3 years ago




2 N. P. STRICKLAND• I have used the following formula for the Hopf invariant of a composite: for α ∈ π n S mand β ∈ π m S i (with n > 1 and m, i > 0) we haveH(β ◦ α) = H(β) ◦ α + (Σ i−1 β) ◦ (Σ m−1 β) ◦ H(α)This is a special case of [1, Corollary III.6.3]. (Note that his γ 2 is our H, and his β#β isour (Σ i−1 β) ◦ (Σ m−1 β), whereas Toda’s β#β is β ∧ β = (Σ i β) ◦ (Σ m β).) It is apparentlywell-known to the experts that the proof in [1] is incorrect. However, the formula is knownto become true after one suspension, and it is also true for easy reasons if either α or βis a suspension. I do not know whether there are any cases in which the formula itself isincorrect.2.1. Abelian groups.2. Mathematica notation• The expression Group[{x 1 , d 1 }, . . . , {x r , d r }] represents an abelian group that is the directsum of cyclic groups of order d i generated by elements x i . The order d i may be Infinity.Expressions of this form are displayed (by default) in traditional notation, for exampleGroup[{x,Infinity},{y,24}] is displayed as Zx ⊕ Z 24 y. One can type FullForm[A] tosee the internal representation.• An isomorphism class of finitely generated abelian groups is represented by an expressionof the form GType[d 1 , . . . , d r ] with 1 < d 1 ≤ . . . ≤ d r ≤ ∞; this represents ⊕ i Z d i, whereZ ∞ just means Z.• If A is an expression of the form Group[. . . ], then GroupType[A] represents the isomorphismclass, GroupOrder[A] gives the order, GroupRank[A] gives the number of cyclic summands,Generators[A] gives the list of generators of these summands, GroupExponent[A] givesthe exponent, and Elements[A] gives the list of all elements of A (if A is finite). Theboolean function FiniteQ[A] is true iff A is a finite group.• The function ElementQ[a,A] will return True if a is visibly an integral linear combinationof the generators of A.• The expression Coset[a, {x 1 , . . . , x r }] refers to the coset of a modulo the subgroup generatedby the elements x i .2.2. Spaces.• The expression Sphere[n] represents the sphere S n , which we officially define to be theone-point compactification of R n . There is no real sign issue in identifying this withI n /∂(I n ), or in identifying S n ∧ S m with S n+m . There are sign issues in identifying S nwith ∆ n /∂(∆ n ) or with the space Sround n := {x ∈ Rn+1 | ‖x‖ = 1}.• We define ΣX = X ∧ S 1 . This is represented in Mathematica as Sigma[X]; the 4-foldsuspension of X (for example) can then be entered as SigmaIterate[4,X], but it will berepresented internally as Sigma[Sigma[Sigma[Sigma[X]]]].• The expression OrthogonalGroup[n] represents the orthogonal group O n .2.3. Homotopy sets.• The expression HomotopySet[X,Y] represents the set of homotopy classes of based mapsfrom X to Y . This will generally have at most one reasonable group structure. At presentwe have no support for nonabelian groups, and no support for multiple group structures.• The expression HomotopyGroup[n,X] refers to π n X. It is automatically converted toHomotopySet[Sphere[n],X].• The expression SpherePi[n,m] refers to Toda’s πn m , which is a naturally defined subgroupof π n S m such that π n S m = πn m ⊕ (odd torsion). We will probably redefine it later to referto π n S m itself. There is ambiguity about this in the code at the moment.• The expression OrthogonalPi[n,m] refers to π n O m .• Given an element a ∈ [X, Y ], we have Source[a]=X and Target[a]=Y. If X = S n wehave SourceSphere[a]=n. Similarly, if Y = S m we have TargetSphere[a]=m. If X = S n

A MATHEMATICA REPRESENTATION OF SOME UNSTABLE HOMOTOPY GROUPS OF SPHERES 3and Y = S m we have Stem[a]=n-m and ST[a]={n,m}. We do not currently distinguishinternally between different zero maps, so Source[0] and so on are undefined.• For general a ∈ [X, Y ], the function Home[a] represents the set [X, Y ]. The functionShowHome[a] displays this in a nicely readable form.• Given a ∈ [X, Y ] and b ∈ [Y, Z], the composite b ◦ a is represented by the expressiono[b,a].2.4. Functions.• The suspension functor is denoted by Sigma. We use the definition ΣX = X ∧ S 1 . We areprimarily interested in Σ as a homomorphism π k S n −→ π k+1 S n+1 .• The iterated suspension Σ n X is represented by SigmaIterate[n,X].• The Hopf invariant H : π k S n −→ π k S 2n−1 is represented by Hopf. We use the James-Hopfdefinition with the left lexicographic ordering.• The connecting map P : π k S 2n+1 −→ π k−2 S n in the EHP sequence is represented byWhiteheadP. I am not sure if we have the sign pinned down.• The unstable J-homomorphism π k O n −→ π k+n S n is represented by JHomomorphism.• The evident map π k O n −→ π k O n+1 is represented by OrthogonalSigma.• The evaluation map π k O n −→ π k S n−1 is represented by OrthogonalHopf (because it iscompatible with Hopf via the J-homomorphism). Note that there is an implicit identificationof the round sphere in R n with our S n−1 = R n−1 ∪ {∞}, so we need to fix thesign.• The connecting map π k S n−1 −→ π k−1 O n−1 in the orthogonal sequence is represented byOrthogonalP. It comes from an map ΩS n−1 = JS n−2 −→ O n−1 extending the reducedreflection map S n−2 −→ SO n−1 , at least up to sign.2.5. Homotopy elements.• ι n : S n → S n is the identity map. Mathematica notation is iota[n].• w[n] refers to the Whitehead product w n = [ι n , ι n ]: S 2n−1 −→ S n .• η 2 : S 3 → S 2 is the complex Hopf map. For k ≥ 2 we put η k = Σ k−2 η 2 . Mathematicanotation is eta[k].• ν 4 : S 7 → S 4 is defined by the properties H(ν 4 ) = ι 7 and 2Σ(ν 4 ) = Σ 2 ν ′ (Lemma 5.4).I think this only defines it modulo 2ν ′ . However, I think we pin it down precisely byrequiring that ν 4 should be equal to the quaternionic Hopf map plus an odd torsion class,and that Σν 4 should be 2-torsion. For n ≥ 4 we put ν n = Σ n−4 ν 4 . Mathematica notationis nu[n].• σ 8 : S 15 → S 8 is the octonionic Hopf map plus an odd torsion class, chosen so that Σσ 8 is2-torsion. Toda defines it (Lemma 5.14) by the property that H(σ 8 ) = ι 15 . For n ≥ 8 weput σ n = Σ n−8 σ 8 . Mathematica notation is sigma[n].• ɛ 3 : S 11 → S 3 is the unique element of the Toda bracket 〈η 3 , Σν ′ , ν 7 〉 1 (see the beginning ofToda’s Chapter VI). For k ≥ 3 we put ɛ k = Σ k−3 ɛ 3 . Mathematica notation is epsilon[k].• In Toda’s Section VI(ii) he defines ν 6 : S 14 → S 6 to be an element of the Toda bracket〈ν 6 , η 9 , ν 10 〉 1 . We will let ν 6 denote the unique element of this Toda bracket that satisfiesH(ν 6 ) = ν 11 . This definition is validated in Section 3. We put ν n = Σ n−6 ν 6 for n ≥ 6.Mathematica notation is nubar[n].• The map µ 3 : S 12 → S 3 is defined in Toda’s Section VI(iii) by a procedure of several steps,involving the cofibre of the map ν ′ : S 6 /8 → S 3 . It turns out that the indeterminacy is{0, η 3 ◦ ɛ 4 } (see the discussion preceeding Lemma 6.5, together with Theorem 7.1). Wealso have π 12 S 3 = Z 2 µ 3 ⊕ Z 2 (η 3 ◦ ɛ 4 ). The element η 3 ◦ ɛ 4 has Adams filtration 4, andwe make the unique choice of µ 3 that has Adams filtration 5. We write µ k = Σ k−3 µ 3 fork ≥ 3. Mathematica notation is mu[k].• We define ζ 5 : S 16 → S 5 to be the unique element of the Toda bracket 〈ν 5 , 8ι 8 , Σσ ′ 〉 1 thatsatisfies H(ζ 5 ) = 8σ 9 and P (ζ 5 ) = ±η 2 ◦µ ′ . This is validated in Section 3, based on Toda’sSection VI(v). For k ≥ 5 we put ζ k = Σ k−5 ζ 5 . Mathematica notation is zeta[k].

fields and galois theory - Neil Strickland - University of Sheffield
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On some representations of the Iwahori subgroup