On the intersection forms of closed 4-manifolds

Publicacions Matemátiques, Vol 36 (1992), 73-83 . Abstract ON THE INTERSECTION FORMS OF CLOSED 4-MANIFOLDS ALBERTO CAVICCHIOLI AND FRIEDRICH HEGENBARTH Given a closed 4-manifold M, let M* be the simply-connected 4-manifold obtained from M by killing the fundamental group . We study the relation between the intersection forms A M and AM- . Finally some topological consequences and examples are described . 1 . Introduction . Let M 4 be a closed connected orientable (PL) 4-manifold with fundamental group II1 . Denote by A M the intersection form of M Am : FH2 (M) x FH2 (M)  -  . Z where FH2 (M) = H2 (M ; Z)/torsion (see for example [5], [10]) . Let M* be the simply-connected closed 4-manifold obtained from M by killing the fundamental group II, (see [6]) . Our purpose is to study what relation links \M to AM . . Thenwe obtain some topological consequences about M* . Finally we give some examples which illustrate the results . 2 . Main results . Let [a] be a generator of II 1 . Since M is orientable, we can extend e¿ : S i - M to an embedding 0 : S 1 x D 3 , M . Recall that there are two ways to extend a since II1(SO(3)) = Z2 . Work performed under the auspicies of the G .N .S .A .G .A . of the C .N .R . and financially supported by the Ministero della Ricerca Scientifica e Tecnologica of Italy within the project "Geometría Reale e Complessa" 74  A . CAVICCI - IIOLI, F . HEGEN13ARTFI 0 Denote by M' = M\O(S I x D 3 ) U D 2 x S 2 the closed 4-manifold obtained from M by surgery on 0 . Since III (M') - 111(M)1[o¿], iterated surgeries on generators of III (M) give a simply-connected closed 4-manifold M* . Problem . Study the relations between AM, A M , and AM, A M . respectively . First we have the following Proposition 1 . If nI (M) has no elements of finite order, then A M . i s isomorphic over the integers to AM . The proof is given for example in [1] . Therefore from now on we will consider manifolds with lI I (M) finite . Proposition 2 . If [a] has finite order, then for some integer a EZ . In any case A M , is indefinite . For there forms there is the following well-known classification : (1)  0  1/ A M ,  even  A M , = pE 8 ® q  1  0 2)  A M ,  odd ==>  A M , - p(1) ®q( -1 ) for some non negative integers p, qE Z . Furthermore, S . K . Donaldson (see [2]) proved the following Theorem 3 . Let M 4 be a closed connected orientable 4-manifold with arbitrary fundamental group . If AM is definite, then AM is isomorphic over the integers to either (1) ® . . . ® (1) or (-1) ® . . . ® (-1) . The parity of Am is related to the second Stiefel-Whitney class 0 Am ®( 1 1) a -even 0 a) - 1 AM e (o 0 - 1) a odd M . (1\ mt . - AM ®  1 0  0 1 INTERSCCTION rORMS or 4-MANtrOLDS  75 w2 (M) E H 2 (M ; Z2) as follows . Using the universal coefficient sequence 0 -> Ext(Hi(M) ; Z2) - H 2 (M ; Z2) -> Hom(H2(M) ; Z2) - 0, it is easily proved that AM is even if and only if w2 (M) E Ext(H1(M) ; Z2) . In particular, if Hl (M) has no 2-torsion, then AM is even if and only if w2 (M) = 0 . Thus proposition 2 implies the following Proposition 4 . If w2 (M) =~ 0 , then 1\ m -  m (D p  1  0  ) 5 - -- r (1) ED S (0 -1 for some non negative integers p, r, s E Z . Further, M* is homeomorphic to the connected sum r(CP~#s(-CP~, being CP 2 the projective complex plane . Now we can also apply theorem 2 of [2] to obtain the following consequence of proposition 2 . Corollary 5 . Let M 4 be a closed connected orientable spin 4-manifold with fundamental group II (M) - Z  ,, . If AM has a positive parí of rank 1, then M* is homeomorphic to either 2(CP 2 )#(2 - v(M)) (-CP 2 ) or 2(S 2 x S 2 ) . In the last case, AM =  o  ~) . Here u(M) denotes the signa ture of Proof . .- By proposition 2, we have either AM . = AM ®  0  1 ) or In the first case, AM . i s even . Since Hl(M*) = 0 has no 2-torsion, theorem 2 of [2] implies that ~M " - ~0 10/ ® CO hence AM . has a positive part of rank 2 . ól - Am E)  o  i) 1 hence AM =  0  1 ) (see [7J, [9]) and M* TOp 2(5 2 x S 2 ) as required . In the 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) Let K = ` {zó + zi + z2 + z3 = 0} CCP 3 be the Kummer surface and let T : CP 3 -> CP 3 be the fixed point free involution defined by T(zo, Z1, z2, z3) = (zl, - zo, z3, - z2) Since T (K) = K, we can consider the orbit space M = K/T, called the Habegger manifold (see [4]) . It is known that I1 1 (M) = Z2 and the intersection form ( 0 1) Ana _ (-E8) ®  1  00 is even with a positive part of rank 1 . Since w2(M) z,¿ 0, proposition 2 gives A M . - ( - E8) ® ( 1  0 ) ® ( 0  0 1  - 10(-1) ® 2(1), -) hence M* - 10(-CP 2 )#2(CP 2 ) by the Freedman classification (see TOP (3l) . We also recall that C . Okonek (see [8}) has shown that all homotopy Enriques surfaces are horneomorphic to the Habegger manifold . 3 .2) Let M' = S(r7 ® 71 (D 17) be the sphere bundle of 77 ® 77 ® 71, where ---> RP 2 is the canonical bundle over the real projective 2-space . Thenwe have Am = 0, w2 (M) ~ 0and II1(M) l- - - Z2, hence and M* - CP 2 #(-CP 2 ) = S 2 x S 2 . TOP  TOP ti 3 .3) Let M` 1 = S(,, ® E 2 ) be the sphere bundle of 77 ® E 2 , where E 2 = E l ® e l ---> RP 2 is the 2-dimensional trivial bundle over RP 2 . Thenwe have AM = 0, w2(M) = 0 and II1(M) = Z2 . It is very-easy to see that H 2 (M ; Z2) --' H 2 (Mo ; Z2) fH 2 (1VI* ; Z2) ,so  ¡so 0 where Mo = M\0(S I xD3), V : S 1 x D 3 - M represents the generator of II1(M) and i : Mo -+ M, i' : Mo - M* are the natural inclusions . Thus w2 (M*) = 0, hence A M * = ( 0  1 ) is even and M* TóP S 2 x S 2 . 4 . Proofs . INTGRSGCTION FORMS OF 4-MANIFOLDS  77 Proof of proposition 2 : For conveniente we assume that II I (M) Z,,,m > 0, with generator [a) = [VIS~xol . For the general case, see remark 1 below . 0 We set Mo = M\O(S I x D 3 ) and consider the cobordism W=MxIUOD 2 xD 3 (I=[0,1]) between M and M' = Mo U D 2 x S 2 . Obviously the pairs (W M), (W, M') are homology equivalent to (D 2 x D 3 , S I x D 3 ) and (D 2 x D 3 , D 2 x S 2 ) respectively . We have the following diagram 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 where i, i', j, k are inclusions . Obviously H2 (M') is a free group of rank rkH2 (M) + 2and H2 (M o ) is free of rank rkH2(M) + 1 since it injects into H2 (M) . Here we often identify an element of H2(M0) with its image under i* . Now we have AM (z* (u), a . (v)) = Am , (z* (u), z* (v)) for every u, v E H2(Mo) . Let e E H2 (Mo) be a primitive element such that ¡ * (e) generates the subgroup TorH2(M) - Z, ;, and suppose that f E H2 (M') maps to the integer m E Z - H2(M',Mo) . Similarly f is chosen to be primitive . Furthermore, denote by V the span of {e, f} in H2 (M') . 78 Lemma 6 . With the above notation, we have ~- _ 0 1 A M , 1 v  1  a where AM , (f, f) = a EZ . A . CAVICCIIIOLI, F . HI3GENBARTI - 1 Proof .- From the diagram, it follows that (1)  Am,(a*(x),y) = Aw(x,j*(y)) for every x E H3 (W M') and y E H2 (M) . Note that and a* [D 2 x D 3 , D 2 x S 2 ] = mi ; (e) = me j* (f) = m[D2 where [, ] denotes the fundamental class . Thus relation (1) implies hence AM , (e, f ) = 1 as required . Furthermore, we have D 3 , S I x D 3 ], . Am , (me, f) _ A m , (a* [D 2 x D 3 , D2 x S 2 ], .f) = 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 the exactness . Thus AM , (e, e) = 0 and the proof of Lemma 6 is completed . Lemma 7 . Let V 1 C H2(M') be the ortlzoyonal complement of V . Then V L C H2 (Mo) and the restrirtion is an isomorphism . ¡ * ¡ v i- : V 1 --, FH2(M) Proof .. To prove that V -L C H2(Mo), we have to show that for every y E H 2 (M') with AM4, e) = AM , (y, .f) = 0, then y E H2(Mo) , i . e . j . (y) =0 . Suppose, on the contrary, j . (y) 7~ 0, i . e . j . (y) = q[D 2 x D 3 , S 1 x D 3 ] for some integer q 7L 0 . Thenwe have 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 contradiction . To prove that i .1  L is mono, let x E V 1 be an element such that i, (x) E TorH 2 (M) -Z, . Thenwe have i, (x) = hi . (e) for some integer h, and so i . (x - he) = 0 . By the exactness, it follows that hence mh'e = x - he, h, h' E Z . But we have (use (1)) ( 2 ) AM,(a'(h'[D2 x D 3 ,D 2 x S 2 ]), f) = Aw(h'[D 2 x D 3 , D 2 xS2  (f» = Aw(h [D2 x D 3 , D 2 . x S 2 ], m[D 2 x D 3 , S' x D 3 ]) = rr¿h' and INTLIISC :CTION rORMS or 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]), f) =AM,(2 :(x-he),f) = AM, (x - he, f ) = AM, (x, f) - hAM, (e, f ) _ -h . Comparing relations (2) and (3) gives mh' = -h, hence mh'e = xhe implies that x = 0 as required . To prove that i, lvl is epi, let z E H2 (M) and let u E H2 (Mo) be an element such that ¡ * (u) = z . We consider the element u' = u -AM , (u, f) e E H2(Mo) . Thenwe have AM , (me, u') = AM , (a' [D 2 x D 3 , D 2 x S 2 ], u , ) since j, o i ; = 0 ; therefore A m , (u', e) = 0 . 80  A . CAVCCI-110L], F . HECLNBARTI-1 Furt heririore i . e . U' =- ¡' * (U') E V' . Finally This completes the proof . By Lemmas 6 and 7, we have the result Proof of Proposition 4 : Suppose now zu2(M) =,,' : 0 . Because (M, Mo) and (M', Mo) are homplogy equivalent to (SI xD3, SI x S 2 ) and (D 2 x S2, SI x S 2 ) respectively, we have also the diagram which implies Am, (u', .f) = AM, (u - Am, (u, f) e, f ) = AM , (U, .f) - AM , (u, f) = 0, ¡ .(u') = ¡,(u) - Am, (u, f )z* (e) = ¡ .(u) = z mod TorH2 (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 injective, relation (4) and w2 (M) =~ 0 give W2 (M) :~ 0, hence A M , is odd . E Remark 1 . Th< ; proof of proposition 2 can be easily generalized to manifolds with arbitrary fundamental groups . Indeed, this follows from Lemma 8 below . Suppose now M a closed connected orientable (PL) 4-manifold with fundamental group rIi . and Let be dis . ioint embeddings which kill r11 . Setting we have Ilv'FERsrCTION ro1ZMs oí . , 4-MANlrol .»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 Lemrna 8 . (1) H,(Mo) = H, (M), H3 (M0) = ®Z p-1 (2) H2(M o ) is o, directt summand of the free . group 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 proof is straightforward . Now we indicate llow Lemrna 8 yields Propositiorl 2 iri the general case . Suppose II 1 (M) finitely gerierated by elements of finite orders, Vence