On the intersection forms of closed 4-manifolds

Publicacions Ma emá iques, Vol 36 (1992), 73-83 . Abs ac ON THE INTERSECTION FORMS OF CLOSED 4-MANIFOLDS ALBERTO CAVICCHIOLI AND FRIEDRICH HEGENBARTH Gi en a closed 4-mani old M, le M* be he simply-connec ed 4-mani old ob ained om M by killing he undamen al g oup . We s udy he ela ion be ween he in e sec ion o ms A M and AM- . Finally some opological consequences and examples a e desc ibed . 1 . In oduc ion . Le M 4 be a closed connec ed o ien able (PL) 4-mani old wi h unda- men al g oup II1 . Deno e by A M he in e sec ion o m o M Am : FH2 (M) x FH2 (M)  -  . Z whe e FH2 (M) = H2 (M ; Z)/ o sion (see o example [5], [10]) . Le M* be he simply-connec ed closed 4-mani old ob ained om M by killing he undamen al g oup II, (see [6]) . Ou pu pose is o s udy wha ela ion links M o AM . . Thenwe ob ain some opological consequences abou M* . Finally we gi e some examples which illus a e he esul s . 2 . Main esul s . Le [a] be a gene a o o II 1 . Since M is o ien able, we can ex end e¿ : S i - M o an embedding 0 : S 1 x D 3 , M . Recall ha he e a e wo ways o ex end a since II1(SO(3)) = Z2 . Wo k pe o med unde he auspicies o he G .N .S .A .G .A . o he C .N .R . and inancially suppo ed by he Minis e o della Rice ca Scien i ica e Tecnologica o I aly wi hin he p ojec "Geome ía Reale e Complessa" 74  A . CAVICCI - IIOLI, F . HEGEN13ARTFI 0 Deno e by M' = M O(S I x D 3 ) U D 2 x S 2 he closed 4-mani old ob ained om M by su ge y on 0 . Since III (M') - 111(M)1[o¿], i e a ed su ge ies on gene a o s o III (M) gi e a simply-connec ed closed 4-mani old M* . P oblem . S udy he ela ions be ween AM, A M , and AM, A M . espec- i ely . Fi s we ha e he ollowing P oposi ion 1 . I nI (M) has no elemen s o ini e o de , hen A M . i s isomo phic o e he in ege s o AM . The p oo is gi en o example in [1] . The e o e om now on we will conside mani olds wi h lI I (M) ini e . P oposi ion 2 . I [a] has ini e o de , hen o some in ege a EZ . In any case A M , is inde ini e . Fo he e o ms he e is he ollowing well-known classi ica ion : (1)  0  1/ A M ,  e en  A M , = pE 8 ® q  1  0 2)  A M ,  odd ==>  A M , - p(1) ®q( -1 ) o some non nega i e in ege s p, qE Z . Fu he mo e, S . K . Donaldson (see [2]) p o ed he ollowing Theo em 3 . Le M 4 be a closed connec ed o ien able 4-mani old wi h a bi a y undamen al g oup . I AM is de ini e, hen AM is isomo phic o e he in ege s o ei he (1) ® . . . ® (1) o (-1) ® . . . ® (-1) . The pa i y o Am is ela ed o he second S ie el-Whi ney class 0 Am ®( 1 1) a -e en 0 a) - 1 AM e (o 0 - 1) a odd M . (1 m . - AM ®  1 0  0 1 INTERSCCTION ORMS o 4-MAN OLDS  75 w2 (M) E H 2 (M ; Z2) as ollows . Using he uni e sal coe icien sequence 0 -> Ex (Hi(M) ; Z2) - H 2 (M ; Z2) -> Hom(H2(M) ; Z2) - 0, i is easily p o ed ha AM is e en i and only i w2 (M) E Ex (H1(M) ; Z2) . In pa icula , i Hl (M) has no 2- o sion, hen AM is e en i and only i w2 (M) = 0 . Thus p oposi ion 2 implies he ollowing P oposi ion 4 . I w2 (M) =~ 0 , hen 1 m -  m (D p  1  0  ) 5 - -- (1) ED S (0 -1 o some non nega i e in ege s p, , s E Z . Fu he , M* is homeomo phic o he connec ed sum (CP~#s(-CP~, being CP 2 he p ojec i e complex plane . Now we can also apply heo em 2 o [2] o ob ain he ollowing conse- quence o p oposi ion 2 . Co olla y 5 . Le M 4 be a closed connec ed o ien able spin 4-mani old wi h undamen al g oup II (M) - Z  ,, . I AM has a posi i e pa í o ank 1, hen M* is homeomo phic o ei he 2(CP 2 )#(2 - (M)) (-CP 2 ) o 2(S 2 x S 2 ) . In he las case, AM =  o  ~) . He e u(M) deno es he signa u e o P oo . .- By p oposi ion 2, we ha e ei he AM . = AM ®  0  1 ) o In he i s case, AM . i s e en . Since Hl(M*) = 0 has no 2- o sion, heo em 2 o [2] implies ha ~M " - ~0 10/ ® CO hence AM . has a posi i e pa o ank 2 . ól - Am E)  o  i) 1 hence AM =  0  1 ) (see [7J, [9]) and M* TOp 2(5 2 x S 2 ) as equi ed . In he second case, Am . = 2(1) ® (2 - u(M))(-1), hence M* Tóp 2(CP 2 )#(2 - a(M))(-CP2) . 76 A . CAVICCHIOLI, F . HECENBARTH 3 . Examples . 3 .1) Le K = ` {zó + zi + z2 + z3 = 0} CCP 3 be he Kumme su ace and le T : CP 3 -> CP 3 be he ixed poin ee in olu ion de ined by T(zo, Z1, z2, z3) = (zl, - zo, z3, - z2) Since T (K) = K, we can conside he o bi space M = K/T, called he Habegge mani old (see [4]) . I is known ha I1 1 (M) = Z2 and he in e sec ion o m ( 0 1) Ana _ (-E8) ®  1  00 is e en wi h a posi i e pa o ank 1 . Since w2(M) z,¿ 0, p oposi ion 2 gi es A M . - ( - E8) ® ( 1  0 ) ® ( 0  0 1  - 10(-1) ® 2(1), -) hence M* - 10(-CP 2 )#2(CP 2 ) by he F eedman classi ica ion (see TOP (3l) . We also ecall ha C . Okonek (see [8}) has shown ha all homo opy En iques su aces a e ho neomo phic o he Habegge mani old . 3 .2) Le M' = S( 7 ® 71 (D 17) be he sphe e bundle o 77 ® 77 ® 71, whe e ---> RP 2 is he canonical bundle o e he eal p ojec i e 2-space . Thenwe ha e Am = 0, w2 (M) ~ 0and II1(M) l- - - Z2, hence and M* - CP 2 #(-CP 2 ) = S 2 x S 2 . TOP  TOP i 3 .3) Le M` 1 = S(,, ® E 2 ) be he sphe e bundle o 77 ® E 2 , whe e E 2 = E l ® e l ---> RP 2 is he 2-dimensional i ial bundle o e RP 2 . Thenwe ha e AM = 0, w2(M) = 0 and II1(M) = Z2 . I is e y-easy o see ha H 2 (M ; Z2) --' H 2 (Mo ; Z2) - H 2 (1VI* ; Z2) ,so  ¡so 0 whe e Mo = M 0(S I xD3), V : S 1 x D 3 - M ep esen s he gene a o o II1(M) and i : Mo -+ M, i' : Mo - M* a e he na u al inclusions . Thus w2 (M*) = 0, hence A M * = ( 0  1 ) is e en and M* TóP S 2 x S 2 . 4 . P oo s . INTGRSGCTION FORMS OF 4-MANIFOLDS  77 P oo o p oposi ion 2 : Fo con enien e we assume ha II I (M) Z,,,m > 0, wi h gene a o [a) = [VIS~xol . Fo he gene al case, see ema k 1 below . 0 We se Mo = M O(S I x D 3 ) and conside he cobo dism W=MxIUOD 2 xD 3 (I=[0,1]) be ween M and M' = Mo U D 2 x S 2 . Ob iously he pai s (W M), (W, M') a e homology equi alen o (D 2 x D 3 , S I x D 3 ) and (D 2 x D 3 , D 2 x S 2 ) espec i ely . We ha e he ollowing diag am 0 --~ H3 (M, Mo) ~-_ Z ---~  H2 (M0)  -~  H2 (M)  - 0 ¡so 0  H3 (W M') -Z --~  H2 (M')  ---,  H 2 (W)  ---, 0 Z = H2 (M, Mo) -~ H2 (W M) HI(Mo) '--- H, (M) -- Z,,, 0 whe e i, i', j, k a e inclusions . Ob iously H2 (M') is a ee g oup o ank kH2 (M) + 2and H2 (M o ) is ee o ank kH2(M) + 1 since i injec s in o H2 (M) . He e we o en iden i y an elemen o H2(M0) wi h i s image unde i* . Now we ha e AM (z* (u), a . ( )) = Am , (z* (u), z* ( )) o e e y u, E H2(Mo) . Le e E H2 (Mo) be a p imi i e elemen such ha ¡ * (e) gene a es he subg oup To H2(M) - Z, ;, and suppose ha E H2 (M') maps o he in ege m E Z - H2(M',Mo) . Simila ly is chosen o be p imi i e . Fu he mo e, deno e by V he span o {e, } in H2 (M') . 78 Lemma 6 . Wi h he abo e no a ion, we ha e ~- _ 0 1 A M , 1  1  a whe e AM , ( , ) = a EZ . A . CAVICCIIIOLI, F . HI3GENBARTI - 1 P oo .- F om he diag am, i ollows ha (1)  Am,(a*(x),y) = Aw(x,j*(y)) o e e y x E H3 (W M') and y E H2 (M) . No e ha and a* [D 2 x D 3 , D 2 x S 2 ] = mi ; (e) = me j* ( ) = m[D2 whe e [, ] deno es he undamen al class . Thus ela ion (1) implies hence AM , (e, ) = 1 as equi ed . Fu he mo e, we ha e D 3 , S I x D 3 ], . Am , (me, ) _ A m , (a* [D 2 x D 3 , D2 x S 2 ], . ) = mAw([D 2 x D 3 , D 2 x S 2 ], [D 2 x D 3 , S l x D 3 ]) = m, m 2 ñM , (e, e) = AM , (me, me) = A M , (a* [D 2 x D 3 , D 2 x S 2 ], a* [D 2 x D 3 , D 2 x S 2 ]) = Aw([D 2 x D 3 , D 2 x S 2 ], j* o a*[D 2 x D 3 , D 2 x S 2]) = 0 since j * o & ; = 0 by he exac ness . Thus AM , (e, e) = 0 and he p oo o Lemma 6 is comple ed . Lemma 7 . Le V 1 C H2(M') be he o lzoyonal complemen o V . Then V L C H2 (Mo) and he es i ion is an isomo phism . ¡ * ¡ i- : V 1 --, FH2(M) P oo .. To p o e ha V -L C H2(Mo), we ha e o show ha o e e y y E H 2 (M') wi h AM4, e) = AM , (y, . ) = 0, hen y E H2(Mo) , i . e . j . (y) =0 . Suppose, on he con a y, j . (y) 7~ 0, i . e . j . (y) = q[D 2 x D 3 , S 1 x D 3 ] o some in ege q 7L 0 . Thenwe ha e AM , (me, y) =Am , (a* [D 2 x D 3 , D2 x S 2 ], y) = Aw([D 2 x D 3 , D 2 x S2],j .(y)) = qAw([D 2 x D 3 , D 2 xS 2 ], [D 2 x D 3 , S 1 x D 3]) = q 7~ 0, hence AM , (e, y) ~¿ 0, whicli is a con adic ion . To p o e ha i .1  L is mono, le x E V 1 be an elemen such ha i, (x) E To H 2 (M) -Z, . Thenwe ha e i, (x) = hi . (e) o some in ege h, and so i . (x - he) = 0 . By he exac ness, i ollows ha hence mh'e = x - he, h, h' E Z . Bu we ha e (use (1)) ( 2 ) AM,(a'(h'[D2 x D 3 ,D 2 x S 2 ]), ) = Aw(h'[D 2 x D 3 , D 2 xS2  ( » = Aw(h [D2 x D 3 , D 2 . x S 2 ], m[D 2 x D 3 , S' x D 3 ]) = ¿h' and INTLIISC :CTION ORMS o 4-MANIFOLDS  79 a'(h'[D 2 x,D 3 , D 2 x S 2 ]) = i ; ( .x - he), Am, (a'(h'[D2 x D 3 ,D 2 x S 2]), ) =AM,(2 :(x-he), ) = AM, (x - he, ) = AM, (x, ) - hAM, (e, ) _ -h . Compa ing ela ions (2) and (3) gi es mh' = -h, hence mh'e = x- he implies ha x = 0 as equi ed . To p o e ha i, l l is epi, le z E H2 (M) and le u E H2 (Mo) be an elemen such ha ¡ * (u) = z . We conside he elemen u' = u -AM , (u, ) e E H2(Mo) . Thenwe ha e AM , (me, u') = AM , (a' [D 2 x D 3 , D 2 x S 2 ], u , ) since j, o i ; = 0 ; he e o e A m , (u', e) = 0 . 80  A . CAVCCI-110L], F . HECLNBARTI-1 Fu he i io e i . e . U' =- ¡' * (U') E V' . Finally This comple es he p oo . By Lemmas 6 and 7, we ha e he esul P oo o P oposi ion 4 : Suppose now zu2(M) =,,' : 0 . Because (M, Mo) and (M', Mo) a e hompl- ogy equi alen o (SI xD3, SI x S 2 ) and (D 2 x S2, SI x S 2 ) espec i ely, we ha e also he diag am which implies Am, (u', . ) = AM, (u - Am, (u, ) e, ) = AM , (U, . ) - AM , (u, ) = 0, ¡ .(u') = ¡,(u) - Am, (u, )z* (e) = ¡ .(u) = z mod To H2 (M) . 0 i , . H 2 (M' ; Z2) AM,-AM® 0 1 (1 a) . H 2 (Mo ; Z2) ---- H 2 (M ; Z2) - 0 i*(za2(M)) = W2 (M0) = i' * (7112(M')) . Since i* is injec i e, ela ion (4) and w2 (M) =~ 0 gi e W2 (M) :~ 0, hence A M , is odd . E Rema k 1 . Th< ; p oo o p oposi ion 2 can be easily gene alized o mani olds wi h a bi a y undamen al g oups . Indeed, his ollows om Lemma 8 below . Suppose now M a closed connec ed o ien able (PL) 4-mani old wi h undamen al g oup Ii . and Le be dis . ioin embeddings which kill 11 . Se ing we ha e Il 'FERs CTION o1ZMs oí . , 4-MANl ol .»s  81 01,02, . . .,Op :S 1 xD 3 >M Mo = M UOj(S 1 x D 3 ) j=1 p M* = Mo U U(D 2 x S 2 ), j=1 Lem na 8 . (1) H,(Mo) = H, (M), H3 (M0) = ®Z p-1 (2) H2(M o ) is o, di ec summand o he ee . g oup H2 (M*) (3) 0 ---, H2 (M0) -+ H2 (M*) - H2 (M*, MO) = (DZ -~ p H2 (M) = H2 (M0) = H2 (M * ) H l (Mo) = Hl (M) - 0 0 ---, H 3 (M)  , H3 (M, Mo) = (1) Z - H2 (Mo) ---, H 2 (M) ---, 0 p Hl (M) - H 3 (M) - H 3 (M, Mo) - ®Z . p The p oo is s aigh o wa d . Now we indica e llow Lem na 8 yields P oposi io l 2 i i he gene al case . Suppose II 1 (M) ini ely ge ie a ed by elemen s o ini e o de s, Vence