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

Publicacions Ma emá iques, Vol 34 (1990), 323-333 . ON THE CONNECTION BETWEEN THE TOPOLOGICAL GENUS OF CERTAIN POLYHEDRA AND THE ALGEBRAIC GENUS OF THEIRHILTON-HOPF QUADRATIC FORMS Abs ac IMRE BOKOR The Hil on-Hop quad a ic o m is de ined o spaces o he homo opy ype o a CW complex wi h one cell each in dimensions 0 and 4n, K cells in dimension 2n and no o he cells . I wo such spaces a e o he same opological genus, hen hei Hil on-Hop quad a ic o ms a e o he same weak algeb aic genus . Fo la ge classes o spaces, such as simply connec ed di e en iable 4-mani olds, he con e se is also ue, as long as he suspensions o he spaces a e also o he same opological genus . This no e allays he conjec u e ha he con e se is ue in gene al by o e ing wo echniques o gene a ing in ini e amilies o coun e examples . I was shown in [1] ha o C,', he class o hose spaces which a e o he same homo opy ype as a CW-complex wi h p ecisely one cell in each o he dimensions 0, 2n and 4n, he e is a Fac o isa ion Theo em wi h espec o he one-poin union (o V-p oduc ) o spaces . The heo em s a es ha as long as he a aching map o he 4n-cell ep esen s a homo opy class o in ini e o de in 7 4n-1(S2n), he homo opy ype o a ini e wedge o such spaces de e mines he homo opy ype o each o he spaces in he wedge . This s ands in con as o he si ua ion whe e he a aching map is o ini e o de , o hen non-cancella ion phenomena occu wi h spaces o he same "genus", as shown in [31, [4], [5] and [6] . The p oo o he Fac o isa ion Theo em applied echniques which sugges a gene alisa ion o he cha ac e isa ion o he genus o a space in C,' o Cñ , he class o spaces o he homo opy ype o a CW-complex wi h one cell in dimensions 0 and 4n and a ini e numbe K o cells in dimension 2n . This la ge class o spaces includes all (2n - 1)-connec ed di e en iable 4n-mani o1ds . Up o homo opy, he spaces in Ch a e mapping cones C o con inuous maps K ; S4n-1  V s2n k=1 324  1 . BOKOR K and hence a e classi ied by 7 4n-1( V S 2n ) . k=1 K We ix K and abb e ia e V S2n o V S2n . k=1 Fo maps , g : Son-1 --> V S2n he se o homo opy classes o con inuous maps be ween hei mapping ones C and C gis in bijec ion wi h he se o homo opy classes o homo opy commu a i e diag ams whe e and S4n-1  i S2n 9 w The se o homo opy classes o maps V S2n --, V S 2n admi s a ing s uc u e isomo phic o M(K ; Z), he ing o K x K in eg al ma ices . The unc ion assigning o each sel -map cp o S 2 n i s deg ee induces one such isomo phism, by mapping cp o he ma ix A(cp) whose (i, j)- h eoe lcien is he deg ee o he composi e map qi o (p oi dj : SU-) V S2n -a V S2n -~ S2n, 2nj : S2n -4 V S 2 n is he . j- h canonical inclusion in he co-p oduc and qi S 2n -> S2n collapses each summand o he base poin excep he i- h one, on which i ac s as he iden i y map . I was shown in [1] ha each homo opy class [ ] E 7 4n-1(V S 2n ) can be ep esen ed by a pai consis ing o he Hil on - Hop quad a ie o m o and he suspension o . The Hil on-Hop quad a ic o m can i sel be ep esen ed as a symme ic in eg al K x K ma ix, HM, whose en ies a e p ecisely he Hil on- Hop in a ian s o , and he suspension by E , a "column o sion ec o " each o whose componen s is an elemen 7 4n(S2n+1 ) . A sui able choice o gene a o s o 7 4n-1(V S2n) p o ides a simple way o compu ing cp o om co and by means o a ma ix calculus : H( , P o ) = A(~P)H( . )(A(~P)) E(cp o . ) = AME( ) ) . and TOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA  325 Thus H( ) is a quad a ic o m which is an in a ian o o ien ed homo opy ype and he assignmen o each E 7 4n-1 ( S 2 n) o i s Hil on-Hop quad a ic o m - ha is, he quad a ic o m wi h ma ix H( ) in he app op ia e basis - is na u al in sel -maps o V SU . (Re an ha he Hil on-Hop quad a ic o m can also be hough o as he in e sec ion o m in he in eg al cohomology o C .) The genus o a nilpo en space - mo e p ecisely, he genus o he homo opy ype o a CW-complex o ini e ype - is de ined o be he se o homo opy ypes o hose nilpo en spaces, each o whose p-localisa ions is homo opy equi alen o he p-localisa ion o he o iginal space . Since we a e only conce ned wi h mapping cones C o maps : Son-1 _> V S2n, i ollows om Lemma 2 .2 o [1] ha he wo spaces C and C 9 a e o he same genus i and only i o each p ime p he e is a homo opy commu a i e diag am s4n-1 i 2n -i s 2n N s4n-1  --------- 4 VS2n wi h bo h he deg ee o 0 and he de e minan o he ma ix p . In e ms o he ma ix no a ion de eloped, his diag am equa ions deg(O)H(g) =A(So)H( )(A(w))` deg(O)ag = A(~p)u , o cp cop ime o is equi alen o wi h deg(0) and de A(cp) cop ime o p, whe e de A is he de e minan o he ma ix A . Since he assignmen o his quad a ic o m is na u al, each o i s in a ian s is, in ac , an in a ian o he mapping cone o . One such in a ian is he genus o a quad a ic o m . An immedia e ques ion is : Wha is he connec ion be ween ¡he opological genus o he mapping cone o and he algeb aic genus o á s Hil on-Hop quad a ic o m H( ) ? The p esen pape is de o ed o a s udy o his ques ion, using he no a ion o [1j . The no ion o genus in he heo y o in eg al quad a ic o ms has a simila de ini ion, namely wo in eg al quad a ic o as 1 and V o dimension K a e o he same genus p ecisely when o e e y p ime p hey a e equi alen o e Z p , he ing o p-adic in ege s . We ecall he de ini ion o equi alence o wo quad a ic o ms o e a ing in i s mo e gene al o m . De ini ion . Le A be a commu a i e ing o cha ac e is ic 0, so ha Z is a sub ing o A . A A-quad a ic o m o dimension K is a unc ion ~D : M --> A om he ee A-module M o ank K o A sa is ying <P(Ax) = A 2 $(x) o e e y 326  I . . BoKOR A E A and x E M . The wo A-quad a ic o ms 1¿ : M -+ A and V : M'  + A o dimension K a e said o be A-equi alen o equi alen o e A i he e is an isomo phism : M --> M' such ha 41 = V o . Obse e ha any in eg al quad a ic o m may be conside ed o be a A- quad a ic o m . Hence i makes sense o speak o A-equi alen e o in eg al quad a ic o ms . The de ini ion o he genus o an in eg al quad a ic o m conce ns he cases A=Z p , he ing o p-adic in ege s, p a p ime numbe . We ansla e his de ini ion in o he language o ma ices, since we shall be compu ing wi h ma ices . Two in eg al quad a ic o ms ep esen ed by symme ic in eg al ma ices H and H' a e o he same genus i and only i o each p ime p he e is an in e ible p-adic in eg al ma ix A such ha H = AH'A' . Obse e ha his algeb aic no ion o genus is de ined in e ms o comple ion whe eas he opological one was de ined in e ms o localisa ion . This di e ence disappea s i only non-singula quad a ic o ms a e conside ed, o hen he ing Z p o p-adic in ege s may be eplaced by Z (p ) he ing o p-local in ege s in he de ini ion o he genus o an in eg al quad a ic o m (c . [2]) . (Recall ha Z( p ) deno es he ing o p-local in ege s, ha is he ing o all a ional numbe s wi h denomina o in educed o m cop ime o p .) In his case he simila i ies be ween he wo concep s o genus a e e en mo e sugges i e . Weakening he algeb aic no ion o genus sligh ly makes he connec ion clea e . De ini ion . Le A a commu a i e ing o cha ac e is ic 0 . Le wo in eg al quad a ic o ms be . gi en, one wi h he ma iz G, he o he wi h he ma iz H in some basis . Then he wo o ms a e said o be weakly A-equi alen i he e a e an in e ible A-ma iz A and m, a uni in A, wi h mG = AHA' . (This condi ion is ob iously independen o he choice o basis .) I A = Z(p), hen we speak simply o weak p-equi alen e . Two non-singula in eg al quad a ic o ms a e o he same weak genus i hey a e weakly Z(p)-equi alen o e e y p ime p . I is an immedia e consequence o he de ini ions ha he weak genus o an in eg al quad a ic o m is an in axian o i s genus . Be o e examining examples o quad a ic o ms o he same weak genus, i should be obse ed ha we may, wi hou loss o gene ali y, es ic ou sel es o in eg al compu a ions, o i is possible o "mul iply up", as he nex lemma makes clea . Lemma 1 .  The in eg al quad a ic o ms G and H a e Z( p ) -equi alen¡ i and only i he e a e an in eg al ma iz A and an in ege m wi h bo h m and de (A) cop ime o p, such ha mG=AHA' P oo .. I m' is a uni in Z( p ) and i A' is an in e ible Z(p)-ma iz such ha m'G= A'H(A')', hen le k be he leas common mul iple o he denomina o s TOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA  327 o he en ies in A' and he denomina o o m' . This k is ce ainly a uni in Z(n), as is he in ege m := k'm' . Mo eo e A := kA' is an in eg al ma ix in e ible o e Z( ), so ha i s de e minan (an in ege ) is a uni in Z(n) . Bu an in ege is in e ible in Z( p ) i and only i i is cop ime o p . Finally, obse e ha mG = k 2 m'G = kA'Hk(A')` = AHA' . The weak genus o a quad a ic o m is equen ly non- i ial . In ac he diagonal bina y quad a ic o ms wi h a pai o dis inc p imes on he diagonal a e always o he same weak genus, , some imes (bu no always) o he same genus, bu ne e equi alen , as he nex esul s show . Theo em 2 . Fo each pai o dis inc p ime numbe s p and q, he in eg al quad a ic o ms wi h ma ices 0  q )  and  p  p q ) a e o ¡he sane weak genus, bu no equi alen . P oo .. Since p(0  ql -_  1  (0 pq) (0  1 p  (p o ) i o he wo o ms a e weakly Z(,)-equi alen o e e y p ime 7~ p . Bu q (p 0  q) - (oq  0l/ (0  q) ( 1  p) , so ha he wo o ms a e weakly Z(n)-equi alen as well . The wo o ms a e equi alen only i he in eg al ma ix equa ion ( 1  0 pq) - (c d) (p0 q) (ab d) has á solu ion . This is only he case i he equa ion pa 2 + qb 2 = 1 has in eg al solu ions, which i clea ly ne e does . 328  I . Boxon Lemma 3 . The wo qu¢d a ic o ms (0 41) and (0 82) a e o he same genus . P oo . . The equali y C2 0) _(-)C1  0)( ó 77 0 41  ' 7 0 82 - shows ha he wo o ms a e Z(p)-equi alen o p :~ 7 and he equali y i ollows ha so ha Bu o any in ege x 4 -') 7 2 ~2 0 - 109 iós   ó s 9 ) ~ 0  82) ( 10  109 0  09 shows ha he wo o ms a e Z( 7 )-equi alen . Lemma 4 . The wo qu¢d a ic o ms C~ 5) and (0 5) a e no o he s¢me genes . P oo .Le kbe an in ege and A he 2 x2 in eg al ma ix Ca d) Then om he ma ix equa ion k2 (0 1 0 5) - (á d) (0 5) (ab d) k 2 -_ 3a 2 + 5b2, k 2 - 3a 2  (mod 5) . x 2 - 0, l  (mod 5), so ha kmus be di isible by 5 . In o he wo ds C0 0 ) and ( 0 15 ) TOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA  32 9 a e no 5-equi alen . Tu ning o he connec ion be ween he opological genus o he mapping cone C o : ,son-1 V S2n and he algeb aic genus o HM, he Hil on-Hop quad a ic o m o , he homo opy commu a i e diag am induces he algeb aic equa ion S4n-1  ~ S2n 9 deg(0)H(g) =AMH( )A(sp) and V and cp a e p-equi alen es i and only i bo h deg(0) and de (A(cp)) a e cop ime o p - in o he wo ds, i and only i H( ) and H(g) a e weakly p- equi alen . The nex heo em summa ises hese conside a ions . Theo em 5 . The mapping ones C and C y a e o he same genus only i H( ) and H(g) a e o ¡he same weak genus . Thus he weak genus o he Hil on-Hop quad a ic o m H( ) is an in a ian o he genus o C , as is he genus o he suspension o C . In ac , hese wo in a ian s cha ac e ise he genus o C in he case ha he bouque o sphe es V S 2 n consis s o p ecisely one sphe e, o in ha case he Hil on-Hop quad a ic o m educes o he classical Hop in a ian o which is a single in ege ; and wo in ege s - in eg al quad a ic o ms o dimension 1 - a e o he sameweak genus i and only i hey ag ee up o sign . Hence he Classi ica ion Theo em o [1] can be e o mula ed as Theo em 6 . Two spaces in C,' a e o he same genus i and only i (i) hei Hil on-Hop quad a ic o ms a e o ¡he same weak genus, and (ii) hei suspensions a e o ¡he same genus . This e o mula ion in i es he conjec u e ha he esul gene alises o he case in which he bouque con ains an a bi a y ( ini e) numbe o 2n-sphe es . Conjec u e . Two spaces in Cñ a e o he same genus i and only i (i) hei Hil on-Hop quad a ic o ms a e o he same weak genus, and (ii) hei suspensions a e o he same genus . O cou se he "only i ' pa o he conjec u e is ce ainly ue . 33 0  I . BOKOR Fo K = 0 he conjec u e is ue because any space in Cñ is homo opy equi alen o S4n, being a simply connec ed Moo e space wi h he app op i- a e homology . Theo em 6 asse s he conjec u e o be ue o K = 1 . The conjec u e is alsó clea ly ue i he o sion subg oup o 7 4 n-1(S 2n ) is i ial, o hen he equa ion in ol ing he o sion componen s imposes no addi ional cons ain , and e e y K x K in eg al ma ix can be ealised as a sel -map o VS' n , o anym>1 . Ne e heless, i K >_ 2 and i he o sion subg oup o 7 4n-1 (S2n) is non- i ial, hen coun e examples o he conjec u e can be sys ema ically con- s uc ed . Two schema a ollow o doing so o n1 {l, 2,4} when he ke nel o he suspension map E : 7 4 .-1(V S2n) , 7 4n ( V S2n+1) consis s p ecisely o hose classes [ ] whose o sion componen is i ial . Coun e example 1 . I he o sion subg oup o 7 4n-1 (S 2 n) i s non- i ial, hen he e a e mapping cones o maps S4n-1 -~ S2n V S2n wi h he ollowing p ope ies . (i) Thei Hil on-Hop quad a ic o ms a e o he same weak genus . (ii) Thei suspensions a e homo opy equi alen . (iii) They a e no hemsel es o he same genus . Coun e example 2 . I he o sion subg oup o 7 4n-1 (S 2 n) is no cyclic, hen he e a e mapping cones o maps S4n-1 , S2n V S2n wi h he ollowing p ope ies . (i) Thei Hil on-Hop quád a ic o ms a e equi alen . (ii) Thei suspensions a e homo opy equi alen . (iii) They a e no hemsel es o he same genus . Cons uc ion o Coun e example 1 . We assume ha .T, he o sion subg oup o 1 4 n -1 (S 2n ), is non- i ial . Le p be a p ime di iso o , he o de o T . Then T has an elemen x o o de p . Le qbe any p ime numbe o he han p . Finally choose , g : S4n-1 -+ S2n V SU wi h he ollowing p ope ies : 2p 0 LLU)  0  2q) and E( )  G) = ( 2  2pq  and E(g) = (0) Then Theo em 2 asse s ha H( ) and H(g) a e o he same weak genus, e i ying (i) . The suspensions a e clea ly homo opically equi alen - in ac bo h a e ho- mo opically equi alen o CE  V S 2 n+ 1 - which es ablishes (ii) . Now conside he homo opy commu a i e diag am S4n-1   ) V S2n S4n-1 V S2n TOPOLOGICAL AND ALGEBRAIC GENUS OF POLYHEDRA  33 1 (whe e z~ has deg ee m and A(~p) =  a  b c d) I ollows om he equa ion ha mH(g) = A(W)H( )(A(so)' mpq = pcz + qdz, so ha p di ides d . Mo eo e , he equa ion A(w)E = MEg shows ha ex = 0, so ha p, which is he o de o x, also di ides c . Bu hen p di ides de (A(cp)) as well, so ha o no such diag am is W a p-equi alence . This es ablishes (iii) he eby es ablishing he i s amily o coun e examples o he conjec u e . Cons uc ion o Coun e example 2 . Fo he second cons uc ion we conside he case when he o sion subg opup T o 7 4n_1(S 2n ) is no cyclic . In ha case i has a subg oup G isomo phic , o Z/pZ ® Z/pZ o some p ime p We iden i y G wi h Z/pZ ® Z/pZ . Choose an in ege wi h 0 < < p . Then he e a e in ege s s and wi h - ps = -1 . We may assume wi hou any loss o gene ali y ha 0 < < p, o ( + kp) - sp = - (s - k)p . F om his i also ollows ha 0 < s < p . Pu Then clea ly ( E GL(2 ; Z), wi h in e se W i ing he elemen s o G as ows means ha he ows o (-' can be ie ued as elemen s o G, and hence aken o ep esen homo opy classes o maps Son-' -b Szn .  The i s ow co esponds o he elemen (- , 0) o G and he second o (s, - ) .  I , u he mo e, he elemen s o Gz a e w i en as columns wi h elemen s o G as en ies, hen we may ega d (-' as an elemen o Gz and so as ep esen ing a homo opy class o maps Son-' , s2n V SI, . pu y :_ (o . Then y is he 2 x 2 uni ma ix 1 iewed as an elemen o Gz and ( de ines an au omo phism o Gz mapping x o y . Now Choose , g : son-i -, szn V Szn wi h he ollowing p ope ies H( ) = ( 0  2 )  and E( ) _ ( s  p ) H(9) _ (~  ~) and E(9) = (  ~)