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On the Bott Periodicity, $ J $-Homomorphisms, Transfer Maps

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On the Bott Periodicity, $ J $-Homomorphisms, Transfer Maps −k On the Bott periodicity, J-homomorphisms, and H∗Q0S Hadi Zare Abstract 0 The Curtis conjecture predicts that the only spherical classes in H∗(Q0S ; Z/2) are the Hopf invariant one and the Kervaire invariant one elements. We consider Sullivan’s decomposition 0 Q0S = J × cokerJ where J is the fibre of ψq − 1 (q = 3 at the prime 2) and observe that the Curtis conjecture holds when 0 we restrict to J. We then use the Bott periodicity and the J-homomorphism SO → Q0S to define 0 some generators in H∗(Q0S ; Z/p), when p is any prime, and determine the type of subalgebras that k they generate. For p = 2 we determine spherical classes in H∗(Ω0 J; Z/2). We determine truncated −k subalgebras inside H∗(Q0S ; Z/2). Applying the machinery of the Eilenberg-Moore spectral sequence we define classes that are not in the image of by the J-homomorphism. We shall make some partial k observations on the algebraic structure of H∗(Ω0 cokerJ; Z/2). Finally, we shall make some comments on the problem in the case equivariant J-homomorphisms. Contents 1 Introduction and statement of results 2 2 The fibre of the Adams operations 4 3 Proof of Theorem 1 5 4 The Hopf invariant one result 6 5 The homology of iterated loops of J-homomorphisms 6 5.1 The case p =2:ProofofTheorem2 .............................. 6 5.2 Proof of Theorem 5: The case of odd primes . ........... 8 arXiv:0911.4478v3 [math.AT] 11 May 2010 k 5.3 The Z/2-homology of Ω JC: Proof of Theorem 5 when p =2................. 9 5.4 An application to the complex transfer . .............. 10 k 6 Spherical classes in H∗(Ω0 J; Z/2): Proof of Theorem 3 11 −k 7 Truncated subalgebras in H∗(Q0S ; Z/2):Proof of Proposition 4 12 7.1 Some remarks on the geometry of QS−k ............................ 13 8 Proof of Theorem 6 15 k 9 Comments on H∗(Ω0 cokerJ; Z/2) 18 10 Equivariant J-homomorphisms 18 1 1 Introduction and statement of results 0 0 i i Let Q0S be the base point component of QS = colim Ω S . We have the following. Theorem 1. The image of the composite 0 h 0 2π∗J ֌ 2π∗Q0S −→ H∗(Q0S ; Z/2) consists only of the image of the Hopf invariant one elements in positive dimensions where h denotes the Hurewicz homomorphism, i.e. only η,ν,σ survive under the above composite. We write pπ∗ for p-primary homotopy where p is a prime number. We shall use f# to denote the mapping induced in homotopy, and f∗ for the mapping induced in homology, where f is a mapping of spaces. Given a class ξi ∈ H∗(X; Z/p), the subindex i will refer to the dimension of the class. We shall use EZ/p for the exterior algebra functor over Z/p, R for the Dyer-Lashof algebra (the algebra of Kudo-Araki operations Qi and their iterations), and A for the Steenrod algebra; all mod p. −k k 0 0 Let Q0S be the base-point component of Ω Q0S . The real J-homomorphism SO → Q1S is defined using the reflection through a hyperplane. After composition with a translation map ∗[−1] : 0 0 Q1S → Q0S , obtained by ‘loop sum’ with a map of degree −1, we obtain 0 ∗[−1] 0 JR : SO −→ Q1S −→ Q0S . On the other hand, the original real Bott periodicity provides a homotopy equivalence Z × BO → Ω8BO. 8 After looping once we have the homotopy equivalence SO → Ω0SO which we refer to as real Bott 8 8 periodicity where Ω0SO denotes the base-point component of Ω SO. Consider the iterated loops of JR as mappings k k −k k 0 Ω JR : Ω SO −→ QS = Ω Q0S . Z k −k −k Z The /p-homology of the spaces Ω SO are completely known. Let xi ∈ Hi(Q0S ; /2) be the image k k of the algebra generator of H∗(Ω0 SO; Z/2) in dimension i under (Ω JR)∗ if there is such a generator, 8j Z −8j 8j e.g. H∗(Ω0 SO; /2) ≃ EZ/2(si : i> 0) and x = (Ω JR)∗si. We then have the following. −k −k Z Theorem 2. The classes xi ∈ Hi(Q0S ; /2) are nontrivial if and only if: (1) If k 6≡ 0 mod 8; or (2) If k = 8j and i 6= 2νj k − 1 where 2νj is the largest power of 2 which divides 34j − 1. I −k The action of the mod 2 Steenrod operations on the classes Q xi is determined by the Nishida relations −k −k together with their action on xi . If k ≡ 0, 1, 2, 4 then the subalgebra inside H∗Q0S spanned by the I −k −k classes Q xi is an exterior algebra. Moreover, if k > 7 then xi belongs to the kernel of the iterated t −k −k+t homology suspension σ∗ : H∗Q0S → H∗+tQ0S for some t 6 7. −n+1 Next, note that any generator α ∈ 2π1Q0S maps nontrivially under the Hurewicz homomorphism −n+1 −n+1 2π1Q0S → H1(Q0S ; Z/2), see Lemma 5.6. We determine how far any such class will survive −n+1 −n+i+1 under the homology suspension H1(Q0S ; Z/2) → Hi+1Q0S when α is a generator in the image of the J-homomorphism, i.e. we determine largest such i. −n −n Theorem 3. The class xd is spherical if and only if d 6 8 and there is an element f ∈ 2πdQ0S n −n −n which is in the image of (Ω JR)# with hf = xd ∈ HdQ0S . Moreover, suppose that n = 8t + k with n ′ ′ n 0 6 k 6 7 and f = (Ω JR)f for some f ∈ 2πdΩ0 SO. Then under the composition n 8t+k ≃ k ≃ JR S 2πdΩ0 SO = 2πdΩ0 SO −→ 2πdΩ0 SO −→ 2πd+kSO −→ 2πd+k the class f ′ maps to a Hopf invariant one element. The precise value for d depends on k, and can be determined as explained in Lemma 6.1. We shall also determine the Hurewicz image of the elements µ8i+1 and ηµ8i+1 (see Lemma 6.2), which are not in the image of the J-homomorphism, and observe that ‘they are detected on very low dimensional spheres’. S −k To state our next result, we fix some notation. Given a generator γ ∈ 2πk ≃ 2π0Q0S let [γ]= hγ −k −k where h : 2π0QS → H0(QS ; Z/2) is the Hurewicz homomorphism. S −k d Proposition 4. Assume that 2πk ≃ 2π0QS has a summand which is the cyclic group of order 2 , k −k and let γ be the generator of this component. Then the homology algebra H∗(Q0S ; Z/2) contains a 2 truncated polynomial subalgebra whose truncation height is at most 2d and the generators of this subalgebra I k are of the form Q [γJ ] where k J k l(J) k −k [γJ ]= Q [γ ] ∗ [−2 γ ] ∈ Hdim(J)Q0S if l(J) < d, k J k −k [γJ ]= Q [γ ] ∈ Hdim(J)Q0S if l(J)= d. We assume both I and J are admissible, and we allow I to be the empty sequence but not J. Moreover, r the action of the Steenrod operations Sq∗ on these subalgebras is determined by the Nishida relations. Note that π∗J is completely known [R86, Proposition 1.5.22]. Hence, we know the type of the trun- cated subalgebras that they give rise to. 0 Next, consider the complex J-homomorphism JC : U → Q0S and its iterated loops. Recall that for primes p > 2 we have im(JC)# ≃ pπ∗J [R86, Theorem 1.5.19] where (JC)# is the mapping induces in homotopy and J is the fibre of ψq − 1 : BU → BU with q a prime such that qi − 1 6≡ 0 mod p for 0 6 i 6 p − 2, and p divides qp−1 − 1 exactly once. Notice that there are many primes like this, and we may choose the least one for simplicity and denote with q(p). Let p > 2 be a prime number, and k > 0 any integer and let ǫ(k) = 1 if k is odd, and ǫ(k) = 0 otherwise. −k k Z k Let w2i+1−ǫ be the image of the generator of H2i+1−ǫ(k)(Ω0 U; /p) under (Ω JC)∗. Notice that these are primitive if k is even. −k −k Z Theorem 5. The classes w2i+1−ǫ ∈ H2i+1−ǫ(k)(Q0S ; /p) are nontrivial if and only if (1) p = 2 and k ≡ 0, 1, 7 mod 8; or (2) p = 2 and k ≡ 2, 3 mod 8 and i is even; or (3) p = 2 and k ≡ 4 mod 8 and i is odd; or (4) p> 2, k = 2n − 1, and p|q(p)n+i − 1; or (5) p> 2, k = 2n, and p|q(p)n+i − 1. I −k The action of the mod p Steenrod operations on the classes Q w2i+1−ǫ is determined by their action on −k I −k w2i+1−ǫ together with the Nishida relations. If k is even, then the classes Q w2i+1−ǫ generate exterior −k Z −k −k Z subalgebra in H∗(Q0S ; /2) whereas w2i+1−ǫ generate an exterior subalgebra in H∗(Q0S ; /2) for I −k p > 2. Moreover, for k even and p > 2 the classes Q w2i+1−ǫ generate a subalgebra whose truncation height is p if I is odd dimensional, and an exterior algebra otherwise.
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