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1 CW Complex, Cellular Homology/Cohomology

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1 CW Complex, Cellular Homology/Cohomology Our goal is to develop a method to compute cohomology algebra and rational homotopy group of fiber bundles. 1 CW complex, cellular homology/cohomology Definition 1. (Attaching space with maps) Given topological spaces X; Y , closed subset A ⊂ X, and continuous map f : A ! y. We define X [f Y , X t Y / ∼ n n n−1 n where x ∼ y if x 2 A and f(x) = y. In the case X = D , A = @D = S , D [f X is said to be obtained by attaching to X the cell (Dn; f). n−1 n n Proposition 1. If f; g : S ! X are homotopic, then D [f X and D [g X are homotopic. Proof. Let F : Sn−1 × I ! X be the homotopy between f; g. Then in fact n n n D [f X ∼ (D × I) [F X ∼ D [g X Definition 2. (Cell space, cell complex, cellular map) 1. A cell space is a topological space obtained from a finite set of points by iterating the procedure of attaching cells of arbitrary dimension, with the condition that only finitely many cells of each dimension are attached. 2. If each cell is attached to cells of lower dimension, then the cell space X is called a cell complex. Define the n−skeleton of X to be the subcomplex consisting of cells of dimension less than n, denoted by Xn. 3. A continuous map f between cell complexes X; Y is called cellular if it sends Xk to Yk for all k. Proposition 2. 1. Every cell space is homotopic to a cell complex. 1 2. Every continuous map between cell complexes is homotopic to a cellular map. Proof. (a)Given σk = (Dk; f) is attached to X. Since @Dk = Sk−1 is compact, f(Sk−1) is covered by interior of finitely many cells. k−1 ◦ f(S ) ⊂ [i(σi) Say σl has the highest dimension among fσig. If l > k − 1, then by Sard’s theorem, ◦ ~ image of f has measure zero in (σl) . Hence can construct a homotopy f ∼ f, with ~ k−1 ~ k−1 ~ f(S ) \ σl = f(S ) \ @σl. Repeat in this way we see f is homotopic to f with ~ k−1 f(S ) ⊂ Xk−1. (b) Similar. Definition 3. (Cellular chain complex) Define cellular chain as the following X cell , f k j 2 k g Ck (X; G) glσl gl G; σl are cells of dimension k : l Also define the boundary operator @ : Ck(X; G) ! Ck−1(X; G) X k 7! k k−1 k−1 σ [σ ; σi ]σi i k k−1 where [σ ; σi ] is called the incidence number, defined to be the degree of the k k composite map Fi (σ = (D ; f)) f W k−1 ∼ k−1 S Xk−1 Xk−1/Xk−2 Si Fi k−1 Si W k−1 − where Si is a bouquet of sphere. Each sphere corresponds to a k 1 cell. Extend @ to be a G − module homomorphism. Proposition 3. @ ◦ @ = 0 Proof. Assume the following two facts without proof. 2 1. f; g : M n ! Sn, f ∼ g , deg f = deg g 2. If A ⊂ X is a closed subset, we have a long exact sequence of homotopy group ···! πk(A) ! πk(X) ! πk(X; A) ! πk−1(A) !··· k k−1 First we construct a map α : πk(Xk;Xk−1) ↠ Ck(X; Z). Given f :(D ;S ) ! (Xk;Xk−1), let Fi be the composite map f W k ∼ k D Xl Xk/Xk−1 Si Fi k Si k k k We define α([f]) = (deg Fi)σi . Note that σ = α([σ ]). @^ ··· πk(Xk−1) πk(Xk)πk(Xk;Xk−1) πk−1(Xk−1) ··· j πk−1(Xk−1;Xk−2) @^ πk−1(Xk−1) By definition @σk = α(j ◦ @^([σk])) @@σk = α(j ◦ @^ ◦ j ◦ @^([σk])) where j ◦ @^ = 0 by exactness. Definition 4. (Cellular homology, cohomology) The cellular cycle, boundary, homology is defined as the following cell , Zn ker @n cell , Bn Im@n+1 cell , Hn Zn/Bn For cellular cohomology, we define cellular cochain k C (X; G) , flinear map from Ck(X; G) to Gg 3 δ : Ck(X; G) ! Ck+1(X; G) α 7! α ◦ @ n , Zcell ker δn n , Bcell Imδn+1 n , n n Hcell Z /B In fact cellular homology/cohomology is isomorphic to the singular one, this can be proved by induction of dimension and the long exact sequence of singular homology. ···! Hi(Xk) ! Hi(Xk+1) ! Hi(Xk+1;Xk) ! Hi−1(Xk) !··· I will give another proof using spectral sequence in the following section. 2 Homology/cohomology of product space, cup product k l If K1;K2 are cell complexes, write K1 = [σ ;K2 = [σ . Then K1 × K2 has cell decomposition [σk × σl. Since we can identify (Dk; f) ∼ (Ik; f), (Ik; f) × (Il; g) ∼ (Ik+l;F ) for some F defines on @Ik+l appropriately. From the construction of F we see i × j i × j − i i × j @(σ1 σ2) = (@σ1) σ2 + ( 1) σ1 (@σ2) Proposition 4. (Kunneth formula) For G arbitrary ring, F arbitrary field, X ∼ Ck(K1 × K2; G) = Cm(K1; G) ⊗ Cn(K2; G) m+n=k X ∼ Hk(K1 × K2; F ) = Hm(K1; F ) ⊗ Hn(K2; F ) m+n=k σm × σl [ σm ⊗ σl X k ∼ m n H (K1 × K2; F ) = H (K1; F ) ⊗ H (K2; F ) m+n=k σ\m × σl [ σ^m ⊗ σ^l 4 When F is not a field, the map is still defined. Proof. The first isomorphism is illustrated above. For the second isomorphism, we choose a ’good’ basis for the vector space Ck. We can always find fxk;i; yk;j; hk;lg basis for Ck, such that @xk;i = yk−1;i;@k;j = 0; @hk;l = 0 I.e. fyk;j; hk;lg basis for Zk fyk;jg basis for Bk fhk;lg basis for Hk P ⊗ For k+l=m Hk(K1; F ) Hl(K2), we have basis k xk ⊗ xl; yk ⊗ xl + (−1) xk ⊗ yl−1; xk ⊗ hl; hk ⊗ xl k k+1 yk ⊗ xl + (−1) xk+1 ⊗ yl−1; (−1) (yk+1 ⊗ yl−1 + yk+1 ⊗ yl−1); yk ⊗ hl; hk ⊗ yl; hk ⊗ hl Hence fhk ⊗ hlg form basis for H(K1 × K2). Thus the isomorphism. Cohomology case is similar. It is easy to check the map is still defined when F is not a field. Now we can introduce multiplicative structure on H∗(X; R). Let 4 be the diagnoal map 4 : X ! X × X x 7! (x; x) Define cup product to be the composite of the following maps Hk(X; R) × Hl(X; R) Hk(X; R) ⊗ Hl(X; R) Hk+l(X × X) ^ 4∗ Hl+k(X) Proposition 5. 1. Cup product is associative 5 2. αi ^ βj = (−1)ijβ ^ α 3. There is a multiplicative identity. Proof. 1. Define 41; 42 : X × X ! X × X × X. 41 :(x; y) 7! (x; x; y) 42 :(x; y) 7! (x; y; y) Hk(X) ⊗ Hl(X) ⊗ Hm(X) Hk+l+m(X × X × X) 4∗ 2 4∗ 1 Hk+l+m(X × X) Hk+l+m(X × X) 4∗ 4∗ Hk+l+m(X) 4∗ ◦ 4∗ 4∗ ◦ 4∗ 4 ◦ 4 4 ◦ 4 While 1 = 2 since 1 = 2 2. It is just orientation. 3.If X is connected, then we can assume there is only one 0−cell ∗.Let p : X × X ! X (x; y) 7! x Then p ◦ 4 = id σbi = (p ◦ 4)∗σbi = 4∗(p∗σbi) = 4∗(σ\i × ∗) 3 Spectral sequence In the section above, we know how to get cell decomposition and homology/cohomology from those of base space and fiber space if the fiber bundle is trivial. For general fiber bundles, we can still derive their cell decomposition from the bases and fibers. Since each cell is contractible, every fiber bundle over a cell is trivial. If thebase [ k [ l space B and the fiber space F have cell decomposition B = σB;F = σF , then 6 [ k × l the total space E has cell decomposition E = σB σF . While in this case we do not have the formula q+j i × j i × j − i i × j @(σE ) = @(σB σF ) = (@σB) σF + ( 1) σB (@σF ) In fact if the base space is simply-connected we will have q+j i × j − j i × j q+j q+j ··· @(σE ) = (@σB) σF + ( 1) σB (@σF ) + @2σE + @3σE + q+j − − where @kσE is a linear combination of the product of the (q k) dimensional cell of the space and (j + k − 1)−dimensional cell of the fiber. In this chapter we introduce spectral sequence to deal with this case. Definition 5. (filtered complex) Given a chain complex C∗, a filtration is an increasing sequence of subset of C∗ · · · ⊂ Fp−1Ci ⊂ FpCi ⊂ Fp+1Ci ⊂ · · · ⊂ Ci With the boundary operator @ sends FpCi to FpCi−1. (FpC∗;@) is called a filtered complex. In our case, we take C∗ to be the sellular chain complex of fiber bundle E. FpCi , fσ 2 Ci(E) j π(σ) ⊂ Bpg where π is the projection map, Bp is the p−skeleton of the base space B. Let GpCi , FpCi/Fp−1Ci, @ : GpCi ! GpCi−1 is well-defined. Define 0 , Ep;q GpCp+q = FpCp+q/Fp+q−1 , 0 ! 0 @0 @ : Ep;q Ep;q−1 ker @ 1 , 0 Ep;q Im @0 In fact, f 2 j 2 g 1 x FpCp+q @x Fp−1Cp+q−1 Ep;q = Fp−1Cp+q + @(FpCp+q+1) 1 2 2 Define @1 on Ep;q as following.
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