%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: irred.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o irred.ps irred.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.01.07:1817 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 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b(surfaces,)j(with)f(b)s(oth)f(their)g(complex)g(and)h(non{complex)g (orien)m(ta-)456 2065 y(tions)d(allo)m(w)m(ed)1009 2029 y Fn(1)1047 2065 y Fp(.)43 b(The)33 b(\014rst)f(coun)m(terexamples)g(w) m(ere)h(constructed)g(in)e(1990)f(b)m(y)456 2181 y(Gompf)36 b(and)j(Mro)m(wk)-5 b(a)39 b([5],)g(and)g(man)m(y)f(others)h(follo)m(w) m(ed.)60 b(Then,)41 b(Gompf)36 b([4])456 2297 y(sho)m(w)m(ed)g(that)e (man)m(y)-8 b(,)34 b(and)h(p)s(ossibly)e(all,)g(these)j(coun)m (terexamples)f(arise)f(from)456 2413 y(symplectic)26 b(four{manifolds.)38 b(Ha)m(ving)26 b(no)g(indication)e(to)i(the)h(con) m(trary)-8 b(,)28 b(man)m(y)456 2530 y(p)s(eople)k(ha)m(v)m(e)i(put)f (forw)m(ard)f(the)h(follo)m(wing:)456 2708 y Fr(Conjecture)k(1.)49 b Fm(Every)42 b(smo)-5 b(oth,)43 b(close)-5 b(d,)43 b(oriente)-5 b(d)41 b(and)g(simply)g(c)-5 b(onne)g(cte)g(d)456 2825 y Fp(4)p Fm({manifold)30 b(is)h(the)h(c)-5 b(onne)g(cte)g(d)31 b(sum)h(of)g(symple)-5 b(ctic)31 b(manifolds,)g(with)h(b)-5 b(oth)31 b(the)456 2941 y(symple)-5 b(ctic)34 b(and)g(the)h(opp)-5 b(osite)34 b(orientations)g(al)5 b(lowe)-5 b(d.)456 3119 y Fp(This)30 b(conjecture)h(is)f(am)m(bitious;)f(it)h(w)m(ould)g(imply) e(the)j(smo)s(oth)e(P)m(oincar)m(\023)-46 b(e)31 b(con-)456 3236 y(jecture.)555 3352 y(Note)38 b(that)g(suc)m(h)i(a)e(connected)i (sum)e(can)g(ha)m(v)m(e)h(summands)f(with)g(de\014nite)456 3468 y(in)m(tersection)43 b(forms,)j(for)e(example)f(copies)h(of)g Fl(C)20 b Fk(P)2419 3432 y Fn(2)2463 3468 y Fp(.)78 b(In)44 b(fact,)j(an)m(y)d(de\014nite)456 3584 y(summand)32 b(has)h(a)f (diagonalizable)e(in)m(tersection)i(form)g(b)m(y)h(Donaldson's)g(theo-) 456 3701 y(rem)i([1],)j(and)e(is)g(therefore)h(homeomorphic)d(to)i Fk(n)p Fl(C)20 b Fk(P)2513 3664 y Fn(2)2595 3701 y Fp(b)m(y)37 b(F)-8 b(reedman's)36 b(clas-)456 3817 y(si\014cation)31 b([3].)555 3933 y(When)37 b(the)f(manifolds)d(under)j(consideration)f (are)g(not)g(simply)g(connected,)456 4049 y(the)g(situation)f(is)g (more)h(complicated.)49 b(Then)37 b(there)e(are)h(ob)m(vious)f(coun)m (terex-)456 4165 y(amples)e(to)h(Conjecture)h(1,)g(e.g.)48 b(rational)32 b(homology)g(spheres)37 b(whic)m(h)d(are)g(not)456 4282 y(homotop)m(y)39 b(spheres.)66 b(Th)m(us,)43 b(one)d(has)g(to)f (allo)m(w)f(de\014nite)h(summands)h(whic)m(h)456 4401 y(are)32 b(more)g(general)g(than)h Fk(n)p Fl(C)20 b Fk(P)1627 4365 y Fn(2)1704 4401 y Fp(or)32 b Fk(n)p 1881 4317 188 4 v Fl(C)21 b Fk(P)2025 4372 y Fn(2)2070 4401 y Fp(.)43 b(The)33 b(natural)f(conjecture)i(is:)456 4579 y Fr(Conjecture)j(2.)49 b Fm(Every)43 b(smo)-5 b(oth,)43 b(close)-5 b(d)41 b(and)h(oriente)-5 b(d)42 b Fp(4)p Fm({manifold)e(is)i(the)456 4696 y(c)-5 b(onne)g(cte)g(d)42 b(sum)i(of)f(symple)-5 b(ctic)44 b(manifolds,)g(with)f(b)-5 b(oth)44 b(the)g(symple)-5 b(ctic)43 b(and)456 4812 y(the)36 b(opp)-5 b(osite)36 b(orientations)f(al)5 b(lowe)-5 b(d,)36 b(and)f(of)i(some)e(manifolds)g (with)h(de\014nite)456 4928 y(interse)-5 b(ction)34 b(forms.)p 456 5022 499 4 v 555 5116 a Fq(1991)28 b Fj(Mathematics)j(Subje)l(ct)f (Classi\014c)l(ation.)43 b Fq(57R55,57R57,53C15.)555 5185 y Fi(1)592 5216 y Fq(The)28 b(4{sphere)e(is)i(the)g(empt)n(y)f (connected)h(sum.)1931 5315 y Fh(1)p eop %%Page: 2 2 2 1 bop 456 236 a Fh(2)1164 b(D.)33 b(K)n(OTSCHICK)456 425 y Fp(This)38 b(has)g(o)s(ccurred)h(to)f(sev)m(eral)h(p)s(eople,)g (esp)s(ecially)e(in)h(the)g(ligh)m(t)f(of)g(the)i(ex-)456 541 y(amples)33 b(constructed)i(in)f([7)o(],)h(and)f(has)g(b)s(een)h (dubb)s(ed)g(the)g(curren)m(t)g(\\minimal)456 658 y(conjecture")f(b)m (y)h(T)-8 b(aub)s(es.)48 b(In)35 b(this)e(note)h(w)m(e)h(sho)m(w)g (that)f(it)e(is)i(false)3009 622 y Fn(2)3048 658 y Fp(.)47 b(Conjec-)456 774 y(ture)33 b(1)f(remains)f(op)s(en.)555 890 y(Recall)g(the)i(follo)m(wing)d(de\014nition:)456 1068 y Fr(De\014nition)36 b(1.)49 b Fp(A)35 b(smo)s(oth)f(closed)h (4{manifold)d Fk(X)43 b Fp(is)35 b Fg(irreducible)f Fp(if)f(for)i(ev-) 456 1184 y(ery)48 b(smo)s(oth)f(connected)i(sum)f(decomp)s(osition)e Fk(X)2488 1157 y Ff(\030)2489 1188 y Fp(=)2619 1184 y Fk(X)2700 1199 y Fn(1)2739 1184 y Fp(#)p Fk(X)2901 1199 y Fn(2)2989 1184 y Fp(one)i(of)f(the)456 1301 y(summands)32 b Fk(X)1020 1316 y Fe(i)1081 1301 y Fp(m)m(ust)g(b)s(e)h(a)f(homotop)m (y)h(sphere.)555 1479 y(W)-8 b(e)33 b(will)d(sho)m(w:)456 1657 y Fr(Theorem)37 b(1.)49 b Fm(Ther)-5 b(e)37 b(exist)h(oriente)-5 b(d)38 b(irr)-5 b(e)g(ducible)37 b Fp(4)p Fm({manifolds)g Fk(X)46 b Fm(with)38 b(in-)456 1773 y(de\014nite)26 b(interse)-5 b(ction)27 b(forms)g(and)g(with)g Fk(\031)2025 1788 y Fn(1)2065 1773 y Fp(\()p Fk(X)8 b Fp(\))27 b(=)h Fl(Z)2430 1788 y Fn(2)2494 1773 y Fm(and)f Fk(b)2717 1732 y Fn(+)2717 1797 y(2)2777 1773 y Fp(\()p Fk(X)8 b Fp(\))27 b Ff(\021)h Fk(b)3115 1732 y Fd(\000)3115 1797 y Fn(2)3175 1773 y Fp(\()p Fk(X)8 b Fp(\))27 b Ff(\021)456 1889 y Fp(0)44 b(\(mo)s(d)32 b(2\))p Fm(.)555 2067 y Fp(This)h(follo)m(ws)e(from)h (Prop)s(osition)e(1)j(in)f(the)h(next)g(section.)456 2245 y Fr(Corollary)i(1.)49 b Fm(Ther)-5 b(e)48 b(exist)g(orientable)g (irr)-5 b(e)g(ducible)47 b Fp(4)p Fm({manifolds)f Fk(X)57 b Fm(with)456 2361 y(inde\014nite)29 b(interse)-5 b(ction)30 b(forms,)g(which)g(ar)-5 b(e)30 b(not)g(almost)g(c)-5 b(omplex)29 b(\(and)h(ther)-5 b(e-)456 2477 y(for)g(e)30 b(not)h(c)-5 b(omplex)30 b(and)g(not)h(symple)-5 b(ctic\))30 b(with)h(r)-5 b(esp)g(e)g(ct)30 b(to)h(either)g(orientation,)456 2594 y(and)45 b(for)h(which)f(the)h(Donaldson)e(and)i(Seib)-5 b(er)g(g{Witten)45 b(invariants)h(ar)-5 b(e)45 b(not)456 2710 y(de\014ne)-5 b(d)33 b(\(or)i(must)g(vanish)f(by)h (de\014nition\).)456 2888 y Fp(If)d(one)h(drops)h(the)f(requiremen)m(t) g(that)g Fk(X)40 b Fp(ha)m(v)m(e)34 b(inde\014nite)f(in)m(tersection)f (form,)456 3004 y(rational)e(homology)g(spheres)35 b(giv)m(e)d(ob)m (vious)h(examples.)555 3120 y(Recall)22 b(that)g(a)h(symplectic)g (4{manifold)c(is)k(called)f(minimal)d(if)i(it)h(con)m(tains)h(no)456 3237 y(symplectically)e(em)m(b)s(edded)j(2{sphere)h(of)e(sel\014n)m (tersection)h Ff(\000)p Fp(1.)41 b(Conjectures)25 b(1)456 3353 y(and)j(2)g(are)h(complemen)m(tary)f(to)g(Gompf)7 b('s)27 b(conjecture)j([4])e(that)g(minimal)d(sym-)456 3469 y(plectic)39 b(4{manifolds)f(are)i(irreducible.)66 b(In)41 b(section)f(3)g(w)m(e)i(deduce)g(from)d(the)456 3585 y(recen)m(t)33 b(w)m(ork)g(of)f(T)-8 b(aub)s(es)33 b([11])f(on)g(the)h(Gromo)m(v)e(and)h(Seib)s(erg{Witten)f(in)m(v)-5 b(ari-)456 3701 y(an)m(ts)25 b(that)f(Gompf)7 b('s)23 b(conjecture)i(is)f(true)h(in)e(man)m(y)i(cases,)i(including)c(all)f (simply)456 3818 y(connected)34 b(manifolds)c(with)i Fk(b)1616 3776 y Fn(+)1616 3842 y(2)1703 3818 y Fk(>)c Fp(1.)43 b(W)-8 b(e)33 b(shall)e(pro)m(v)m(e:)456 3996 y Fr(Theorem)37 b(2.)49 b Fm(L)-5 b(et)28 b Fk(X)36 b Fm(b)-5 b(e)28 b(a)g(minimal)f(symple)-5 b(ctic)28 b Fp(4)p Fm({manifold)f(with)h Fk(b)3117 3954 y Fn(+)3117 4020 y(2)3176 3996 y Fp(\()p Fk(X)8 b Fp(\))28 b Fk(>)456 4112 y Fp(1)p Fm(.)42 b(If)29 b Fk(X)790 4084 y Ff(\030)791 4116 y Fp(=)895 4112 y Fk(X)976 4127 y Fn(1)1016 4112 y Fp(#)p Fk(X)1178 4127 y Fn(2)1246 4112 y Fm(is)g(a)g(smo)-5 b(oth)29 b(c)-5 b(onne)g(cte)g(d)28 b(sum)h(de)-5 b(c)g(omp)g(osition) 27 b(of)i Fk(X)8 b Fm(,)30 b(then)456 4228 y(one)j(of)h(the)g Fk(X)995 4243 y Fe(i)1057 4228 y Fm(is)g(an)g(inte)-5 b(gr)g(al)34 b(homolo)-5 b(gy)33 b(spher)-5 b(e)33 b(whose)g (fundamental)h(gr)-5 b(oup)456 4344 y(has)34 b(no)g(non{trivial)g (\014nite)h(quotient.)456 4522 y Fp(This)46 b(strengthening)h(of)e (Prop)s(osition)g(1)h(in)f([7])i(go)s(es)f(a)g(long)f(w)m(a)m(y)i(to)m (w)m(ards)456 4639 y(con\014rming)31 b(a)h(conjecture)i(made)e(there.) 456 4817 y Fr(Corollary)j(2.)49 b Fm(Minimal)41 b(symple)-5 b(ctic)40 b Fp(4)p Fm({manifolds)f(with)i Fk(b)2768 4775 y Fn(+)2768 4841 y(2)2867 4817 y Fk(>)e Fp(1)i Fm(and)g(with)456 4933 y(r)-5 b(esidual)5 b(ly)34 b(\014nite)h(fundamental)f(gr)-5 b(oups)34 b(ar)-5 b(e)35 b(irr)-5 b(e)g(ducible.)p 456 5025 499 4 v 555 5086 a Fi(2)592 5116 y Fq(W)e(e)24 b(do)e(not)h(ha)n (v)n(e)f(an)n(y)g(prop)r(osal)f(for)h(a)h(new)f(\\minimal)h (conjecture")f(in)h(the)g(non{simply)456 5216 y(connected)k(case.)p eop %%Page: 3 3 3 2 bop 1295 236 a Fh(IRREDUCIBLE)32 b(F)n(OUR{MANIF)n(OLDS)798 b(3)456 425 y Fp(See)33 b(section)g(3)f(for)g(a)g(result)h(in)f(and)g (commen)m(ts)h(on)f(the)h(case)h Fk(b)2851 384 y Fn(+)2851 450 y(2)2938 425 y Fp(=)28 b(1.)555 541 y(F)-8 b(or)38 b(K\177)-49 b(ahler)39 b(surfaces,)j(Theorem)d(2)g(and)g(Corollary)e(2) i(could)f(b)s(e)i(deduced)456 658 y(easily)26 b(from)f(the)j(reduction) e(of)h(the)g(Seib)s(erg{Witten)f(equation)g(to)h(the)g(K\177)-49 b(ahler)456 774 y(v)m(ortex)27 b(equation)g(and)f(the)i(study)f(of)g (e\013ectiv)m(e)g(divisors)f(on)h(complex)f(surfaces,)456 890 y(due)33 b(to)f(Kronheimer{Mro)m(wk)-5 b(a)32 b(and)h(Witten)f ([13].)935 1088 y(2.)55 b Fo(Irreducibility)38 b(of)g(quotient)g(manif) n(olds)555 1262 y Fp(Theorem)32 b(1)f(will)e(follo)m(w)g(from)h(the)i (follo)m(wing)d(application)f(of)j(the)h(co)m(v)m(ering)456 1378 y(tric)m(k)g(in)m(tro)s(duced)h(in)f([6]:)456 1546 y 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b(As)456 2295 y Fk(\031)511 2310 y Fn(1)550 2295 y Fp(\()p Fk(Y)22 b Fp(\))732 2267 y Ff(\030)733 2299 y Fp(=)837 2295 y Fk(G)27 b Fp(is)f(\014nite,)i(it)e(cannot)h(b)s(e)f(a)h (non{trivial)d(free)j(pro)s(duct)g(and)g(w)m(e)h(ma)m(y)456 2411 y(assume)33 b Fk(\031)848 2426 y Fn(1)887 2411 y Fp(\()p Fk(M)10 b Fp(\))1096 2383 y Ff(\030)1097 2415 y Fp(=)1201 2411 y Fk(G)32 b Fp(and)h Fk(\031)1555 2426 y Fn(1)1595 2411 y Fp(\()p Fk(N)10 b Fp(\))1787 2383 y Ff(\030)1788 2415 y Fp(=)1892 2411 y Ff(f)p Fp(1)p Ff(g)p Fp(.)555 2527 y(Let)42 b Fk(d)f Fp(b)s(e)g(the)h(order)g(of)e Fk(G)p Fp(.)70 b(The)42 b(connected)h(sum)f(decomp)s(osition)d(of)i Fk(Y)456 2643 y Fp(induces)33 b(a)e(connected)j(sum)e(decomp)s(osition) f Fk(X)2303 2616 y Ff(\030)2304 2647 y Fp(=)p 2408 2563 105 4 v 2408 2643 a Fk(M)11 b Fp(#)p Fk(dN)f Fp(,)33 b(where)p 3074 2563 V 33 w Fk(M)43 b Fp(is)32 b(the)456 2759 y(univ)m(ersal)44 b(co)m(v)m(ering)g(of)g Fk(M)10 b Fp(.)79 b(As)45 b(either)f Fk(b)2091 2718 y Fn(+)2091 2784 y(2)2151 2759 y Fp(\()p Fk(X)8 b Fp(\))47 b Ff(\024)h Fp(1)c(or)g Fk(X)52 b Fp(is)44 b(assumed)h(to)456 2876 y(ha)m(v)m(e)d(a)f(non{trivial)d(Donaldson)i(or)h(Seib)s(erg{Witten)g (in)m(v)-5 b(arian)m(t,)41 b(it)g(follo)m(ws)456 2992 y(that)e(the)i(in)m(tersection)f(form)e(of)i Fk(N)50 b Fp(is)39 b(negativ)m(e)i(de\014nite.)65 b(By)41 b(Donaldson's)456 3108 y(theorem)c([1])h(it)e(is)i(diagonalizable)c(o)m(v)m(er)39 b Fl(Z)p Fp(,)d(and)i(therefore)g(either)g(trivial)d(or)456 3224 y(o)s(dd.)555 3341 y(On)26 b(the)h(other)g(hand,)g(the)g(in)m (tersection)f(form)f(of)h Fk(N)37 b Fp(m)m(ust)26 b(b)s(e)h(ev)m(en,)i (b)s(ecause)456 3457 y(it)41 b(is)h(a)g(direct)h(summand)e(of)h(the)h (in)m(tersection)g(form)e(of)h Fk(X)8 b Fp(,)45 b(whic)m(h)e(is)f(ev)m (en)456 3573 y(b)s(ecause)32 b Fk(X)39 b Fp(is)30 b(spin.)43 b(W)-8 b(e)31 b(conclude)h Fk(b)1882 3588 y Fn(2)1922 3573 y Fp(\()p Fk(Y)21 b Fp(\))27 b(=)h(0.)43 b(As)31 b Fk(N)42 b Fp(is)30 b(simply)g(connected,)456 3689 y(it)h(is)h(a)h (homotop)m(y)f(sphere)i(and)e Fk(Y)54 b Fp(is)32 b(irreducible.)p 3374 3689 4 66 v 3378 3627 59 4 v 3378 3689 V 3436 3689 4 66 v 555 3855 a(T)-8 b(o)40 b(obtain)e(examples)i(as)g(in)f(the)h (statemen)m(t)g(of)f(Theorem)h(1,)h(tak)m(e)g(for)e Fk(X)456 3972 y Fp(the)c(F)-8 b(ermat)33 b(surface)i(of)f(degree)i Fk(d)31 b Ff(\021)g Fp(2)44 b(\(mo)s(d)32 b(4\))i(in)g Fl(C)20 b Fk(P)2635 3935 y Fn(3)2715 3972 y Fp(with)34 b Fk(d)c Ff(\025)i Fp(6.)49 b(This)456 4088 y(is)32 b(the)h(surface)g (de\014ned)h(in)e(homogeneous)h(co)s(ordinates)f(b)m(y)1473 4245 y Fk(x)1528 4204 y Fe(d)1591 4245 y Fp(+)22 b Fk(y)1741 4204 y Fe(d)1803 4245 y Fp(+)g Fk(z)1950 4204 y Fe(d)2013 4245 y Fp(+)g Fk(t)2146 4204 y Fe(d)2215 4245 y Fp(=)27 b(0)33 b Fk(:)-1971 b Fp(\(1\))456 4402 y(It)37 b(is)g(simply)f (connected,)41 b(and)d(spin)f(b)s(ecause)i(its)e(canonical)f(class)i (is)f(the)h(re-)456 4518 y(striction)f(of)h(\()p Fk(d)25 b Ff(\000)i Fp(4\))p Fk(H)45 b Fp(to)38 b Fk(X)8 b Fp(,)40 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(haracteristic)p eop %%Page: 4 4 4 3 bop 456 236 a Fh(4)1164 b(D.)33 b(K)n(OTSCHICK)456 425 y Fp(and)h(the)h(signature)e(under)i(\014nite)f(unrami\014ed)g(co)m (v)m(erings,)h(one)g(can)f(calculate)456 541 y Fk(b)497 500 y Fd(\006)497 566 y Fn(2)556 541 y Fp(\()p Fk(Y)21 b Fp(\))40 b(=)877 502 y Fn(1)p 877 519 36 4 v 877 576 a(2)922 541 y Fp(\()p Fk(b)1001 500 y Fd(\006)1001 566 y Fn(2)1060 541 y Fp(\()p Fk(X)8 b Fp(\))27 b Ff(\000)h Fp(1\))39 b(whic)m(h)i(are)f(b)s(oth)f(p)s(ositiv)m(e)h(\(b)s(ecause)h Fk(d)f Ff(\025)h Fp(6\))e(and)456 659 y(ev)m(en)34 b(\(b)s(ecause)g Fk(d)27 b Ff(\021)h Fp(2)g(mo)s(d)e(4\).)555 775 y(This)33 b(completes)f(the)h(pro)s(of)f(of)g(Theorem)h(1.)456 956 y Fm(R)-5 b(emark)34 b(1.)48 b Fp(W)-8 b(ang)24 b([12)o(])g(has)g (sho)m(wn)h(that)f(the)g(quotien)m(ts)g(of)f(simply)g(connected)456 1072 y(minimal)g(algebraic)j(surfaces)j(of)e(general)g(t)m(yp)s(e)i(b)m (y)f(free)g(an)m(ti{holomorphic)d(in-)456 1188 y(v)m(olutions)j(ha)m(v) 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2300 y(tic)39 b Fp(4)p Fm({manifold)e(with)i Fk(b)1365 2259 y Fn(+)1365 2324 y(2)1425 2300 y Fp(\()p Fk(X)8 b Fp(\))35 b Fk(>)h Fp(1)p Fm(.)58 b(Supp)-5 b(ose)38 b(a)h(non{trivial)f (\014nite)h(gr)-5 b(oup)39 b Fk(G)456 2416 y Fm(acts)f(fr)-5 b(e)g(ely)38 b(by)h(orientation{pr)-5 b(eserving)36 b(di\013e)-5 b(omorphisms)36 b(of)i Fk(X)8 b Fm(.)55 b(Then)37 b(the)456 2532 y(quotient)e Fk(Y)49 b Fp(=)27 b Fk(X=G)34 b Fm(is)h(an)g (orientable)f(irr)-5 b(e)g(ducible)34 b Fp(4)p Fm({manifold.)456 2713 y(Pr)-5 b(o)g(of.)41 b Fp(If)e Fk(Y)975 2686 y Ff(\030)976 2718 y Fp(=)1092 2713 y Fk(M)10 b Fp(#)p Fk(N)51 b Fp(with)39 b Fk(\031)1690 2728 y Fn(1)1730 2713 y Fp(\()p Fk(M)10 b Fp(\))1949 2686 y Ff(\030)1950 2718 y Fp(=)2066 2713 y Fk(G)39 b Fp(and)h Fk(\031)2434 2728 y Fn(1)2473 2713 y Fp(\()p Fk(N)10 b Fp(\))2677 2686 y Ff(\030)2678 2718 y Fp(=)2794 2713 y Ff(f)p Fp(1)p Ff(g)p Fp(,)40 b(then)g Fk(X)3367 2686 y Ff(\030)3368 2718 y Fp(=)p 456 2755 105 4 v 456 2835 a Fk(M)10 b Fp(#)p Fk(dN)g 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Fm(R)-5 b(emark)34 b(3.)48 b Fp(In)35 b(another)f(direction,)g(the)h (assumption)e Fk(b)2592 4942 y Fn(+)2592 5008 y(2)2652 4983 y Fp(\()p Fk(X)8 b Fp(\))30 b Fk(>)g Fp(1)k(can)h(prob-)456 5099 y(ably)44 b(b)s(e)i(remo)m(v)m(ed)g(from)d(Theorem)j(2)f(and)g (Corollary)e(2.)81 b(T)-8 b(o)45 b(do)g(this)g(one)456 5216 y(needs)27 b(to)e(understand)i(ho)m(w)f(the)g(nec)m(k{stretc)m (hing)h(in)e(the)h(pro)s(of)f(of)g(Theorem)h(2)p eop %%Page: 6 6 6 5 bop 456 236 a Fh(6)1164 b(D.)33 b(K)n(OTSCHICK)456 425 y Fp(and)h(the)h(p)s(erturbations)f(in)g(T)-8 b(aub)s(es's)36 b(argumen)m(ts)f([9,)f(11])g(in)m(teract)h(with)f(the)456 541 y(c)m(ham)m(b)s(er)c(structure)h(of)e(the)h(Seib)s(erg{Witten)f(in) m(v)-5 b(arian)m(ts)28 b(for)h(manifolds)f(with)456 658 y Fk(b)497 616 y Fn(+)497 682 y(2)584 658 y Fp(=)f(1.)44 b(W)-8 b(e)33 b(will)d(return)j(to)f(this)g(question)h(in)f(the)h (future.)555 774 y(Ho)m(w)m(ev)m(er,)38 b(some)d(results)h(ab)s(out)f (the)g(case)h(when)g Fk(b)2497 733 y Fn(+)2497 798 y(2)2589 774 y Fp(=)c(1)j(can)g(b)s(e)h(deduced)456 890 y(from)46 b(Theorem)i(2.)88 b(F)-8 b(or)47 b(example,)k(all)45 b(manifolds)g(with)j(non{tivial)d(\014nite)456 1006 y(fundamen)m(tal)31 b(groups)i(are)g(dealt)f(with)g(b)m(y)h(the)g(follo)m(wing:)456 1170 y Fr(Corollary)i(3.)49 b Fm(L)-5 b(et)26 b Fk(X)33 b Fm(b)-5 b(e)26 b(a)f(minimal)f(symple)-5 b(ctic)25 b Fp(4)p Fm({manifold)e(with)j Fk(b)3117 1129 y Fn(+)3117 1195 y(2)3176 1170 y Fp(\()p Fk(X)8 b Fp(\))28 b(=)456 1287 y(1)i Fm(and)h Fk(b)762 1302 y Fn(1)802 1287 y Fp(\()p Fk(X)8 b Fp(\))27 b Ff(\024)h Fp(1)p Fm(.)43 b(If)31 b Fk(\031)1375 1302 y Fn(1)1415 1287 y Fp(\()p Fk(X)8 b Fp(\))30 b Fm(is)h(a)g(non{trivial)f(r)-5 b(esidual)5 b(ly)30 b(\014nite)h(gr)-5 b(oup,)32 b(then)456 1403 y Fk(X)42 b Fm(is)35 b(irr)-5 b(e)g(ducible.)456 1567 y(Pr)g(o)g(of.)41 b Fp(Supp)s(ose)30 b Fk(X)1250 1539 y Ff(\030)1251 1571 y Fp(=)1355 1567 y Fk(M)10 b Fp(#)p Fk(N)g Fp(.)44 b(W)-8 b(e)30 b(ma)m(y)f(assume)h(that)f Fk(N)40 b Fp(has)30 b(negativ)m(e)g(def-)456 1683 y(inite)39 b(in)m(tersection)i(form)e(and)i(its)g(fundamen)m(tal)e(group)i(has)g (no)g(non{trivial)456 1799 y(\014nite)34 b(quotien)m(t.)49 b(Residual)33 b(\014niteness)i(then)g(implies)d(that)i Fk(N)45 b Fp(is)34 b(simply)f(con-)456 1915 y(nected,)38 b(and)f Fk(\031)1045 1930 y Fn(1)1085 1915 y Fp(\()p Fk(M)10 b Fp(\))1300 1888 y Ff(\030)1301 1920 y Fp(=)1412 1915 y Fk(\031)1467 1930 y Fn(1)1506 1915 y Fp(\()p Fk(X)e Fp(\).)55 b(By)37 b(assumption,)g Fk(X)44 b Fp(has)37 b(a)f(\014nite)h(co)m(v)m(er)p 3356 1835 89 4 v 38 w Fk(X)456 2037 y Fp(of)i(degree)j Fk(d)e(>)h Fp(1)f(whic)m(h)h(is)f (di\013eomorphic)f(to)p 2321 1957 105 4 v 40 w Fk(M)10 b Fp(#)p Fk(dN)g Fp(,)43 b(where)p 3005 1957 V 42 w Fk(M)51 b Fp(is)40 b(a)g Fk(d)p Fp({)456 2153 y(fold)e(co)m(v)m(er)i(of)f Fk(M)10 b Fp(.)64 b(But)p 1426 2073 89 4 v 39 w Fk(X)47 b Fp(is)39 b(minimal)c(symplectic)k(b)s(ecause)h Fk(X)47 b Fp(is,)41 b(and)e(so)456 2269 y(Corollary)28 b(2)i(implies)d(that)j Fk(N)40 b 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(questions)h(whic)m(h)g(this)e(pap)s(er)i(answ)m(ers,)j(and)d(Cli\013) 456 3067 y(T)-8 b(aub)s(es)33 b(for)f(con)m(v)m(ersations)i(and)f (corresp)s(ondence.)1660 3265 y Fo(References)497 3422 y Fq(1.)41 b(S.)27 b(K.)g(Donaldson,)f Fj(A)n(n)j(applic)l(ation)i(of)f (gauge)g(the)l(ory)g(to)f(four{dimensional)j(top)l(olo)l(gy)p Fq(,)603 3522 y(J.)c(Di\013eren)n(tial)f(Geometry)g Fb(18)g Fq(\(1983\),)g(279{315.)497 3622 y(2.)41 b(S.)26 b(K.)g(Donaldson,)g Fj(The)j(orientation)g(of)g(Y)-6 b(ang{Mil)t(ls)30 b(mo)l(duli)f(sp)l (ac)l(es)g(and)f Fq(4)p Fj({manifold)603 3721 y(top)l(olo)l(gy)p Fq(,)h(J.)f(Di\013eren)n(tial)g(Geometry)e Fb(26)i Fq(\(1987\),)e (397{428.)497 3821 y(3.)41 b(M.)34 b(H.)h(F)-7 b(reedman,)35 b Fj(The)i(top)l(olo)l(gy)g(of)g(four{dimensional)h(manifolds)p Fq(,)g(J.)33 b(Di\013eren)n(tial)603 3920 y(Geometry)27 b Fb(17)g Fq(\(1982\),)g(357{453.)497 4020 y(4.)41 b(R.)25 b(E.)f(Gompf,)i Fj(A)h(new)g(c)l(onstruction)f(of)i(symple)l(ctic)g (manifolds)p Fq(,)f(Annals)e(of)g(Math.)f(\(to)603 4120 y(app)r(ear\).)497 4219 y(5.)41 b(R.)21 b(E.)f(Gompf)h(and)f(T.)h(S.)g (Mro)n(wk)-5 b(a,)20 b Fj(Irr)l(e)l(ducible)k(four{manifolds)i(ne)l(e)l (d)e(not)e(b)l(e)i(c)l(omplex)p Fq(,)603 4319 y(Annals)k(of)f(Math.)h Fb(138)f Fq(\(1993\),)f(61{111.)497 4419 y(6.)41 b(D.)26 b(Kotsc)n(hic)n(k,)e Fj(On)i(c)l(onne)l(cte)l(d)h(sum)g(de)l(c)l(omp)l (ositions)i(of)f(algebr)l(aic)h(surfac)l(es)f(and)g(their)603 4518 y(fundamental)i(gr)l(oups)p Fq(,)f(In)n(ternat.)e(Math.)g(Res.)h (Notices)f Fb(1993)p Fq(,)g(179{182.)497 4618 y(7.)41 b(D.)31 b(Kotsc)n(hic)n(k,)f(J.)h(W.)g(Morgan)e(and)i(C.)f(H.)i(T)-7 b(aub)r(es,)31 b Fj(F)-6 b(our{manifolds)35 b(without)e(sym-)603 4717 y(ple)l(ctic)42 b(structur)l(es)e(but)h(with)h(non{trivial)g(Seib) l(er)l(g{Witten)g(invariants)p Fq(,)i(Math.)d(Re-)603 4817 y(searc)n(h)26 b(Letters)h Fb(2)h Fq(\(1995\),)e(119{124.)497 4917 y(8.)41 b(P)-7 b(.)33 b(B.)h(Kronheimer)e(and)h(T.)h(S.)g(Mro)n (wk)-5 b(a,)33 b Fj(The)j(genus)f(of)h(emb)l(e)l(dde)l(d)g(surfac)l(es) g(in)f(the)603 5016 y(pr)l(oje)l(ctive)d(plane)p Fq(,)c(Math.)g (Researc)n(h)e(Letters)h Fb(1)h Fq(\(1994\),)e(797{808.)497 5116 y(9.)41 b(C.)26 b(H.)h(T)-7 b(aub)r(es,)26 b Fj(The)j(Seib)l(er)l (g{Witten)f(invariants)h(and)g(symple)l(ctic)g(forms)p Fq(,)f(Math.)e(Re-)603 5216 y(searc)n(h)g(Letters)h Fb(1)h Fq(\(1994\),)e(809{822.)p eop %%Page: 7 7 7 6 bop 1295 236 a Fh(IRREDUCIBLE)32 b(F)n(OUR{MANIF)n(OLDS)798 b(7)456 425 y Fq(10.)40 b(C.)32 b(H.)g(T)-7 b(aub)r(es,)33 b Fj(Mor)l(e)i(c)l(onstr)l(aints)e(on)h(symple)l(ctic)g(manifolds)i(fr) l(om)e(Seib)l(er)l(g{Witten)603 525 y(invariants)p Fq(,)29 b(Math.)f(Researc)n(h)e(Letters)h Fb(2)g Fq(\(1995\),)g(9{14.)456 624 y(11.)40 b(C.)e(H.)g(T)-7 b(aub)r(es,)40 b Fj(The)f(Seib)l(er)l (g{Witten)h(and)f(the)g(Gr)l(omov)h(invariants)p Fq(,)h(researc)n(h)35 b(an-)603 724 y(nouncemen)n(t.)456 824 y(12.)40 b(S.)d(W)-7 b(ang,)39 b Fj(A)f(vanishing)i(the)l(or)l(em)f(for)g(Seib)l(er)l (g{Witten)g(invariants)p Fq(,)h(preprin)n(t)c(April)603 923 y(1995.)456 1023 y(13.)k(E.)34 b(Witten,)i Fj(Monop)l(oles)i(and)e (four{manifolds)p Fq(,)i(Math.)c(Researc)n(h)f(Letters)g Fb(1)h Fq(\(1994\),)603 1123 y(769{796.)555 1391 y Fa(Ma)-6 b(thema)g(tisches)41 b(Institut,)f(Universit)2091 1384 y(\177)2088 1391 y(at)f(Basel,)h(Rheinspr)n(ung)g(21,)g(4051)456 1490 y(Basel,)30 b(Switzerland)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF