertex that corresponds to A, and one k-cell for each k-cell of X that does not lie in A. Cellular homology The aim of this section is to provide an effective tool to calculate homology groups on a cell complex. The main tool for this calculation is called cellular homology. Note, that one can generalise the axioms for a reduced homology theory from Definition 6.8 almost literally to the case of cell complexes. Now, given a cell complex X and a reduced homology theory H~ we define the cellular chain groups as 7.2 Definition. Given a cell complex X we define the cellular chain groups as k k−1 Ck(X) = H~k(X =X ) 1 with boundary maps dk : Ck(X) ! Ck−1(X) k k−1 given by the composition dk = (pk−1)∗ ◦ @k. Where @k : H~k(X =X ) ! k−1 H~k−1(X ) is the corresponding homomorphism from the long exact se- k k−1 k−1 k−1 k−2 quence for the pair (X ;X ) and pk−1 : X ! X =X is the contraction map. For the case k = 0 the above definition still makes sense when setting X0=X−1 := X0 t ∗ = X0 _ S0: The cellular homology groups are defined by cell Hk (X) = ker dk= im dk+1: To make sense to the last definition we need to have im dk ⊂ ker dk−1. This follows from 7.3 Lemma. For the boundary maps defined above we have dk ◦ dk+1 = 0; i.e. im dk ⊂ ker dk−1. Proof. By definition of d we have the following commutative diagram k+1 k H~k+1(X =X ) dk+1 @k+1 ( k (pk)∗ k k−1 @k k−1 H~k(X ) / H~k(X =X ) / H~k−1(X ) dk (pk−1)∗ ) k−1 k−2 H~k−1(X =X ) Note, that horizontal line is a part from the long exact sequence for (Xk;Xk−1). In particular, we have @k ◦ (pk)∗ = 0 and hence also dk ◦ dk+1 = 0. At a first glance the definition of cellular homology seems to be useless to calculate homology groups, since it itself involves the homology of Xk=Xk−1, W k k W k but these spaces are naturally homeomorphic to α(¯eα=@eα) =: α Sα a wedge sum of k-spheres. Hence, by Theorem 6.13 we obtain ∼ M ~ k Ck(X) = Hk(Sα) α where α is running over the index set of k-cells of X. Every component of the k above direct sum gives rise to a generator corresponding to 1 2 Hk(Sα) = Z, 2 k which we denote by eα (by abuse of notation). Hence, Ck(X) can be written as M k Z · eα: α It now remains to investigate the boundary maps to make the calculation of cellular homology groups effective. 7.4 Proposition. The boundary maps as defined above are given by their values on generators k X k−1 dk(eα) = deg('αβ) · eβ : β Here β is running over all indices of (k − 1)-cells and 'αβ is the composition k−1 fα k−1 p k−1 k−2 ∼ _ k−1 p¯β k−1 S −! X −! X =X = Sγ −! Sβ γ withp ¯β being the projection to the β-component of the wedge sum as in Theorem 6.13 and Remark 6.14. Proof. Note that the characteristic fα is by definition a map of pairs fα : k k−1 k k−1 (D ;S ) ! (X ;X ) hence it gives rise to the commutative diagram (Exactness axiom) ∼ ~ k @ ~ k−1 ∼ Z = Hk(S ) / Hk−1(S ) = Z (f¯α)∗ (fα)∗ k k−1 @k k−1 H~k(X =X ) / H~k−1(X ) k k ∼ W k ¯ k Note, that X =X = γ Sγ and fα is exactly the inclusion of Sα into this ¯ k ~ k k−1 wedge sum. In particular, (fα)∗(1) equals the generator eα in Hk(X =X ) by Remark 6.14. From Exercise 7.5 we know that the first row of the diagram is an isomorphism (which we can assume maps 1 to 1 after choosing the isomorphisms to Z accordingly). By definition of 'αβ we may extend the above diagram to the right (' ) ~ k @ ~ k−1 αβ ∗ ~ k−1 Hk(S ) / Hk−1(S ) / Hk−1(Sβ ) O (f¯α)∗ (fα)∗ (¯pβ )∗ (p ) k k−1 @k k−1 k−1 ∗ k−1 k−2 H~k(X =X ) / H~k−1(X ) / H~k−1(X =X ) k−1 k−2 ∼ W k−1 Note, that X =X = γ Sγ for this to make sense. Now, by definition pf dk we have k k ¯ dk(eα) = (pk−1)∗ ◦ @k(eα) = (pk−1)∗ ◦ @k ◦ (fα)∗(1): 3 By commutativity of the left part of the diagram this gives k dk(eα) = (pk−1)∗ ◦ (fα)∗(1): k−1 k−2 When expressing an element of H~k−1(X =X ) in the basis elements k−1 k−1 eγ the coefficient at eβ is given by the natural projection of the direct k−1 sum to the component Z · eβ ~ k−1 k−2 ∼ M k−1 k−1 Hk−1(X =X ) = Z · eγ −! Z · eβ ; γ but from Remark 7.16 we know that this projection coincides with (¯pβ)∗. Now, by commutativity of the right part of the diagram it follows that k (¯pβ)∗ ◦ dk(eα) = 'αβ(1) = deg('αβ): 7.5 Remark. One need's to be a bit careful when interpreting the propo- sition and it's proof for the case k = 1. Interpreting the statement of the 1 theorem correctly this means d1 sends eα to 0 if the boundary consists of only one point and if the boundary consists of two points the image is the difference of the corresponding basis elements of C0(X). When taking X0=X−1 = X0=; := X0 t ∗ this follows indeed from the above proof. Using this we are indeed able to calculate the cellular homology groups for cell complexes. 2 ∼ 2 1 7.6 Example (Sphere). Using the identification S = D =S one obtains a cell decomposition of the sphere into the image of the interior (a 2-cell) and the complement, which is just a point (a 0-cell). ( k = 0; 2 Hcell(S2) = Z k 0 else; since all the boundary maps are necessarily trivial. 2 Alternatively we may use the identification S2 =∼ D =∼ from Exercise 1.7, where (x; z) ∼ (x0; y0) if and only if x = x0 and y = jy0j. This induces a cell 2 1 decomposition into one 2-cell e1, the image of the interior, one 1-cell e2, the 1 0 0 image of S n f(1; 0); (−1; 0)g and two 0-cells e3; e4, the images of (1; 0) and (−1; 0). Hence, the cellular chain complex looks as follows. d2 d1 2 0 ! Z −! Z −! Z ! 0: 4 1 0 0 From Remark 7.5 we see that d1(e2) = e3 − e4. In particular, the kernel of d1 is 0. For d2, note that '1;2 is given by ( eαi 0 α π ' (eαi) = 6 6 1;2 −αi e π 6 α 6 2kπ: One obtains deg('1;2) = 0 and d2 ≡ 0. This leads to the same results for the homology as in the above calculation. We saw that for this example the cellular homology groups didn't depend on the chosen cell decomposition and that they coincide with the simplicial homology groups. 7.7 Example (Torus). We can consider the cell decomposition of the unit square into one 2-cell (the interior which is isomorphic to S2) four 1- cells (the edges) and four 0-cells (vertices) all with the obvious characteristic maps. Via the quotient map the 1-cells get identified in pairs and 0-cells are mapped to the same point. This induces a cellular decomposition of 2 1 1 0 the torus with one 2-cell e1, two 1-cells e2; e3 and one 0-cell e4.