aa r X i v : . [ m a t h . A T ] O c t BOUNDING SIZE OF HOMOTOPY GROUPS OF SPHERES
GUY BOYDE
Abstract.
Let p be prime. We prove that, for n odd, the p -torsion part of π q ( S n )has cardinality at most p p − q − n +3 − p ) , and hence has rank at most 2 p − ( q − n +3 − p ) .For p = 2 these results also hold for n even. The best bounds proven in the existingliterature are p q − n +1 and 2 q − n +1 respectively, both due to Hans-Werner Henn. Themain point of our result is therefore that the bound grows more slowly for largerprimes. As a corollary of work of Henn, we obtain a similar result for the homotopygroups of a broader class of spaces. Introduction
Our goal is to bound the size of π q ( S n ). Serre [Ser53] showed that these groups arefinitely generated abelian, and that they contain a single Z -summand when q = n , orwhen n is even and q = 2 n −
1, and are finite otherwise. We may therefore restrictour attention to the p -torsion summand for each prime p . When p is odd it suffices toconsider n odd, because, again by work of Serre [Ser51], π q ( S n ( p ) ) ∼ = π q − ( S n − p ) ) ⊕ π q ( S n − p ) ) , where X ( p ) denotes the localisation of the space X at p .For us, the rank of a finitely generated module will be the size of a minimal gener-ating set. Selick [Sel82] proved that the rank of π q ( S n ( p ) ), regarded as a Z ( p ) -module,is at most 3 q . B¨odigheimer and Henn [BH83] prove that the base- p logarithm of thecardinality of the p -torsion part of π q ( S n ) is at most 3 ( q − n ) , for n odd and all primes p .By the same method, they prove that the rank of π q ( S n ( p ) ) satisfies the same bound. In[Hen86], Henn further improved this bound to 2 q − n +1 . In [Iri87], Iriye states the bound3 q − n p − , which is similar to our Theorem 1.1, but gives no proof.We will use the same machinery as all three of those papers, namely the EHP se-quences of James [Jam57] and Toda [Tod56]. In particular, the new ideas in this paperare primarily combinatorial. A sub-exponential bound is not known, and [BH83] notethat to produce such a bound, one would have to introduce additional information fromtopology.Our main result is as follows. Denote by s p ( n, q ) the base- p logarithm of the cardi-nality of the p -torsion part of π q ( S n ). Theorem 1.1.
For all natural numbers q , and n odd, s p ( n, q ) ≤ p − ( q − n +3 − p ) . Mathematics Subject Classification.
Primary 55Q40.
Key words and phrases. homotopy groups of spheres, EHP Sequence. If p = 2 then this bound holds also for n even.As a corollary we obtain the weaker but simpler bound s p ( n, q ) ≤ q − np − . The 3 − p which appears in the original statement reflects the classical fact that the first p -torsionclasses appear in the (2 p − q − n in base 2 p − . The main advantage of this bound over itspredecessors is that as p becomes large, the base of the exponential approaches 1, soour bound grows more slowly for larger primes.As in [BH83], we also obtain a bound on the rank, but we prefer to regard it asfollowing from Theorem 1.1, by using that the rank of a finite p -torsion group is atmost log p of its cardinality. Corollary 1.2.
For n odd, the rank of the p -torsion part of π q ( S n ) is at most 2 q − np − . If p = 2, this bound holds also for n even.The bound proven by Henn in [Hen86] was a lemma used to establish results aboutthe rank of the p -torsion part of π q ( X ) for X any simply connected space of finite type.Our improvement to the bound feeds directly into the main theorem of that paper togive that a certain constant c p which appears there may be assumed to be at least( ) p − (Henn shows that it is at least ). This has the following corollary. Corollary 1.3.
Let X be a simply connected space of finite type, and suppose thatthe radius of convergence of the power series P ∞ q =1 dim Z /p ( H q (Ω X ; Z /p )) · x q is 1. Then P ∞ q =1 dim Z /p ( π q ( X ; Z /p ) ⊗ Z /p ) · x q has radius of convergence at least ( ) p − . In partic-ular, the rank of the p -torsion part of π q ( X ) is at most 2 qp − for all but perhaps finitelymany q .The hypotheses of the above corollary are satisfied if, for example, the dimension of H q (Ω X ; Z /p ) is bounded above by a polynomial in q .I would like to thank my supervisor, Stephen Theriault, for all of his help and sup-port. I would also like to thank Hans-Werner Henn for his helpful correspondence -in particular for drawing my attention to the methods of his paper [Hen86], and formaking me aware of the paper [FP87] of Flajolet and Prodinger, which lead indirectlyto the idea for this paper. 2. Approach
It will be convenient for us to think of a stem in the homotopy groups of spheres as acombinatorial object, so we say that the k -th stem is the set { ( n, q ) ∈ N × N | q − n = k } .The negative stems, for example, are those for which k < p , and assume henceforth that all spaces are localized at p . For a space X , let J k ( X ) denote the k -th stage of the James Construction on X . For n odd, wehave the following ( p -local) fibrations, from [Tod56] and [Jam57]: J p − ( S n − ) −→ Ω S n −→ Ω S p ( n − , and S n − −→ Ω J p − ( S n − ) −→ Ω S p ( n − − . OUNDING SIZE OF HOMOTOPY GROUPS OF SPHERES 3
The long exact sequences on homotopy groups induced by these fibrations are called
EHP-sequences . The following inequalities are proven in the first lemma of [BH83].The first inequality is obtained by considering the EHP-sequences, assuming that allgroups are finite, and the second inequality accounts for the possibility that one of thegroups is not finite, using knowledge of the relative homotopy groups π q (Ω S n , S n − ),as in for example Appendix 2 of [Hus94]. When p = 2 the situation is simpler: theabove fibrations are just the odd and even cases of S n − −→ Ω S n −→ Ω S n − . Lemma 2.1.
For all q , and n odd:(1) s p ( n, q ) ≤ s p ( p ( n − , q )+ s p ( p ( n − − , q − s p ( n − , q −
2) if q = p ( n − s p ( n, q ) ≤ s p ( n − , q −
2) if q = p ( n − p = 2, we no longer need to restrict to odd n , and the first inequality can bereplaced by s ( n, q ) ≤ s (2 n − , q ) + s ( n − , q − . These inequalities will be used to prove Theorem 1.1. It is worth noting that theextent to which the inequalities fail to be equalities is measured by the size of theimages of the boundary maps (equivalently, the kernels of the suspensions) in the EHPsequences. In some sense, therefore, the extent to which our bound fails to be sharp ismeasuring the aggregate size of the images of EHP boundary maps.3.
Limitations of our approach
If in Lemma 2.1, one replaces the inequalities with equalities, and regards this as aninductive definition of integers t p ( n, q ), then necessarily t p ( n, q ) is the best upper boundthat can be obtained for s p ( n, q ) using that lemma. In [Hen86], Henn defines inductivelyintegers b ( n, k ). He shows that t ( n, q ) = b ( q − , n ) (note that n has switched roles).In [FP87], Flajolet and Prodinger study a combinatorially defined sequence H n . Bydefinition, t (2 , q ) = b ( q − ,
2) = H q − , a fact I was made aware of by Henn. Flajoletand Prodinger obtain an asymptotic estimate H q ∼ K · ν q , giving formulas for K and ν and computing both to 15 decimal places. To 3 decimal places, K is 0.255, and ν is1.794. They remark that H q (which is equal to t (2 , q + 2)) is at least F q , where F q isthe q -th Fibonacci number. We will not do so here, but one can show by induction that t p (3 , p + j (4 p + 5)) ≥ F j +1 for j ≥
0. Inductively applying t p ( n − , q − ≤ t p ( n, q )then gives that t p ( n, p + j (4 p + 5) + n − ≥ F j +1 for all odd n ≥
3. We thereforehave the following.
Corollary 3.1.
Any bound on the base- p logarithm of the cardinality of the p -primarypart of π p + j (4 p +5)+ n − ( S n ) (or the rank of that group) which can be obtained fromLemma 2.1 is greater than or equal to F j +1 . In particular, any base for an exponentialbound which can be obtained from Lemma 2.1 is at least φ p +5 , where φ denotes thegolden ratio. GUY BOYDE Proof of Theorem 1.1
Proof of Theorem 1.1.
We will actually prove the slightly stronger result that s p ( n, q ) ≤ ⌊ p − ( q − n +3 − p ) ⌋ . The floor function forces the exponent to be an integer, which will beuseful in the proof. We will use a (slightly modified) strong double induction overstems. More precisely, the proof of the result for ( n, q ) will use the result for all ( m, r )with r − m < q − n (that is, on lower stems) and for (1 , q − n + 1) (that is, the entryat the base of the stem on which ( n, q ) lies). A proof using the other lower entries onthe same stem in the induction is possible, but results in a more unwieldy inductivehypothesis. The case p = 2, n even will be treated at the end.Suppose first that q ≤ n . In this case, π q ( S n ) is torsion free (indeed, it is zerofor q < n ) so s p ( n, q ) = 0. This proves the result for all non-positive stems. Thehigher homotopy groups of S are trivial (since it has a contractible universal cover) so s p (1 , q ) = 0 for all q . This proves the base case of each stem.It remains only to treat an inductive step on a positive stem. Thus, let ( n, q ) ∈ N × N with n odd, and suppose that the result is proven for all ( m, r ) with r − m < q − n .Consider the two inequalities of Lemma 2.1. We wish to apply the first inequalityinductively down the stem to bound s p ( n, q ) by a sum of terms on lower stems and s p (1 , q − n + 1), which is zero by the discussion above. The only complicating factor isthe second inequality, which may require us to add one to our bound at certain steps.However, the second case of Lemma 2.1 can occur at most once per stem, so at worst wewill have to add one to the bound that we would obtain if the first case of the Lemmaheld everywhere. More precisely, we obtain s p ( n, q ) ≤ ( n − X i =0 ( s p ( p ( n − i −
1) + 1 , q − i ) + s p ( p ( n − i − − , q − i − . Each s p ( m, r ) is an integer. Therefore, for those ( m, r ) for which we inductivelyhave s p ( m, r ) ≤ ⌊ p − ( r − m +3 − p ) ⌋ , we actually have the sightly stronger statement that s p ( m, r ) ≤ ⌊ ⌊ p − ( r − m +3 − p ) ⌋ ⌋ . Including this fact into the above inequality, we find that( ∗ ) s p ( n, q ) ≤ ( n − X i =0 ( ⌊ ⌊ p − ( q − i − ( p ( n − i − − p ) ⌋ ⌋ + ⌊ ⌊ p − ( q − i − − ( p ( n − i − − − p ) ⌋ ⌋ ) . Notice that the value of the floor function on an integer power of 2 is given by ⌊ i ⌋ = ( i i ≥ i < . We now bound this summation by another where the nonzero exponents are distinctintegers. More precisely, adding 1 − p − to the exponent of the second term in ( ∗ )(inside the floor function) gives that OUNDING SIZE OF HOMOTOPY GROUPS OF SPHERES 5 s p ( n, q ) ≤ ( n − X i =0 ( ⌊ ⌊ p − ( q − i − ( p ( n − i − − p ) ⌋ ⌋ + ⌊ ⌊ p − ( q − (2 i +1) − ( p ( n − (2 i +1) − − p ) ⌋ ⌋ )= 1 + n − X i =0 ( ⌊ ⌊ p − ( q − i − ( p ( n − i − − p ) ⌋ ⌋ . In particular, s p ( n, q ) is at most one greater than a sum of powers of 2. It suffices toshow that those powers of 2 that are not killed off by the outer floor function are alldistinct and strictly smaller than 2 ⌊ p − ( q − n +3 − p ) ⌋ , because P k − i =0 i = 2 k − i by 1 changes the exponent by 1.To see that they are strictly smaller than 2 ⌊ p − ( q − n +3 − p ) ⌋ , consider the largest poweroccurring in the summation, which is the i = n − ⌊ p − ( q − n + 3 − p ) − ⌋ = ⌊ p − ( q − n + 3 − p ) ⌋ −
1, as required. This completes theproof for n odd.It remains to treat the case p = 2, n even. Since the simplification at p = 2 in Lemma2.1 holds for all n , the above proof may be repeated without restricting to n odd, anddoing so gives the result for all n . (cid:3) References [BH83] Carl-Friedrich B¨odigheimer and Hans-Werner Henn,
A remark on the size of π q ( S n ),Manuscripta Math. (1983), no. 1, 79–83.[FP87] Philippe Flajolet and Helmut Prodinger, Level number sequences for trees , Discrete Math. (1987), no. 2, 149–156.[Hen86] Hans-Werner Henn, On the growth of homotopy groups , Manuscripta Math. (1986), no. 2,235–245.[Hus94] Dale Husemoller, Fibre bundles , third ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994.[Iri87] Kouyemon Iriye,
On the ranks of homotopy groups of a space , Publ. Res. Inst. Math. Sci. (1987), no. 1, 209–213.[Jam57] I. M. James, On the suspension sequence , Ann. of Math. (2) (1957), 74–107.[Sel82] Paul Selick, A bound on the rank of π q ( S n ), Illinois J. Math. (1982), no. 2, 293–295.[Ser51] Jean-Pierre Serre, Homologie singuli`ere des espaces fibr´es. Applications , Ann. of Math. (2) (1951), 425–505.[Ser53] , Groupes d’homotopie et classes de groupes ab´eliens , Ann. of Math. (2) (1953),258–294.[Tod56] Hirosi Toda, On the double suspension E , J. Inst. Polytech. Osaka City Univ. Ser. A. (1956), 103–145. Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UnitedKingdom
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