Applications of the Serre Spectral Sequence

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1 Applications of the Serre Spectral Seuence Floris van Doorn November, 25 Serre Spectral Seuence Definition A Spectral Seuence is a seuence (E r p,, d r ) consisting of An R-module E r p, for p, and r Differentials d r p, : E r p, E r p r,+r such that d 2 r = where E r+ is defined to be the homology of (E r, d r ) That is, E r+ p, = ker(d r p,)/ im(d r p+r, r+) The variable r is called the page, p the filtration degree, the complementary degree and p + the total degree Theorem 2 (Serre Spectral Seuence) Let F X B be a fibration such that B is path-connected and π (B) acts trivially on H (F ; G) Then H p (B; H (F ; G)) = H p+ (X; G) This means that there is a spectral seuence (Ep,, r d r ) where Ep, 2 H p (B; H (F ; G)) and there is a filtration Fp+ F p+ p+ = H p+ (X; G) such that Ep, Fp+/F p p p+ Note that if B is simply connected, then conditions of the theorem are satisfied Examples Example 3 Suppose X = B F, where B is path-connected, and suppose that G is a field Then π (B) acts trivially on H (F ; G) and we have H n (X; G) = H p (B; G) H (F ; G) (Künneth formula) p+=n = p+=n H p (B; H (F ; G)) (Univ Coeff Th for homology) This means that all entries in the second page survive until page infinity The other extreme is if X is contractible, where almost nothing will survive, as we will see in the next examples In the next example, we will use that S = K(, ) and CP = K(, 2) Example 4 Consider the path space fibration of B = CP, that is ΩB P B B and note that S = ΩB Since B is simply connected, we can apply the Serre Spectral Seuence with coefficients in We know that E 2 p, H p (B; H (S )) and H (S ) = for > and for =, This means that the page E 2 looks like this

2 H (B) H 2 (B) H 3 (B) H 4 (B) H (B) H 2 (B) H 3 (B) H 4 (B) p Moreover, we have E p, = for p = = and otherwise From this we can conclude that H i (CP, ) = if i even if i odd Example 5 In this example we will compute the homology groups of the loop space of the sphere, ΩS n for n 2 We use the fibration ΩS n P S n S n and we can apply the Serre Spectral Seuence, since S n is simply connected Now H p (S n ; G) = G for p =, n and otherwise This means that only the and the n column can be nonzero 3n 3 H 3n 3 (ΩS n ) H 3n 3 (ΩS n ) 2n 2 H 2n 2 (ΩS n ) H 2n 2 (ΩS n ) n H n (ΩS n ) H n (ΩS n ) After some reasoning, we get that H i (ΩS n, ) = n p for n i otherwise 2 Serre Class Theorem Definition 2 We say that a space X is abelian if the action of π (X) on π n (X) is trivial for all n Note that every simply connected space is abelian Definition 22 A Serre Class is a class C of abelian groups containing the trivial group such that for every SES A B C we have B C iff A, C C In this document I call a Serre class nice if for every A, B C also A B and Tor(A, B) are in C (this name is made up by me) Lemma 23 The following classes are nice Serre classes FG, the class of finitely generated abelian groups 2

3 T P for some set P of primes This is the class of torsion abelian groups whose elements have orders divisible only by primes in P F P, the finite groups in T P Note that P is the set of all primes T P becomes the class of all torsion abelian groups and F P becomes the class of all finite abelian groups Theorem 24 Let X be a path-connected and abelian space, and let C be a nice Serre class Then (n > )(π n (X) C) (n > )(H n (X) C) Corollary 25 The homotopy groups of a finite simply connected CW-complex are finitely generated In particular, the homotopy groups of spheres are finitely generated Recall the following definition and theorem Definition 26 The Hurewicz homomorphism is the homomorphism h : π n (X) H n (X) defined by h([f]) = f (γ), where γ is a generator of H n (S n ) Theorem 27 (Hurewicz) Let n 2 and X a (n )-connected space Then H i (X) = for i < n and the Hurewicz homomorphism h : π n (X) H n (X) is an isomorphism We will now generalize this theorem Theorem 28 Let X be a path-connected and abelian space, and let C be a nice Serre class Suppose that π i (X) C for i < n Then the Hurewicz homomorphism h : π n (X) H n (X) is an isomorphism modulo C, which means that its kernel and cokernel are in C 3 Cohomology Serre Spectral Seuence There is a Serre Spectral Seuence for cohomology which is completely analogous Of course, the arrows in these spectral seuences are reversed Theorem 3 Let F X B be a fibration such that B is path-connected and π (B) acts trivially on H (F ; G) Then H p (B; H (F ; G)) = H p+ (X; G) However, we can now also use the cup product if the underlying group is a ring Theorem 32 There is a bilinear product Er p, satisfying E s,t r E p+s,+t r for r (written as concatenation) For x Er p, we define x = p + Then we have d r (xy) = (d r x)y + ( ) x x(d r y) This means that the product on level r induces a product on level r +, which coincides with the given bilinear product at level r + The product in E is induces from the products at the finite levels At page 2, the product is up to a factor ( ) s induced from the cup product under the correspondence E p, 2 H p (B; H (F ; R)) In the RHS the cup product sends (φ, ψ) to φ ψ and coefficients are also multiplied via the cup product The cup product in H (X; R) restricts to maps on the filtrations Fp m Fs n Fp+s m+n uotient maps Fp m /Fp+ m Fs n /Fs+ n Fp+s m+n /F m+n Under the correspondence Ep, p+s+ the product in the LHS corresponds to these maps in the RHS ab = ( ) a b ba and d(x n ) = nx n dx if x is even In particular, if we apply the cup E p, 2 E, 2 E p, 2 to a pair of units, we get a unit 3 which induce Fp p+ /F p+ p+

4 3 Examples Example 33 Consider the path space fibration of B = CP again By the universal coefficient theorem, we know that the cohomology groups are the same as the homology groups Now let s compute the cup 2 Then x is the unit for product structure Let x 2i be the generator of E 2i, 2 and a the generator of E, multiplication, and ax 2i are generators of E 2i, 2 a ax 2 ax 4 x x 2 x p All arrows are isomorphisms We may assume that d 2 a = x 2 Then we compute d 2 (ax 2i ) = x 2 x 2i so we may assume that x 2 x 2i = x 2i+2 This gives x 2i = x i 2 Hence H (CP, ) [x 2 ] Example 34 We will compute the cup product structure of H (ΩS n ; ) using the path space fibration of S n for n 2 The additive structure is the same as for homology, and we can name the generators as in the figure, where a = 3n 3 a 3 a 3 x 2n 2 a 2 a 2 x n a a x a a x n p We may assume that d(a k+ ) = a k x and note that a k x = xa k We distinguish two cases If n is odd we compute by induction to i + j that a i a j = ( ) i+j i ai+j Hence H (ΩS n, ) Γ [a ], where the divided polynomial algebra Γ R [α] is the uotient of the free R-algebra R[α, α 2, ] by the relations α i α j = ( ) i+j i αi+j If n is even, then we compute a 2 = and by induction on k we compute a a 2k = a 2k+ and a a 2k+ = and a k 2 = k!a 2k Now H (ΩS n, ) Λ [a ] Γ [a 2 ] where the exterior algebra Λ R [α, α 2, ] is the free R-module with basis finite products α i α ik for i < < i k where multiplication is defined as α i α j = α j α i and αi 2 = For the next example, we use the following Remark 35 We can factor any map f : A B as a homotopy euivalence followed by a fibration: A E f B Here E f = (a, γ) A B I γ() = f(a)} In HoTT we would have E f = Σ(x : A)Σ(y : B), f(a) = y 4

5 Example 36 In this example we will proof that the p-torsion subgroups of π i (S 3 ) is for i < 2p and p for i = 2p Start with a map S 3 K(, 3) inducing an isomorphism on π 3 Turn this into a fibration with fiber F By the LES of homotopy groups of a fibration we get that F is 3-connected and π i (F ) = π i (S 3 ) for i > 3 Now convert the map F S 3 into a fibration By the LES we see that the fiber is K(, 2) = CP F K(, 3) CP X S 3 f We now use the Serre Spectral Seuence of this last fibration We know the homology groups of S 3 and CP, so we know the second page looks like this Here the arrows are not all isomorphisms 6 a 3 a 3 x 4 a 2 a 2 x 2 a ax x 3 p Since X is 3-connected, d : a x must be an iso, so we may assume da = x Then d(a n ) = na n x Now we know what groups survive until E, to compute H i n if i = 2n + n if i = 2n > (X; ) = hence H i (X; ) = if i = 2n, if i = 2n The Hurewicz Theorem modulo C now implies that the first p-torsion in π (X), hence also in π (S 3 ) is a p in π 2p For p = 2 we get the stronger result that π 4 (S 3 ) = 2, hence also π n+ (S n ) = 2 for n 3 5