The Hilton-Hopf quadratic form is defined for spaces of the homotopy type of a CW complex with one cell each in dimensions 0 and 4n, K cells in dimension 2n and no other cells. If two such spaces are of the same topological genus, then their Hilton-Hopf quadratic forms are of the same weak algebraic genus. For large classes of spaces, such as simply connected differentiable 4-manifolds, the converse is also true, as long as the suspensions of the spaces are also of the same topological genus. This note allays the conjecture that the converse is true in general by offering two techniques for generating infinite families of counterexamples

Bokor, Imre

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Publicacions Matemátiques, Vol 34 (1990), 323-333 .ON THE CONNECTION BETWEEN THETOPOLOGICAL GENUS OF CERTAINPOLYHEDRA AND THE ALGEBRAIC GENUS OFTHEIR HILTON-HOPF QUADRATIC FORMSAbstractIMRE BOKORThe Hilton-Hopf quadratic form is defined for spaces of the homotopytype of a CW complex with one cell each in dimensions 0 and 4n, Kcells in dimension 2n and no other cells . If two such spaces are of thesame topological genus, then their Hilton-Hopf quadratic forms are of thesame weak algebraic genus . For large classes of spaces, such as simplyconnected differentiable 4-manifolds, the converse is also true, as long asthe suspensions of the spaces are also of the same topological genus . Thisnote allays the conjecture that the converse is true in general by offeringtwo techniques for generating infinite families of counterexamples .It was shown in [1] that for C,', the class of those spaces which are of thesame homotopy type as a CW-complex with precisely one cell in each of thedimensions 0, 2n and 4n, there is a Factorisation Theorem with respect to theone-point union (or V-product) of spaces . The theorem states that as long asthe attaching map of the 4n-cell represents a homotopy class of infinite order in7r4n-1(S2n), the homotopy type of a finite wedge of such spaces determines thehomotopy type of each of the spaces in the wedge . This stands in contrast to thesituation where the attaching map is of finite order, for then non-cancellationphenomena occur with spaces of the same "genus", as shown in [31, [4], [5] and[6] .The proof of the Factorisation Theorem applied techniques which suggesta generalisation of the characterisation of the genus of a space in C,' to Cñ,the class of spaces of the homotopy type of a CW-complex with one cell indimensions 0 and 4n and a finite number K of cells in dimension 2n. This largerclass of spaces includes all (2n - 1)-connected differentiable 4n-manifo1ds . Upto homotopy, the spaces in Ch are mapping cones Cf of continuous mapsKf ; S4n-1	V s2nk=1324	1 . BOKORKand hence are classified by 7r4n-1( V S2n ) .k=1KWe fix K and abbreviate V S2n to V S2n.k=1For maps f, g : Son-1 --> V S2n the set of homotopy classes of continuousmaps between their mapping tones Cf and Cg is in bijection with the set ofhomotopy classes of homotopy commutative diagramswhereandS4n-1 fi v S2n9wThe set of homotopy classes of maps V S2n --, V S2 n admits a ring structureisomorphic to M(K; Z), the ring of K x K integral matrices . The functionassigning to each self-map cp of S2n its degree induces one such isomorphism,by mapping cp to the matrix A(cp) whose (i, j)-th eoeflcient is the degree of thecomposite mapqi o (p o irdj :SU -) V S2n -a V S2n -~ S2n,2nj : S2n -4 V S2nis the.j-th canonical inclusion in the co-product andqi S2n -> S2ncollapses each summand to the base point except the i-th one, on which it actsas the identity map.It was shown in [1] that each homotopy class [f] E 7r4n-1(V S2n) can berepresented by a pair consisting of the Hilton - Hopf quadratie form of f and thesuspension of f. The Hilton-Hopf quadratic form can itself be represented as asymmetric integral K x K matrix, HM, whose entries are precisely the Hilton-Hopf invariants of f, and the suspension by Ef , a "column torsion vector" eachof whose components is an element 7r4n(S2n+1 ) .A suitable choice of generators for 7r4n-1(V S2n) provides a simple way ofcomputing cp o f from co and f by means of a matrix calculus :H(,P o f) = A(~P)H(.f )(A(~P)) tE(cp o .f) = AME(f)) .andTOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA	325Thus H(f) is a quadratic form which is an invariant of oriented homotopytype and the assignment to each f E 7r4n-1 (v S2n) of its Hilton-Hopf quadraticform - that is, the quadratic form with matrix H(f) in the appropriate basis -is natural in self-maps of V SU . (Retan that the Hilton-Hopf quadratic formcan also be thought of as the intersection form in the integral cohomology ofCf.)The genus of a nilpotent space - more precisely, the genus of the homotopytype of a CW-complex of finite type - is defined to be the set of homotopy typesof those nilpotent spaces, each of whose p-localisations is homotopy equivalentto the p-localisation of the original space . Since we are only concerned withmapping cones Cf of maps f : Son-1 _> V S2n, it follows from Lemma 2.2 of[1] that the two spaces Cf and C9 are of the same genus if and only if for eachprime p there is a homotopy commutative diagrams4n-1 i 2n-i v sNs4n-1 ---------4 V S2nwith both the degree of 0 and the determinant of the matrixp . In terms of the matrix notation developed, this diagramequationsdeg(O)H(g) =A(So)H(f)(A(w))`deg(O)ag = A(~p)uf,of cp coprime tois equivalent towith deg(0) and det A(cp) coprime to p, where det A is the determinant of thematrix A.Since the assignment of this quadratic form is natural, each of its invariantsis, in fact, an invariant of the mapping cone of f . One such invariant is thegenus of a quadratic form. An immediate question is : What is the connectionbetween ¡he topological genus of the mapping cone of f and the algebraic genusof áts Hilton-Hopf quadratic form H(f) ?The present paper is devoted to a study of this question, using the notationof [1j .The notion of genus in the theory of integral quadratic forms has a similardefinition, namely two integral quadratic forras 1 and V of dimension K areof the same genus precisely when for every prime p they are equivalent overZp , the ring of p-adic integers . We recall the definition of equivalence of twoquadratic forms over a ring in its more general form .Definition . Let A be a commutative ring of characteristic 0, so that Z isa subring of A. A A-quadratic form of dimension K is a function ~D : M --> Afrom the free A-module M of rank K to A satisfying <P(Ax) = A 2 $(x) for every326	I . . BoKORA E A and x E M. The two A-quadratic forms 1¿ : M -+ A and V : M' + Aof dimension K are said to be A-equivalent or equivalent over A if there is anisomorphism r : M --> M' such that 41 = V o r .Observe that any integral quadratic form may be considered to be a A-quadratic form . Hence it makes sense to speak of A-equivalente of integralquadratic forms . The definition of the genus of an integral quadratic formconcerns the cases A = Zp , the ring of p-adic integers, p a prime number .We translate this definition into the language of matrices, since we shallbe computing with matrices . Two integral quadratic forms represented bysymmetric integral matrices H and H' are of the same genus if and only if foreach prime p there is an invertible p-adic integral matrix A such thatH = AH'A'.Observe that this algebraic notion of genus is defined in terms of completionwhereas the topological one was defined in terms of localisation. This differencedisappears if only non-singular quadratic forms are considered, for then the ringZp of p-adic integers may be replaced by Z(p ) the ring of p-local integers in thedefinition of the genus of an integral quadratic form (cf . [2]) . (Recall thatZ(p ) denotes the ring of p-local integers, that is the ring of all rational numberswith denominator in reduced form coprime to p.) In this case the similaritiesbetween the two concepts of genus are even more suggestive . Weakening thealgebraic notion of genus slightly makes the connection clearer .Definition. Let A a commutative ring of characteristic 0 . Let two integralquadratic forms be .given, one with the matriz G, the other with the matriz Hin some basis . Then the two forms are said to be weakly A-equivalent if thereare an invertible A-matriz A and m, a unit in A, with mG = AHA'. (Thiscondition is obviously independent of the choice of basis.) If A = Z(p), then wespeak simply of weak p-equivalente . Two non-singular integral quadratic formsare of the same weak genus if they are weakly Z(p)-equivalent for every primep .It is an immediate consequence of the definitions that the weak genus of anintegral quadratic form is an invaxiant of its genus .Before examining examples of quadratic forms of the same weak genus, itshould be observed that we may, without loss of generality, restrict ourselvesto integral computations, for it is possible to "multiply up", as the next lemmamakes clear .Lemma 1 . The integral quadratic forms G and H are Z(p ) -equivalen¡ if andonly if there are an integral matriz A and an integer m with both m and det(A)coprime to p, such thatmG=AHA'Proof. If m' is a unit in Z(p ) and if A' is an invertible Z(p)-matriz such thatm'G = A'H(A')', then let k be the least common multiple of the denominatorsTOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA	327of the entries in A' and the denominator of m' . This k is certainly a unit inZ(n), as is the integer m := k'm' . Moreover A := kA' is an integral matrixinvertible over Z(r), so that its determinant (an integer) is a unit in Z(n) . Butan integer is invertible in Z(p ) if and only if it is coprime to p . Finally, observethatmG = k2m'G = kA'Hk(A')` = AHA' .The weak genus of a quadratic form is frequently non-trivial . In fact thediagonal binary quadratic forms with a pair of distinct primes on the diagonalare always of the same weak genus, , sometimes (but not always) of the samegenus, but never equivalent, as the next results show .Theorem 2. For each pair of distinct prime numbers p and q, the integralquadratic forms with matrices0 q ) and p pq)are of ¡he sane weak genus, but not equivalent .Proof. Sincep(0 ql -_1(0 pq) (0 1p (p o) i othe two forms are weakly Z(,)-equivalent for every prime r 7~ p . Butq (p0 q) -(oq0l/ (0 q) (1 p),so that the two forms are weakly Z(n)-equivalent as well . The two formsare equivalent only if the integral matrix equation(1 0pq)-(c d)(p0q)(abd)has á solution . This is only the case if the equationpa2 + qb2 = 1has integral solutions, which it clearly never does .328	I . BoxonLemma 3. The two qu¢dratic forms(0 41)and(0 82)are of the same genus .Proof.. The equalityC20) _(-)C10)( ó770 41 ' 7 0 82 -shows that the two forms are Z(p)-equivalent for p :~ 7 and the equalityit follows thatso thatBut for any integer x4f-')72~2 0 -109iós ó s 9 ) ~0 82) (10 109009shows that the two forms are Z(7)-equivalent .Lemma 4. The two qu¢dratic formsC~ 5)and(0 5)are not of the s¢me genes .Proof.Let k be an integer and A the 2 x 2 integral matrixCa d)Then from the matrix equationk2(0 105) -(ád) (0 5)(abd)k2 -_ 3a2 + 5b2,k2 - 3a2 (mod 5) .x2 - 0, fl (mod 5),so that k must be divisible by 5. In other wordsC00 ) and ( 0 15 )TOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA	329are not 5-equivalent .Turning to the connection between the topological genus of the mapping coneCf of f : ,son-1 V S2n and the algebraic genus of HM, the Hilton-Hopfquadratic form of f, the homotopy commutative diagraminduces the algebraic equationS4n-1 f~ v S2n9deg(0)H(g) =AMH(f)A(sp) tand V and cp are p-equivalentes if and only if both deg(0) and det(A(cp)) arecoprime to p - in other words, if and only if H(f) and H(g) are weakly p-equivalent . The next theorem summarises these considerations .Theorem 5. The mapping tones Cf and Cy are of the same genus only ifH(f) and H(g) are of ¡he same weak genus .Thus the weak genus of the Hilton-Hopf quadratic form H(f) is an invariantof the genus of Cf, as is the genus of the suspension of Cf . In fact, thesetwo invariants characterise the genus of Cf in the case that the bouquet ofspheres V S2n consists of precisely one sphere, for in that case the Hilton-Hopfquadratic form reduces to the classical Hopf invariant of f which is a singleinteger ; and two integers - integral quadratic forms of dimension 1 - are of thesame weak genus if and only if they agree up to sign . Hence the ClassificationTheorem of [1] can be reformulated asTheorem 6. Two spaces in C,' are of the same genus if and only if(i) their Hilton-Hopf quadratic forms are of ¡he same weak genus, and(ii) their suspensions are of ¡he same genus .This reformulation invites the conjecture that the result generalises to thecase in which the bouquet contains an arbitrary (finite) number of 2n-spheres .Conjecture . Two spaces in Cñ are of the same genus if and only if(i) their Hilton-Hopf quadratic forms are of the same weak genus, and(ii) their suspensions are of the same genus .Of course the "only if' part of the conjecture is certainly true .330	I. BOKORFor K = 0 the conjecture is true because any space in Cñ is homotopyequivalent to S4n, being a simply connected Moore space with the appropri-ate homology. Theorem 6 asserts the conjecture to be true for K = 1 . Theconjecture is alsó clearly true if the torsion subgroup of 7r4n-1(S2n ) is trivial,for then the equation involving the torsion components imposes no additionalconstraint, and every K x K integral matrix can be realised as a self-map ofVS'n,foranym>1.Nevertheless, if K >_ 2 and if the torsion subgroup of 7r4n-1 (S2n) is non-trivial, then counterexamples to the conjecture can be systematically con-structed . Two schemata follow for doing so for n 1 {l, 2,4} when the kernelof the suspension map E : 7r4.-1(VS2n) , 7r4n (VS2n+1) consists precisely ofthose classes [f] whose torsion component is trivial .Counterexample 1 . If the torsion subgroup of 7r4n-1 (S 2n) is non-trivial,then there are mapping cones of maps S4n-1 -~ S2n V S2n with the followingproperties .(i) Their Hilton-Hopf quadratic forms are of the same weak genus .(ii) Their suspensions are homotopy equivalent .(iii) They are not themselves of the same genus .Counterexample 2 . If the torsion subgroup of 7r4n-1 (S2n) is not cyclic,then there are mapping cones of maps S4n-1 , S2n V S2n with the followingproperties .(i) Their Hilton-Hopf quádratic forms are equivalent .(ii) Their suspensions are homotopy equivalent .(iii) They are not themselves of the same genus .Construction of Counterexample 1 . We assume that .T, the torsionsubgroup of 1r4n-1 (S2n ), is non-trivial . Let p be a prime divisor of t, the orderof T . Then T has an element x of order p . Let q be any prime number otherthan p. Finally choose f, g : S4n-1 -+ S2n V SU with the following properties :2p 0LLU) 0 2q) and E(f) G)= ( 2 2pq and E(g) = (0)Then Theorem 2 asserts that H(f) and H(g) are of the same weak genus,verifying (i) .The suspensions are clearly homotopically equivalent - in fact both are ho-motopically equivalent to CE  V S2n+ 1 - which establishes (ii) .Now consider the homotopy commutative diagramS4n-1 f ) V S2nS4n-1 V S2nTOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA	33 1(wherez~ has degree m and A(~p) = a bc d)It follows from the equationthatmH(g) = A(W)H(f)(A(so)'mpq = pcz + qdz,so that p divides d . Moreover, the equationA(w)Ef = MEgshows that ex = 0, so that p, which is the order of x, also divides c . But thenp divides det(A(cp)) as well, so that for no such diagram is W a p-equivalence .This establishes (iii) thereby establishing the first family of counterexamples tothe conjecture .Construction of Counterexample 2 . For the second construction weconsider the case when the torsion subgropup T of 7r4n_1(S2n ) is not cyclic . Inthat case it has a subgroup G isomorphic, to Z/pZ ® Z/pZ for some prime pWe identify G with Z/pZ ® Z/pZ .Choose an integer r with 0 < r < p. Then there are integers s and t withrt - ps = -1. We may assume without any loss of generality that 0 < t < p,for r(t + kp) - sp = rt - (s - rk)p . From this it also follows that 0 < s < p .PutThen clearly ( E GL(2; Z), with inverseWriting the elements of G as rows means that the rows of (-' can be vievued aselements of G, and hence taken to represent homotopy classes of maps Son-' -bSzn . The first row corresponds to the element (-t, 0) of G and the secondto (s, -r) . If, furthermore, the elements of Gz are written as columns withelements of G as entries, then we may regard (-' as an element v of Gz andso as representing a homotopy class of maps Son-' , s2n V SI, . put y :_ (o .Then y is the 2 x 2 unit matrix 1 viewed as an element of Gz and ( defines anautomorphism of Gz mapping x to y. Now Choose f, g : son-i -, szn V Sznwith the following propertiesH(f) =( 0 2 )and E(f ) _ ( st p)H(9) _ (~ ~) and E(9) =(v~)332	I. BOKORThe Hilton-Hopf quadratic forms H(f) and H(g) are identical .Equally, the suspensions ECf and ECy are homotopy equivalent . For exam-ple, any self-map (p of S2n+1 V S2n+1 with A(cp) --. ( together with the identitymap of S4n, defines a suitable homotopy equivalence from ECf to ECg .Now consider the homotopy commutative diagramthatso thatS4n-1 f , V S2n01 1 ;ps4n-1 ) V s2nwhere ~, has degree m and A(cp) = ab(e dIt follows fromMH(g) = il(P) .( .f )(A(~P)) tA(cp)(A(cp))' = ml,( a b`p_- -ab Aawith \2 = 1 .A fürther direct computation using Oy = cpx shows thatap - rb = a(bt -}- as) = 0 (modp),from which it follows successively that p divides both b and a, since r, s and Aare all coprime to p . Thus co cannot be a p-equivalence, which establishes theconstruction for the second family of counterexamples .Remarks. These counterexamples also dispose of a number of possible re-finements of the conjecture, for the spaces constructed satisfy more stringentrequirements than set out in the conjecture . The second recipe provides amethod for constructing counterexamples with the Hilton-Hopf forms actuallycoinciding and the suspensions being homotopically equivalent . All that isneeded for this is that the torsion subgrcup of 7r4n-1(S2n ) not be cyclic, as isthe case, for example, for n = 5, 8, 9 . It should however be noted that in noneof the counterexamples constructed does the Hilton-Hopf quadratic form havedeterminant 1, so that none of the spaces is a manifold.It remains to be seen whether the conjecture be true for manifolds and alsoto find some invariant, additional to (i) and (ii) df the discredited conjecture,which will guarantee that Cf and Cg are of the same genus' .TOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA	333References1 . BOKOR, I ., "Genes and cancellation in homotopy theory," (to be pub-lished) .2. CASSELS, J.W., "Rational quadratic forms," London, Academic Press,1978 .3. FREYD, P ., Stable homotopy II, AMS Proc. of Symp . in Pure Math . 18(1970),161-183 .4. HILTON, P.J ., The grothendiek group of compact polyhedra, FundamentaMathematicw 61 (1967), 199-214.5. MOLNAR, E ., Relation between wedge cancellation and localization forcomplexes with two cells, J. of P. and Appl . Alg. 3 (1972), 77-81 .6. WILKERSON, C., Genus and cancellation, Topology 14 (1975), 29-36 .75 Nount StCoogee 2034AUSTRALIARebut el 17 de Desembre de 1989

On the connection between the topological genus of certain polyhedra and the algebraic genus of their Hilton-Hopf quadratic forms

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On the connection between the topological genus of certain polyhedra and the algebraic genus of their Hilton-Hopf quadratic forms

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