ON THE MISLIN GENUS OF CERTAIN CIRCLE BUNDLES AND NONCANCELLATION

© Hindawi Publishing Corp.

ON THE MISLIN GENUS OF CERTAIN CIRCLE BUNDLES AND NONCANCELLATION

PETER HILTON and DIRK SCEVENELS (Received 1 February 1999)

Abstract.In an earlier paper, the authors proved that a process described much earlier for passing from a finitely generated nilpotent groupNof a certain kind to a nilpotent spaceXof finite type produced a bijection of Mislin generaᏳ(N)Ᏻ(X). The present paper is concerned with related results obtained by weakening the restrictions onNand generalizing the homotopical nature of the spacesXto be associated with a givenN.

Keywords and phrases. Mislin genus, circle bundles, noncancellation, nilpotent groups, nilpotent spaces.

2000 Mathematics Subject Classification. Primary 55P60, 20F18.

1. Introduction. The genus of a finitely generated nilpotent group N was intro- duced in [11] as the set of isomorphism classes of finitely generated nilpotent groups M such that the localizations Mp and Np are isomorphic at every primep. Analo- gously the Mislin genus of a connected nilpotent spaceX(of the homotopy type of a CW-complex) of finite type (cf. [4]) is defined as the set of homotopy types of nilpotent spacesY of finite type such that the localizationsYpandXpare homotopy equivalent at every primep.

In [10] the author, generalizing results in [13], provided a powerful tool for the calculation of the Mislin genusᏳ(X)of a (connected) nilpotent spaceXof finite type which was an H0-space (i.e., its rationalization is anH-space) and had only finitely many nonvanishing homotopy groups. It turns out thatᏳ(X)may then be given the structure of a finite abelian group. Much earlier, in [1, 2], the Mislin genusᏳ(N)of a finitely generated nilpotent groupNsatisfying three conditions had been calculated.

The conditions are stated in terms of the natural short exact sequence

TN

// //

_N

// //

_{FN, (1.1)}

whereTN is the torsion subgroup ofNwithFN the torsionfree quotient; they are (i) TN andFN are commutative;

(ii) relation (1.1) splits for the actionω:FN

//

_AutTN; (iii) ω(FN)lies in the center of AutTN.

It was observed in [1] that, in the presence of (i), (iii) is equivalent to

(iii) for allξinFN, there exists an integerusuch thatξ·a=ω(ξ)(a)=uafor all a∈TN. (Here,TN is writtenadditively.)

A finitely generated nilpotent group satisfying (i), (ii), (iii) is also said to belong toᏺ1.

Now it was known (see [3, 11]) that ifFN is commutative, thenᏳ(N)may also be given the structure of a finite abelian group. Moreover, a procedure was described in [2], and exploited in [6], whereby one might associate with the groupN, in the case that FN is cyclic, a circle bundleX, over a baseMwhich depends only onᏳ(N), inducing an injectionᏳ(N)

// //

_Ᏻ(X). Using [10, McGibbon’s formula], it was shown in [6] that the injection is indeed a bijection; in fact, the abelian group structures onᏳ(N)and Ᏻ(X)coincide. However, it should be remarked that the result in [6] was based on the assumption that one had chosenXto have vanishing homotopy groups in dimensions i≥3.

The restriction onN, thatFN be cyclic, is reasonable if one wishes to construct and calculate examples of nontrivial genera, since a result of [8] implies thatᏳ(N)is trivial ifFNis not cyclic. On the other hand, from the point of view of obtaining information about the genus of a certain spaceX, it is significant that we can extend the association N

//

_Xto the case ofFN not cyclic and hence, via McGibbon’s formula, show that Ᏻ(X)=0 in this case. We do this in Section 2. However, by refining the result quoted from [8], we obtain very precise information about the nature of the spaces Y of which we can claim the triviality of the genus. The improvement of the result from [8] is described in Section 6, and the spaces in question are specified in Theorems 2.3 and 2.4.

In Section 3, we describe an enlargement of the domains of validity of our main result in [6] and our results in Section 2. The injection Ᏻ(N)

// //

_Ᏻ(X)referred to earlier only requires that the baseMof the circle bundleXhave specified 2-type (i.e., we specifyπ1M, π2M and the action of π1M onπ2M as functions ofN), and says nothing about the homotopy groups ofMin dimensionsi≥3 beyond requiring that M (and henceX) be nilpotent. On the other hand, we have no reason to expect the injection to be a bijection in this generality. We describe in this section how we may ease the restriction onX(that its higher homotopy groups vanish) and still preserve the bijective property proved in [6] (whenFN is cyclic) and in Section 2 (whenFN is not cyclic).

In Section 4, we extend the method of [6] and this paper fromᏳ(N), whereNis a finitely generated nilpotent group inᏺ1, toᏳ(N^k), whereN^kis the direct product ofk copies ofN. This extension is significant becauseN^k, withk≥2, does not inherit prop- erty (iii) fromN, thoughᏳ(N^k)is still a finite abelian group sinceN^kinherits property (i) fromN. The calculation ofᏳ(N^k)fork≥2, was carried out in [7]—it was already known that there is a surjectionᏳ(N)

// //

_Ᏻ(N^k_{)given by ˜N}

//

N˜×N^k−1(cf. [1])—

and effectively we show in Section 4 that exactly the same process of passing to the appropriate quotient groups takes place in analysing the surjectionᏳ(X)

// //

_Ᏻ(X^k_), whereX^kis the topological product ofkcopies ofX, the circle bundle associated with N. As a consequence, we know thatᏳ(N^k)

//

_Ᏻ(X^k₎is bijective. Of course, we can here allow the generalization of the construction ofXfromNdiscussed in Section 3.

We conjecture that we may further generalize the results of this section by replacing N^kbyN1×N2×···×Nk, where eachNiis a finitely generated nilpotent group satisfy- ing conditions (i), (ii), and (iii); more precisely, we conjecture that we obtain a bijection Ᏻ(N1×N2×···×Nk)Ᏻ(X1×X2×···×Xk). The abelian groupᏳ(N1×N2×···×Nk) was calculated in [5].

In Section 5, we use the equivalence betweenᏳ(N^k)andᏳ(X^k)to transfer noncan- cellation phenomena from the category of finitely generated nilpotent groups to that of nilpotent spaces of finite type. The original noncancellation phenomena were de- scribed in [9]; they are a ready consequence of the calculation ofᏳ(N^k).

2. The caseFN noncyclic. Here we generalize the procedure used in [6] to study the case when FN is not cyclic. Thus we have, as in Section 1, a finitely generated nilpotent groupNbelonging toᏺ1and fitting into a split short exact sequence

TN

// //

_N

// //

_{FN, (2.1)}

but now we assume thatFN is free abelian of rank r ≥2. We then know that we can write

FN=

ξ1,ξ2,...,ξr

, (2.2)

where, fori=1,2,...,r,

ξi·a=uia, ∀a∈TN, (2.3)

and the order ofuimodmisti, wherem=expTN, and

t1|t2| ··· |tr. (2.4) As in [1], we know that the genus ofNis given by

Ᏻ(N)

Z/t1∗/{±1}. (2.5)

However, [8, Theorem 1.1] tells us that, in fact,t1=1 or 2 (givenr≥2), so that we have the following.

Theorem2.1. Ifr≥2, thenᏳ(N)is trivial.

Actually, as will be shown in Section 6, we can improve on [8, Theorem 1.1] and deduce (see Theorem 6.1) the following.

Theorem2.2. Ifr≥2, the quantitiest1,t2,...,tr of (2.4) satisfy

t1=t2= ··· =tr−2=1, tr−1=1or2. (2.6) In fact, the casetr−1=2 is a highly exceptional case—though it certainly does occur.

Let us consider the group N0 in ᏺ1, where expTN0= m and FN0 is cyclic with generatorξandξ·a=uafor alla∈TN0. Suppose that the order ofumodmist, so that

ᏳN0

(Z/t)^∗/{±1}. (2.7)

Then, withtr=t, the groupNwe have been discussing, witht1=t2= ··· =tr−1=1, tr=t, is justC^r−1×N0, whereCis cyclic infinite, so that

ᏳC^r−1×N0

=0 forr≥2. (2.8)

Now suppose thatN0gives rise, as in [6], to the circle bundleX0over a baseM, where π1X0=C,π2X0=TN0, andπiX0=0 fori≥3. Then, as proved in [6],

ᏳX0

(Z/t)^∗/{±1}. (2.9)

We can now carry out a process very like that in [6]. UsingX0as our base, we con- struct a trivial circle bundle overX0and the argument of [6] show that

ᏳC×N0

ᏳS¹×X0

. (2.10)

Iterating this procedure, we finally obtain the following.

Theorem2.3. We haveᏳ((S¹)^r−1×X0)=0forr≥2.

Let us now turn to the exceptional case wheretr−1=2 in Theorem 2.2. We first recall when the exceptional case arises. Using the notation above (and in Section 6), let

m=p₁ⁿ¹p₂ⁿ²···p_λ^nλ, (2.11) wherep1< p2<···< pλare primes. Moreover, we know that eachuiis of the form

ui=1+cipⁿ₁ⁱ¹p₂ⁿⁱ²···pⁿ_λ^iλ, withnij≥1 forj=1,2,...,λ, (2.12) whereciis prime top1p2···pλ. Then we may find ourselves in the exceptional case of Theorem 6.1, withε=2, ifp1=2 andn1≥3.

Let us then begin with a groupNinᏺ1such thatr=2 andt1=2,t2=t. We construct a nilpotentCW-complexMaccording to the following homotopical specifications:

• π1M=C2×C= η,ξ;

• π2M=TN, withη·a=2a, andξ·a=ta, for alla∈TN;

• πiM=0 fori≥3.

ThenH²(M;Z)⊇Ext(H1M,Z)=Ext(Z/2,Z)=Z/2= g, and we interpretg as the homotopy class of a map, which, by abuse, we also designate asg:M

//

_K(Z,2).

We then usegto induce a circle bundleXoverM; thus,

S¹

//

_X ^h

//

_M ^g

//

_{K(Z,2). (2.13)}

Thenhinduces an isomorphismh∗:πiXπiM fori≥2; moreover,π1X=C×C andhmaps one generator ofπ1Xontoη, the other being mapped ontoξ. As in [6]

(cf. [12, Theorem 1.8]), we have N

S¹,ΩX

fr=π2X # π1X, (2.14) the semidirect product for the action of π1X onπ2X. It again follows, just as we argued in [6], thatᏳ(N)Ᏻ(X). However,Ᏻ(N)=0, so we conclude that

Ᏻ(X)=0. (2.15)

However, we may now putXin the role ofMand repeat the construction using the zero element ofH²(X;Z). We obtain, of course, the trivial bundleS¹×X, corresponding to the nilpotent groupC×N, where, of course,Ᏻ(C×N)=0. Continuing in this way, we finally prove the following.

Theorem2.4. LetN be as above, and let X be the associated circle bundle con- structed as in (2.13). Then, for anyr∈N,

ᏳS¹_r

×X

=0. (2.16)

3. Carrying the calculation ofᏳ(X)further. Let us remove a key item in the spec- ification ofMin [6] and in Section 2. Thus we first assume thatFN in (2.1) is cyclic, but we no longer insist thatπiM=0 fori≥3. In fact, we impose no restriction at all on the homotopy groups ofMin dimensionsi≥3 asπ1M-modules, beyond insisting thatMbe nilpotent. We may then constructXexactly as we did in [6], and it remains true that we may find, for eachN$ in the genus ofN, a spaceX$ in the genus ofX, and that we obtain thereby an embedding ofᏳ(N)inᏳ(X). (Indeed, the method of passing fromNtoXwas carried out in [2] in the sense described in this paragraph, that is, without requiring thatπiM=0 for 1≥3.)

However, at this level of generality we certainly cannot claim thatᏳ(N) andᏳ(X) coincide, i.e., the embedding referred to above is a bijection.

Since we certainly want to apply the key formula of McGibbon, we must insist thatX be anH0-space with only finitely many nonzero homotopy groups. We will be content to present here a generalization of the theorem of [6] which requires virtually no change in the arguments. We use the notation of Section 1; and denote byT the set of primespsuch thatNhasp-torsion.

Theorem3.1. LetN∈ᏺ1with FN cyclic and letMbe constructed according to the following specifications:

• π1M=Ct= η;

• π2M=TN , withη·a=ua, for alla∈TN ;

• πiM=0, for almost alliand is always a (finite)T-group ifi≥2.

One then constructsXas in[6]and the mappingN$

//

_{X$, for$∈(Z/t)}^∗, induces a bijection ofᏳ(N)withᏳ(X).

Proof. We only need to remark that X is anH0-space, indeed,X is rationally equivalent toS¹, and thatXhas only finitely many nonzero homotopy groups since πiXπiMfori≥2. Thus we may apply McGibbon’s sequence

s-EquX ^d

//

_(Z/s)^∗_/{±1}

// //

_{Ᏻ(X) (3.1)} to calculateᏳ(X), except thats may be replaced by a largerT-number ˜s having s as a factor. However, both ans-equivalence and an ˜s-equivalence of Xare just aT- equivalence; and the calculation ofᏳ(X)from (3.1) is unaffected by replacingsin the middle term of (3.1) by any otherT-number having sas a factor. Thus we still ob- tain, from the calculations in [6], the conclusionᏳ(X)(Z/t)^∗/{±1}, whence, finally, Ᏻ(N)Ᏻ(X)(Z/t)^∗/{±1}.

4. Products. In this section, we will calculate the Mislin genus of the productX^kof kcopies ofXfork≥2, whereXis a circle bundle associated with a nilpotent group Ninᏺ1by the methods of [6] or Section 2.

We start by a very general result.

Lemma4.1. Letk≥1. IfXis a connected, nilpotent, finite typeH0-space with at most a finite number of nonzero homotopy groups, then there is an epimorphism

ρ:Ᏻ(X)

//

_ᏳX^k_{. (4.1)}

Proof. By [10, McGibbon’s result], and using the notation of the paper (except that we replace McGibbon’stbys, since we have usedt for another purpose), we know that there are exact sequences

s-EquX ^d

//

_(Z/s)∗/{±1}_$

// //

_Ᏻ(X),

˜

s-EquX^{k d˜}

//

_Z/˜s^∗_/{±1}^$

// //

_ᏳX^k_. ^(4.2) By definition it is easily seen that ˜s=s, so that there is a diagram

s-EquX ^d

//

σ

(Z/s)^∗/{±1}_$

// //

_Ᏻ(X)

˜

s-EquX^{k d˜}

//

_Z/˜s^∗_/{±1}^$

// //

_ᏳX^k_,

(4.3)

whereσis given byσ (φ)=φ×id. Of course, ˜d◦σ=din (4.3), so that an epimorphism ρ:Ᏻ(X)

// //

_Ᏻ(X^k₎is induced in (4.3).

In particular, we infer from Theorems 2.3 and 2.4 the following result.

Proposition4.2. LetY be any of the spaces discussed in Theorems 2.3 and 2.4.

ThenᏳ(Y^k)=0fork≥1.

In the case of a groupNinᏺ1withFN cyclic, the genus of the direct productN^k was calculated in [7]. More precisely, suppose, according to [1], that

Ᏻ(N)(Z/t)^∗/{±1}, (4.4)

where

t=p^$₁¹···p^$_λ^λ, (4.5)

withp1< p2<···< pλprime numbers, and$i≥1 fori∈ {1,...,λ}. Starting with the exact sequence

T-AutN^k

//

_(Z/e)^∗_/{±1}

// //

_ᏳNk

, (4.6)

whereT is the set of primes occuring in the torsion ofN, so thattis aT-number, it is shown in [7] that

ᏳN^k Ᏻ(N)

H , (4.7)

where, after the identification (4.4),Hconsists of those residue classesmmodtsuch that

m≡ ±1 modp_i^$i ∀i∈ {1,...,λ}. (4.8) Repeating the construction of the circle bundle X associated with N (cf. [6]), and considering products of these, we obtain the torus fibration

(S¹)^k

//

_X^k

//

_M^{k g×···×g}

//

_K(Z,2)×···×K(Z,2). (4.9) Again, as in [6], we have thatN^k[S¹,Ω(X^k)]fr, the group of free homotopy classes;

and the short exact sequences

TN^k

// //

_Nk

// //

_FNk, π2X^k

// //

_S¹_,ΩX^k

fr

// //

_π1Xk (4.10) may be identified. To complete the calculation ofᏳ(X^k), we have to compare the exact sequence (cf. [10])

s-EquX^{k d}

//

_(Z/s)^∗_/{±1}

// //

_ᏳX^k _(4.11) with (4.6). Since s=expTN (cf. [6]), we infer that s-EquX^k =T-EquX^k. It remains to analyze d in (4.11). Again, as in [6], it is clear that any f ∈s-EquX^k induces a commutative diagram

π2X^k

// //

f∗

S¹,Ω X^k

fr

// //

f∗

π1X^k

f∗

π2X^k

// //

_S¹_,ΩX^k_fr

// //

_π1Xk

(4.12)

or, equivalently,

TN^k

// //

N^k

// //

f∗

FN^k

f∗

TN^k

// //

_Nk

// //

_FN^k_,

(4.13)

wheref∗:FN^k

//

_FN^k_{is aT}-automorphism. Thus detf∗≡mmodt, wherem∈H, as defined in (4.8), by [7, Proposition 2.2]. Conversely, given any residue classmmodt satisfying (4.8), we can construct aT-equivalence f:X^k

//

_X^k _{such thatd(f )=} det(f∗ :π1X^k

//

_π1X^k₎₌m. Indeed, we use the construction of the homomor- phismψ:FN^k

//

_FN^kin [7, proof of Theorem 1.3] to obtain a map

ψ:K(Z,2)×···×K(Z,2)

//

K(Z,2)×···×K(Z,2), (4.14) making the diagram

M^{kg×···×g}

//

K(Z,2)×···×K(Z,2)

ψ

M^{kg×···×g}

//

K(Z,2)×···×K(Z,2)

(4.15)

commutative. This yields a commutative diagram S¹_k

//

ψ

X^k

//

f

M^k

//

K(Z,2)×···×K(Z,2)

ψ

S¹_k

//

_Xk

//

_Mk

//

K(Z,2)×···×K(Z,2),

(4.16)

andf is the desiredT-equivalence. Concluding, we thus have proved the following theorem.

Theorem4.3. LetN∈ᏺ1with FN cyclic, and letXbe the associated circle bundle, as constructed in[6]. ThenᏳ(X^k)fork≥2is obtained fromᏳ(X)(Z/t)^∗/{±1}by factoring out those residue classesmmodtsatisfying (4.8). Indeed,Ᏻ(X^k)Ᏻ(N^k).

Observe that, instead of constructing the circle bundleXand using the arguments of [6], we could have used the arguments of Section 3, thus allowingXto have some nonzero higher homotopy groups, and that the results of Proposition 4.2 and Theorem 4.3 would remain true.

5. Noncancellation phenomena. LetN∈ᏺ1withFN cyclic, and letXbe the asso- ciated circle bundle. According to [1, 6],

Ᏻ(N)Ᏻ(X)(Z/t)^∗

{±1} , (5.1)

wheretis as in (4.5). Furthermore, a complete set{X$|$∈(Z/t)^∗}of homotopy types of spaces inᏳ(X), in bijective correspondence with the complete set{N$|$∈(Z/t)^∗} of isomorphism classes inᏳ(N), was described in [6]. Using the results of the previous section, we derive some consequences regarding noncancellation phenomena, paral- leling results obtained in [9].

Theorem5.1. Letk≥2and letm1,...,mk,m₁,...,m_k∈(Z/t)^∗. The following con- ditions are equivalent:

• Xm1×···×Xm_kXm₁×···×Xm_k;

• Nm1×···×NmkN_m1×···×N_mk;

• there exists a residue classmmodtsatisfying (4.8) such that

m1···mk≡mm₁···m_k modt. (5.2) Proof. This is essentially a consequence of our proof of Theorem 4.3.

Corollary5.2. Letk≥1and let m1,m2∈(Z/t)^∗. ThenXm1×X^kXm2×X^k if and only if there exists a residue class mmodt satisfying (4.8) such that m1 ≡ mm2modt.

As a final consequence, we obtain the following result (cf. [13]).

Corollary5.3. IfX$∈Ᏻ(X), thenX_$^φ(t)/2X^φ(t)/2, whereφdenotes Euler’s to- tient function.

6. Appendix: on the structure of certain nilpotent groups. In this appendix, we sharpen [8, Theorem 1.1] which asserts that, in the notation of Section 2, withr≥2, we must havet1=1 or 2.

First, we point out that the condition that (2.1) splits, is not required in the conclu- sion of the theorem. This condition appeared (implicitly) in [8, proof of Lemma 2.1], in the description of the terms of the lower central series ofN. However, we know thatNis nilpotent if and only ifTN isFN-nilpotent (sinceTN,FN are commutative);

and, even without the splitting hypothesis, the description given in fact describes the

“lower central”FN-series ofTN. Thus the conclusion of [8, Lemma 2.1] still holds.

On the other hand, it doesnot follow, without the splitting hypothesis, from the fact thatt1=1 or 2 thatᏳ(N)is trivial. For the calculation ofᏳ(N)as(Z/t1)^∗/{±1}, carried out in [1], makes essential use of the splitting hypothesis. Thus it remains an interesting open problem to determineᏳ(N)whereNis a finitely generated nilpotent group satisfying hypothesis (i) and (iii) of Section 1. Further, the construction of a spaceXsatisfyingN[S¹,ΩX]frrequires the splitting hypothesis.

In fact, we can substantially improve on the conclusion of [8, Theorem 1.1], even without the splitting hypothesis. We recall that we expressm=expTN as

m=p₁ⁿ¹p₂ⁿ²···p_λ^nλ, (6.1) wherep1< p2<···< pλare primes. We then consider two cases. Case 2 (the excep- tional case) is given byp1=2,n1≥3; while Case 1 (the general case) is simply the complement of Case 2. We now argue just as in [8], except that we consider the pairs (ti,ti+1),i=1,2,...,r−1, instead of just the pair(t1,t2). By doing so, and by incor- porating the argument implicit in [8] thatr ≥3 ift=2, we conclude the following theorem.

Theorem6.1. (i)In the general case, withr≥2, t1,t2,...,tr

=

1,1,...,1,tr

. (6.2)

(ii) In the exceptional case, withr≥2, t1,t2,...,tr

=

1,1,...,ε,tr

, whereε=1or2. (6.3) Of course, the conclusion of Theorem 6.1 is entirely consistent with the statement in [8] thattcan take any value ifr=1.

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Peter Hilton: Department of Mathematical Sciences, SUNY, Binghamton, NY13902- 6000, USA

Current address: Department of Mathematics, University of Central Florida, Orlando, FL32816-1364, USA

E-mail address:[email protected]

Dirk Scevenels: Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnen- laan200B, B-3001Heverlee, Belgium

E-mail address:[email protected]