%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: book.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o book.ps book.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2001.06.12:1514 %%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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(orien)m(ted)f(surfaces)h(in)e(a)h(closed)456 1672 y(orien)m(ted)48 b(4{manifold)c FF(X)56 b FM(de\014ne)49 b(a)e(symmetric)g(bilinear)f (form)h FF(Q)3075 1687 y FH(X)3190 1672 y FM(on)h(its)456 1788 y(second)25 b(homology)-8 b(,)25 b(called)e(the)i Fr(in)m(tersection)f(form)g FM(of)g FF(X)8 b FM(,)26 b(whic)m(h)f(is)f(dual)g(to)g(the)456 1904 y(cup)g(pro)s(duct)f(in)g (cohomology)-8 b(.)39 b(It)23 b(is)g(a)g(consequence)k(of)c(P)m(oincar) m(\023)-46 b(e)23 b(dualit)m(y)g(that)456 2020 y(the)37 b(in)m(tersection)f(form)f(is)h(unimo)s(dular)e(and)j(in)e(particular)g (non{degenerate)456 2137 y(on)d(the)h(lattice)e FF(H)1138 2152 y FB(2)1177 2137 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))p FF(=)p FM(T)-8 b(orsion)m(.)605 2344 y FL(Example)37 b(4.1.)49 b FM(Consider)36 b FF(X)k FM(=)32 b FA(C)20 b FF(P)2080 2308 y FB(2)2124 2344 y FM(.)52 b(Its)35 b(homology)f(is)g(generated)i(b)m(y)456 2460 y(the)27 b(class)h(of)f(a)g(complex)g(line,)g(whose)i(self{in)m(tersection)e(n)m (um)m(b)s(er)h(is)f(1)g(b)s(ecause)456 2577 y(t)m(w)m(o)35 b(di\013eren)m(t)g(lines)f(in)m(tersect)i(in)e(one)h(\(p)s(ositiv)m (e\))f(p)s(oin)m(t.)50 b(Th)m(us)36 b FF(Q)3012 2593 y Fv(C)9 b FH(P)3109 2574 y Fy(2)3185 2577 y FM(=)32 b(\(1\).)456 2693 y(If)42 b(w)m(e)i(rev)m(erse)g(the)f(orien)m(tation,) h(i.e.)e(w)m(e)h(consider)g FA(C)20 b FF(P)2619 2657 y FB(2)2706 2693 y FM(with)43 b(the)g(orien)m(ta-)456 2809 y(tion)38 b(opp)s(osite)i(to)f(the)i(one)f(induced)h(b)m(y)f(the)h (complex)e(structure,)k(then)e(the)456 2925 y(in)m(tersection)32 b(n)m(um)m(b)s(er)h(c)m(hanges)h(sign.)605 3133 y(More)h(generally)-8 b(,)34 b(if)f FF(X)42 b FM(is)34 b(orien)m(ted)h(and)p 2179 3053 89 4 v 34 w FF(X)42 b FM(denotes)36 b(the)f(same)f(manifold) 456 3249 y(with)e(the)h(orien)m(tation)e(rev)m(ersed,)k(then)e FF(Q)p 2048 3221 68 3 v 26 x FH(X)2143 3249 y FM(=)28 b FG(\000)p FF(Q)2401 3264 y FH(X)2469 3249 y FM(.)605 3365 y(By)35 b(the)f(w)m(ork)h(of)f(Whitehead{Milnor)e(and)j(F)-8 b(reedman)33 b(resp)s(ectiv)m(ely)-8 b(,)36 b(the)456 3482 y(in)m(tersection)d(form)g(determines)g(the)h(homotop)m(y)g(t)m (yp)s(e)g(and)g(ev)m(en)h(the)f(homeo-)456 3598 y(morphism)26 b(t)m(yp)s(e)j(of)e(a)h(simply)e(connected)k(smo)s(oth)d(4{manifold.)39 b(Ho)m(w)m(ev)m(er,)31 b(as)456 3714 y(disco)m(v)m(ered)h(b)m(y)f (Donaldson,)f FF(Q)1640 3729 y FH(X)1739 3714 y FM(do)s(es)h(not)f (determine)g(the)h(di\013eomeorphism)456 3830 y(t)m(yp)s(e,)i(and)g(w)m (e)g(shall)f(exhibit)g(examples)g(to)g(this)h(e\013ect)g(in)f(later)f (c)m(hapters.)605 3947 y(F)-8 b(reedman)27 b(also)g(pro)m(v)m(ed)i (that)e(ev)m(ery)j(unimo)s(dular)25 b(form)h(o)s(ccurs)j(as)f(the)g (in-)456 4063 y(tersection)23 b(form)e(of)h(a)g(simply)g(connected)i (top)s(ological)19 b(4{manifold.)37 b(That)23 b(this)456 4179 y(is)36 b(not)h(true)g(for)g(di\013eren)m(tiable)f(manifolds)e (follo)m(ws)i(already)g(from)g(Rokhlin's)456 4295 y(theorem)42 b(1.15,)i(and)f(further)f(restrictions)g(w)m(ere)i(found)f(b)m(y)g (Donaldson.)72 b(In)456 4411 y(this)38 b(Chapter)h(w)m(e)g(pro)m(v)m(e) h(the)f(kno)m(wn)h(restrictions)e(imp)s(osed)f(on)i FF(Q)3062 4426 y FH(X)3168 4411 y FM(b)m(y)g(the)456 4528 y(requiremen)m(t)30 b(that)g FF(X)38 b FM(b)s(e)30 b(smo)s(oth.)42 b(These)32 b(restrictions)d(ha)m(v)m(e)j(to)d(do)h(with)g(the)456 4644 y(in)m(tegral)h(structure)i(underlying)f FF(Q)1790 4659 y FH(X)1858 4644 y FM(.)605 4760 y(If)c(w)m(e)h(w)m(ere)g(to)e (tak)m(e)i(homology)c(with)j(co)s(e\016cien)m(ts)h(in)e(a)g(\014eld)h (of)f(c)m(haracter-)456 4876 y(istic)34 b(zero,)i(suc)m(h)h(as)f FA(R)46 b FM(or)35 b FA(Q)12 b FM(,)42 b(then)36 b(the)g(in)m (tersection)f(form)f(w)m(ould)i(b)s(e)f(deter-)456 4993 y(mined,)29 b(up)h(to)g(isomorphism)e(o)m(v)m(er)j(the)f(ground)g (\014eld,)g(b)m(y)h(its)e(rank)i FF(r)s(k)s FM(\()p FF(Q)3236 5008 y FH(X)3303 4993 y FM(\))d(=)456 5109 y FF(r)s(k)s FM(\()p FF(H)676 5124 y FB(2)715 5109 y FM(\()p FF(X)8 b FM(\)\))34 b(=)g FF(b)1103 5124 y FB(2)1143 5109 y FM(\()p FF(X)8 b FM(\))37 b(and)f(signature)h FF(\033)t FM(\()p FF(Q)2140 5124 y FH(X)2207 5109 y FM(\))e(=)g FF(\033)t FM(\()p FF(X)8 b FM(\))34 b(=)h FF(b)2801 5068 y FB(+)2801 5133 y(2)2860 5109 y FM(\()p FF(X)8 b FM(\))25 b FG(\000)g FF(b)3193 5068 y FC(\000)3193 5133 y FB(2)3253 5109 y FM(\()p FF(X)8 b FM(\).)456 5225 y(Clearly)45 b(all)f(p)s(ossible)h(ranks)i(and)f(signatures)g(are)g(realised)f(b)m (y)i(the)g(simply)1912 5375 y FE(47)p eop %%Page: 48 2 48 47 bop 456 235 a FE(48)860 b(4.)34 b(NON{EXISTENCE)f(RESUL)-6 b(TS)456 443 y FM(connected)47 b(smo)s(oth)e(4{manifolds)e FF(p)p FA(C)20 b FF(P)2027 407 y FB(2)2072 443 y FM(#)p FF(q)p 2200 360 188 4 v 4 w FA(C)g FF(P)2343 414 y FB(2)2434 443 y FM(as)46 b(w)m(e)h(v)-5 b(ary)46 b(the)g(natural)456 559 y(n)m(um)m(b)s(ers)33 b FF(p)g FM(and)f FF(q)t FM(.)605 676 y(Ov)m(er)i(the)f(in)m(tegers,)g(forms)f(are)g(not)h(determined)f (b)m(y)i(their)e(rank)h(and)g(sig-)456 792 y(nature.)43 b(F)-8 b(or)32 b(example,)g(the)h(h)m(yp)s(erb)s(olic)f(form)1677 1027 y FF(H)j FM(=)1896 887 y Fw(\022)1970 967 y FM(0)82 b(1)1970 1083 y(1)g(0)2150 887 y Fw(\023)456 1262 y FM(is)32 b(not)g(isomorphic)f(o)m(v)m(er)i FA(Z)d FM(to)j(the)g(diagonal)d(form) 1420 1497 y(\(1\))21 b FG(\010)i FM(\()p FG(\000)p FM(1\))28 b(=)1999 1357 y Fw(\022)2073 1437 y FM(1)121 b(0)2073 1554 y(0)83 b FG(\000)p FM(1)2331 1357 y Fw(\023)2453 1497 y FF(;)456 1732 y FM(although)45 b(they)j(ha)m(v)m(e)g(the)f(same) f(rank)h(and)g(signature.)85 b(These)48 b(forms)e(are)456 1849 y(distinguished)30 b(b)m(y)i(their)f(parit)m(y)-8 b(.)42 b(The)32 b(form)e FF(H)39 b FM(is)31 b Fr(ev)m(en)p FM(,)i(meaning)d(that)g(ev)m(ery)456 1965 y(v)m(ector)35 b(has)f(ev)m(en)h(sel\014n)m(tersection)g(n)m(um)m(b)s(er,)g(and)f (\(1\))22 b FG(\010)i FM(\()p FG(\000)p FM(1\))34 b(is)f(not.)47 b(F)-8 b(orms)456 2081 y(whic)m(h)33 b(are)f(not)h(ev)m(en)h(are)e (called)g Fr(o)s(dd)p FM(.)605 2197 y(The)45 b(parit)m(y)e(of)g(the)h (in)m(tersection)g(form)e(is)h(closely)h(related)f(to)g(the)h(spin)456 2314 y(condition)31 b(on)h(a)h(manifold:)605 2501 y FL(Lemma)38 b(4.2.)49 b FK(If)35 b FF(X)43 b FK(is)35 b(spin,)g(then)g FF(Q)2065 2516 y FH(X)2169 2501 y FK(is)g(even.)46 b(The)35 b(c)-5 b(onverse)34 b(holds)h(if)456 2617 y FF(H)537 2632 y FB(1)576 2617 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))32 b FK(is)j(fr)-5 b(e)g(e)34 b(of)h FM(2)p FK({torsion.)605 2804 y FL(Pr)n(oof.)42 b FM(Ev)m(ery)31 b(class)f(in)f FF(H)1685 2819 y FB(2)1724 2804 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))27 b(can)i(b)s(e)h(represen)m(ted)i(b)m(y)e(a)g(smo)s (othly)456 2920 y(em)m(b)s(edded)j(closed)g(orien)m(ted)g(surface)g (\006)28 b FG(\032)g FF(X)8 b FM(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)1575 3099 y FF(T)14 b(X)8 b FG(j)1763 3114 y FB(\006)1845 3099 y FM(=)27 b FF(T)14 b FM(\006)23 b FG(\010)f FF(\027)39 b(;)456 3279 y FM(where)33 b FF(\027)39 b FM(is)33 b(the)g(normal)d (bundle)j(of)f(\006)h(in)f FF(X)8 b FM(.)605 3395 y(Recall)31 b(from)g(Chapter)i(1)f(that)g FF(X)40 b FM(is)32 b(spin)g(if)f(its)h (second)h(Stiefel{Whitney)456 3511 y(class)23 b FF(w)745 3526 y FB(2)784 3511 y FM(\()p FF(X)8 b FM(\))23 b(is)g(zero.)40 b(As)24 b FF(X)31 b FM(and)24 b(\006)f(are)g(orien)m(ted,)j(the)d (Whitney)h(sum)f(form)m(ula)456 3627 y(for)32 b FF(w)675 3642 y FB(2)746 3627 y FM(giv)m(es)1041 3806 y FG(h)p FF(w)1150 3821 y FB(2)1189 3806 y FM(\()p FF(X)8 b FM(\))p FF(;)17 b FM([\006])p FG(i)28 b FM(=)f FG(h)p FF(w)1801 3821 y FB(2)1840 3806 y FM(\(\006\))p FF(;)17 b FM([\006])p FG(i)23 b FM(+)f FG(h)p FF(w)2423 3821 y FB(2)2462 3806 y FM(\()p FF(\027)6 b FM(\))p FF(;)17 b FM([\006])p FG(i)33 b FF(:)456 3986 y FM(On)39 b(the)h(righ)m(t{hand)f(side)h(the)g (\014rst)h(summand)e(is)g(zero)h(b)s(ecause)h(\006)f(is)f(spin,)456 4102 y(and)32 b(the)g(second)h(summand)e(is)h(just)g(the)g(mo)s(d)f(2)g (degree)i(of)f FF(\027)6 b FM(,)32 b(i.e.)g(the)g(mo)s(d)f(2)456 4218 y(reduction)h(of)g(the)h(sel\014n)m(tersection)h(n)m(um)m(b)s(er)f (of)f(\006.)605 4334 y(Th)m(us)e(w)m(e)g(see)f(that)g FF(Q)1426 4349 y FH(X)1522 4334 y FM(is)f(ev)m(en)i(if)e(and)g(only)g (if)g FG(h)p FF(w)2510 4349 y FB(2)2549 4334 y FM(\()p FF(X)8 b FM(\))p FF(;)17 b FM([\006])p FG(i)28 b FM(v)-5 b(anishes)29 b(for)456 4451 y(all)h(\006.)44 b(This)33 b(pro)m(v)m(es)h(the)f(\014rst)g(claim.)605 4567 y(Con)m(v)m(ersely)-8 b(,)42 b(assume)d(that)f FF(Q)1762 4582 y FH(X)1868 4567 y FM(is)g(ev)m(en,)j(so)d(that)h FF(w)2641 4582 y FB(2)2680 4567 y FM(\()p FF(X)8 b FM(\))j(:)35 b FF(H)2999 4582 y FB(2)3038 4567 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))35 b FG(!)456 4683 y FA(Z)525 4698 y FB(2)610 4683 y FM(is)48 b(the)h(zero)g(homomorphism.)88 b(As)49 b(the)g(torsion)f(in)f FF(H)2800 4698 y FB(2)2840 4683 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))46 b(and)i(in)456 4799 y FF(H)545 4763 y FB(2)584 4799 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))44 b(is)i(the)i(same)f(as)g(that)g(in)f FF(H)2023 4814 y FB(1)2062 4799 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))44 b(b)m(y)k(P)m(oincar)m(\023)-46 b(e)47 b(dualit)m(y)f(and)456 4916 y(the)34 b(univ)m(ersal)h(co)s(e\016cien)m(t)g(theorem,)f(the)h (assumption)f(ab)s(out)g(absence)i(of)e(2{)456 5032 y(torsion)d(in)h FF(H)976 5047 y FB(1)1015 5032 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))30 b(implies)g(that)j FF(w)1930 5047 y FB(2)1969 5032 y FM(\()p FF(X)8 b FM(\))32 b(v)-5 b(anishes.)p 3374 5032 4 66 v 3378 4969 59 4 v 3378 5032 V 3436 5032 4 66 v 456 5225 a(The)33 b(lemma)d(is)j(sharp,)g(as)g(sho)m(wn)g(b)m(y) h(the)f(follo)m(wing:)p eop %%Page: 49 3 49 48 bop 1382 235 a FE(1.)35 b(THE)f(INTERSECTION)f(F)n(ORM)824 b(49)605 435 y FL(Example)37 b(4.3.)49 b FM(Let)33 b FF(X)k FG(\032)29 b FA(C)20 b FF(P)1840 399 y FB(3)1918 435 y FM(b)s(e)33 b(a)g(smo)s(oth)f(quartic)h(surface)h(de\014ned)456 551 y(o)m(v)m(er)27 b FA(R)37 b FM(whic)m(h)27 b(has)f(no)h(real)e(p)s (oin)m(ts,)i(sa)m(y)g FF(X)36 b FM(=)27 b FG(f)p FM(\()p FF(z)2339 566 y FB(0)2379 551 y FF(;)17 b(:)g(:)g(:)32 b(;)17 b(z)2659 566 y FB(3)2699 551 y FM(\))p FG(j)p FM(\006)p FF(z)2884 515 y FB(4)2880 576 y FH(i)2952 551 y FM(=)27 b(0)p FG(g)p FM(.)41 b(Then)456 667 y(complex)36 b(conjugation)g(acts)i(freely)f(on)h FF(X)45 b FM(and)37 b(the)h(quotien)m(t)f FF(Y)59 b FM(is)36 b(not)h(spin)456 783 y(but)30 b(has)g(ev)m(en)i(in)m(tersection)e(form)f FF(Q)1849 798 y FH(Y)1937 783 y FM(=)f FF(H)c FG(\010)18 b FF(E)2313 798 y FB(8)2353 783 y FM(,)30 b(where)h FF(E)2761 798 y FB(8)2831 783 y FM(is)f(the)g(negativ)m(e)456 900 y(de\014nite)i(ev)m(en)j(form)c(asso)s(ciated)h(with)h(the)g FF(E)2182 915 y FB(8)2254 900 y FM(Dynkin)f(diagramme.)605 1074 y(The)37 b(manifold)c FF(Y)57 b FM(is)36 b(a)f(complex)h(surface,) h(called)e(the)h Fr(Enriques)h(surface)p FM(,)456 1191 y(although)d(this)i(is)f(not)h(ob)m(vious)g(from)f(our)h(description.) 53 b(One)36 b(can)h(c)m(hec)m(k)h(the)456 1307 y(assertion)f(that)h(it) f(is)g(not)h(spin)g(from)f(the)h(construction,)h(or)f(one)g(can)g(in)m (v)m(ok)m(e)456 1423 y(Rokhlin's)31 b(theorem)i(1.15.)605 1539 y(The)f(form)e FF(E)1105 1554 y FB(8)1176 1539 y FM(is)h(but)h(the)f(simplest)g(one)g(of)g(the)h(non{diagonalisable)c (def-)456 1655 y(inite)34 b(forms,)j(meaning)e(that)h(it)f(is)h(not)g (equiv)-5 b(alen)m(t)36 b(o)m(v)m(er)h(the)f(in)m(tegers)h(to)f(the)456 1772 y(standard)29 b(form)f(\()p FG(\000)p FM(1\))15 b FG(\010)g FF(:)i(:)g(:)g FG(\010)e FM(\()p FG(\000)p FM(1\).)43 b(Other)30 b(examples)f(are)g FF(E)2818 1787 y FB(8)2873 1772 y FG(\010)15 b FM(\()p FG(\000)p FM(1\))31 b(whic)m(h)456 1888 y(has)k(rank)h(9)f(and)h(is)e(o)s(dd,)i(or)f (higher{rank)h(ev)m(en)g(forms)f(lik)m(e)g FF(E)2875 1903 y FB(8)2938 1888 y FG(\010)25 b FF(E)3112 1903 y FB(8)3187 1888 y FM(or)35 b(the)456 2004 y(Leec)m(h)43 b(lattice)d(whic)m(h)i(is)f(ev)m(en)i(of)e(rank)h(24.)70 b(The)42 b(classi\014cation)e(of)h(de\014nite)456 2120 y(unimo)s(dular)30 b(forms)h(o)m(v)m(er)j FA(Z)c FM(is)i(not)g (understo)s(o)s(d.)605 2237 y(Notice)k(that)h(b)m(y)h(Lemma)d(4.2)h (and)h(Rokhlin's)f(theorem)h(1.15,)g FF(E)3109 2252 y FB(8)3185 2237 y FM(or)g(the)456 2353 y(Leec)m(h)51 b(lattice)e(cannot) h(b)s(e)g(the)g(in)m(tersection)g(forms)g(of)f(simply)g(connected)456 2469 y(smo)s(oth)42 b(4{manifolds,)g(though)h(they)h(could)e(b)s(e)i (the)f(in)m(tersection)g(forms)f(of)456 2585 y(manifolds)c(with)i (2{torsion)g(in)g FF(H)1771 2600 y FB(1)1810 2585 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\).)65 b(Of)41 b(course,)j(Rokhlin's)c(theorem) 456 2702 y(sa)m(ys)35 b(nothing)e(ab)s(out)h FF(E)1376 2717 y FB(8)1439 2702 y FG(\010)24 b FF(E)1612 2717 y FB(8)1651 2702 y FM(.)48 b(All)33 b(these)i(forms)f(are)g(ruled)g(out)g (b)m(y)h(the)f(fol-)456 2818 y(lo)m(wing)g(result)h(whic)m(h)h(sa)m(ys) g(that)f(among)f(de\014nite)i(forms)e(only)h(the)h(standard)456 2934 y(ones)29 b(arise)g(from)e(smo)s(oth)h(manifolds,)g(without)g(an)m (y)i(assumption)e(on)h(the)g(\014rst)456 3050 y(homology)h(or)j(the)g (fundamen)m(tal)e(group.)605 3225 y FL(Theorem)38 b(4.4.)49 b FM(\(Donaldson\))23 b FK(If)i(a)f(smo)-5 b(oth)25 b(close)-5 b(d)24 b(oriente)-5 b(d)24 b FM(4)p FK({manifold)456 3341 y(has)34 b(a)h(de\014nite)f(interse)-5 b(ction)34 b(form,)g(then)h(this)g(form)f(diagonalizes)f(over)i FA(Z)p FK(.)605 3516 y FM(W)-8 b(e)33 b(shall)e(pro)m(v)m(e)j(this)e (in)g(Section)g(2)h(b)s(elo)m(w.)605 3632 y(In)42 b(the)g(inde\014nite) f(case,)j(the)e(classi\014cation)e(of)h(unimo)s(dular)e(symmetric)456 3748 y(bilinear)d(forms)j(is)g(kno)m(wn.)64 b(By)40 b(the)f(so{called)f (Hasse{Mink)m(o)m(wski)k(theorem)456 3864 y(they)35 b(are)f(diagonal)e (when)k(o)s(dd,)f(and)f(isomorphic)f(to)h FF(aH)d FG(\010)24 b FF(bE)2897 3879 y FB(8)2971 3864 y FM(when)36 b(ev)m(en,)456 3981 y(where)43 b FF(a;)17 b(b)43 b FG(2)h FA(Z)39 b FM(and)i FF(a)j FG(\025)f FM(1.)71 b(W)-8 b(e)42 b(ha)m(v)m(e)h (already)e(seen)i(that)f(the)g(diagonal)456 4097 y(forms)32 b(are)i(realised)f(b)m(y)h(connected)h(sums)f(of)f FA(C)20 b FF(P)2336 4061 y FB(2)2381 4097 y FM(,)34 b(so)g(that)f(w)m(e)i(are)e (left)g(with)456 4213 y(the)456 4370 y FD(Question:)75 b FM(Whic)m(h)48 b(ev)m(en)i(inde\014nite)e(forms)g(are)g(realised)g(b) m(y)h(smo)s(oth)f(4{)456 4487 y(manifolds?)605 4644 y(W)-8 b(e)47 b(consider)g(the)g(spin)f(case)i(\014rst.)86 b(By)47 b(Rokhlin's)f(theorem)g(if)g FF(aH)39 b FG(\010)456 4760 y FF(bE)569 4775 y FB(8)646 4760 y FM(is)e(the)h(in)m(tersection)g (form)e(of)h(a)g(spin)h(manifold,)e(then)i FF(b)g FM(m)m(ust)g(b)s(e)g (ev)m(en.)456 4876 y(One)e(can)g(realise)f FF(H)43 b FM(b)m(y)37 b FF(S)1478 4840 y FB(2)1541 4876 y FG(\002)25 b FF(S)1709 4840 y FB(2)1748 4876 y FM(,)37 b(and)f(3)p FF(H)c FG(\010)24 b FM(2)p FF(E)2389 4891 y FB(8)2465 4876 y FM(b)m(y)36 b(the)h FF(K)7 b FM(3)35 b(surface,)j(the)456 4993 y(quartic)47 b(surface)h(in)e FA(C)20 b FF(P)1418 4956 y FB(3)1510 4993 y FM(men)m(tioned)47 b(in)g(the)h(Example)f(4.3.) 87 b(Both)47 b(these)456 5109 y(manifolds)c(are)i(simply)f(connected)j (and)f(spin.)82 b(F)-8 b(orming)43 b(connected)k(sums)456 5225 y(and)30 b(c)m(hanging)h(orien)m(tations,)f(one)h(can)g(ralise)e FF(mH)e FG(\010)19 b FM(2)p FF(nE)2686 5240 y FB(8)2756 5225 y FM(for)30 b(all)f FF(m)f FG(\025)g FM(3)p FG(j)p FF(n)p FG(j)p FM(.)p eop %%Page: 50 4 50 49 bop 456 235 a FE(50)860 b(4.)34 b(NON{EXISTENCE)f(RESUL)-6 b(TS)456 435 y FM(The)31 b(follwing)d(conjecture)k(sa)m(ys)g(that)e(no) h(other)f(forms)g(should)h(b)s(e)g(realised)e(b)m(y)456 551 y(spin)j(manifolds:)466 669 y FB(11)p 466 685 71 4 v 483 742 a(8)546 708 y FD({Conjecture:)68 b FM(If)45 b FF(X)52 b FM(is)44 b(a)h(smo)s(oth)e(closed)i(orien)m(ted)g(spin)g (4{manifold,)456 833 y(then)33 b FF(b)719 848 y FB(2)759 833 y FM(\()p FF(X)8 b FM(\))27 b FG(\025)1066 793 y FB(11)p 1066 810 V 1084 867 a(8)1147 833 y FG(j)p FF(\033)t FM(\()p FF(X)8 b FM(\))p FG(j)p FM(.)605 989 y(If)38 b(one)g(relaxes)g(the)g(spin)g(condition)e(and)i(only)g(requires)g (that)g(the)g(in)m(ter-)456 1106 y(section)i(form)g(b)s(e)h(ev)m(en,)j (then)e(more)e(forms)g(are)g(p)s(ossible)g(as)h(sho)m(wn)h(b)m(y)g(the) 456 1222 y(existence)33 b(of)f(the)g(Enriques)h(surface.)44 b(F)-8 b(orming)30 b(connected)k(sums)e(of)f FF(S)3180 1186 y FB(2)3241 1222 y FG(\002)21 b FF(S)3405 1186 y FB(2)456 1338 y FM(and)37 b(the)h(Enriques)h(surface,)h(one)e(can)g (realise)f FF(aH)c FG(\010)26 b FF(bE)2649 1353 y FB(8)2727 1338 y FM(for)37 b(all)e FF(a)i FG(\025)g(j)p FF(b)p FG(j)g FM(as)456 1454 y(in)m(tersection)29 b(forms)g(of)h(smo)s(oth)f (4{manifolds.)39 b(Again,)30 b(it)e(is)i(p)s(ossible)f(that)g(no)456 1571 y(other)j(forms)g(o)s(ccur:)456 1728 y FD(Conjecture:)40 b FM(If)25 b FF(X)32 b FM(is)25 b(a)g(smo)s(oth)f(closed)h(orien)m(ted) g(4{manifold)d(with)j(an)g(ev)m(en)456 1844 y(in)m(tersection)32 b(form,)g(then)h FF(b)1500 1859 y FB(2)1540 1844 y FM(\()p FF(X)8 b FM(\))27 b FG(\025)1847 1805 y FB(10)p 1847 1821 V 1865 1878 a(8)1928 1844 y FG(j)p FF(\033)t FM(\()p FF(X)8 b FM(\))p FG(j)p FM(.)605 2001 y(Both)28 b(conjectures)h(are)f (satis\014ed)g(b)m(y)h(all)c(compact)j(complex)f(surfaces.)43 b(Us-)456 2117 y(ing)37 b(Theorem)h(4.5)g(b)s(elo)m(w)g(and)g(the)h (Hitc)m(hin{Thorp)s(e)e(inequalit)m(y)-8 b(,)39 b(cf.)f(Chap-)-3484 b Fm(T)-7 b(ak)n(e)35 b(this)i(up)-40 2217 y(in)27 b(Ch.)37 b(9...)456 2233 y FM(ter)45 b FD(??)p FM(,)j(one)d(sees)i(that)e(the)h (conjectures)g(hold)f(for)f(Einstein)h(4{manifolds,)456 2349 y(or)33 b(for)h(Riemannian)e(manifolds)g(satisfying)h(certain)h (pinc)m(hing)f(conditions)g(on)456 2466 y(their)38 b(sectional)h(curv) -5 b(atures.)64 b(Th)m(us)41 b(connected)g(sums)f(of)e(complex)h (surfaces)456 2582 y(and)32 b(Einstein)h(manifolds)d(also)h(satisfy)i (the)g(conjectures.)605 2698 y(In)d(this)f(Chapter)i(w)m(e)f(shall)f (pro)m(v)m(e)h(the)h(follo)m(wing)26 b(results)31 b(in)d(the)i (direction)456 2814 y(of)i(the)745 2775 y FB(11)p 745 2791 V 762 2849 a(8)825 2814 y FM({Conjecture)i(b)s(ey)m(ond)g(Theorem) e(4.4:)605 2978 y FL(Theorem)38 b(4.5.)49 b FK(L)-5 b(et)27 b FF(X)34 b FK(b)-5 b(e)27 b(a)g(smo)-5 b(oth)26 b(close)-5 b(d)26 b(oriente)-5 b(d)26 b(spin)g FM(4)p FK({manifold.)456 3094 y(If)34 b(its)h(interse)-5 b(ction)34 b(form)h(is)f FF(mH)c FG(\010)23 b FM(2)p FF(nE)2039 3109 y FB(8)2113 3094 y FK(with)35 b FF(n)28 b FG(6)p FM(=)f(0)p FK(,)35 b(then)g FF(m)28 b FG(\025)g FM(3)p FK(.)605 3257 y FM(In)i(other)g(w)m (ords,)h(the)1456 3218 y FB(11)p 1456 3234 V 1473 3291 a(8)1536 3257 y FM({Conjecture)g(holds)e(for)g(manifolds)f(with)h FF(b)3176 3216 y FB(+)3176 3281 y(2)3263 3257 y FG(\024)g FM(2.)456 3373 y(This)48 b(w)m(as)h(pro)m(v)m(ed)h(b)m(y)f(Donaldson)e (using)h(Y)-8 b(ang{Mills)46 b(theory)-8 b(,)53 b(under)c(the)456 3489 y(additional)30 b(assumption)j(that)g FF(H)1727 3504 y FB(1)1766 3489 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))31 b(is)i(free)g(of)g(2{torsion.)45 b(In)33 b(Section)h(3)456 3606 y(w)m(e)f(shall)e(giv)m(e)i(a)f(pro)s(of)g(using)g(the)h(monop)s (ole)e(mo)s(duli)f(spaces.)605 3722 y(In)j(Section)f(4)h(w)m(e)g(shall) e(pro)m(v)m(e:)605 3885 y FL(Theorem)38 b(4.6.)49 b FM(\(F)-8 b(uruta\))33 b FK(L)-5 b(et)34 b FF(X)42 b FK(b)-5 b(e)34 b(a)g(smo)-5 b(oth)34 b(close)-5 b(d)33 b(orientable)g(spin)456 4001 y FM(4)p FK({manifold.)43 b(If)34 b(its)h(interse)-5 b(ction)34 b(form)g(is)h FF(mH)30 b FG(\010)23 b FM(2)p FF(nE)2581 4016 y FB(8)2620 4001 y FK(,)35 b(then)g FF(m)28 b FG(\025)g FM(2)p FF(n)p FK(.)605 4165 y FM(Equiv)-5 b(alen)m(tly)d(,)1560 4324 y FF(b)1601 4339 y FB(2)1641 4324 y FM(\()p FF(X)8 b FM(\))28 b FG(\025)1949 4256 y FM(10)p 1949 4301 98 4 v 1973 4392 a(8)2056 4324 y FF(\033)t FM(\()p FF(X)8 b FM(\))32 b FF(:)605 4528 y FD(1.1.)56 b(A)25 b(few)g(w)m(ords)g(on)g(the)g(fundamen)m(tal)h (group.)49 b FM(By)22 b(doing)f(surgery)456 4644 y(on)j(connected)i (sums)f(of)f(copies)g(of)g FF(S)1818 4608 y FB(1)1863 4644 y FG(\002)5 b FF(S)2011 4608 y FB(3)2076 4644 y FM(one)24 b(can)h(realise)e(ev)m(ery)k(\014nitely)c(pre-)456 4760 y(sen)m(table)37 b(group)f(as)g(the)h(fundamen)m(tal)f(group)g(of) g(a)g(smo)s(oth)g(closed)g(orien)m(ted)456 4876 y(spin)c(4{manifold.) 605 4993 y(W)-8 b(e)32 b(ha)m(v)m(e)g(seen)g(that)f(the)h(absence)h(of) d(2{torsion)g(in)g(the)i(Ab)s(elianised)e(fun-)456 5109 y(damen)m(tal)44 b(group)h FF(H)1264 5124 y FB(1)1303 5109 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))43 b(can)i(constrain)g(the)h (in)m(tersection)f(form)f(of)h FF(X)8 b FM(.)456 5225 y(Ho)m(w)m(ev)m(er,)35 b(the)e(free)g(part)f(of)g FF(H)1634 5240 y FB(1)1673 5225 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))30 b(has)j(no)g(suc)m(h)h(e\013ect:)p eop %%Page: 51 5 51 50 bop 1429 236 a FE(2.)35 b(PR)n(OOF)d(OF)i(THEOREM)f Fg(??)871 b FE(51)605 435 y FL(Lemma)38 b(4.7.)49 b FK(If)39 b(a)h(biline)-5 b(ar)39 b(form)g(o)-5 b(c)g(curs)40 b(as)f(the)h (interse)-5 b(ction)39 b(form)h(of)456 551 y(some)35 b(close)-5 b(d)34 b(smo)-5 b(oth)35 b FM(4)p FK({manifold)f FF(X)8 b FK(,)36 b(then)f(it)h(also)f(o)-5 b(c)g(curs)36 b(as)f(the)h(interse)-5 b(c-)456 667 y(tion)46 b(form)g(of)g(a)g FM(4)p FK({manifold)e FF(Y)68 b FK(with)46 b(pr)-5 b(escrib)g(e)g(d)45 b(\014rst)i(Betti)f(numb)-5 b(er.)79 b(In)456 783 y(p)-5 b(articular,)34 b(one)h(c)-5 b(an)34 b(r)-5 b(e)g(quir)g(e)35 b FF(b)1655 798 y FB(1)1695 783 y FM(\()p FF(Y)21 b FM(\))28 b(=)f(0)35 b FK(and)f FF(Q)2330 798 y FH(Y)2419 783 y FM(=)27 b FF(Q)2599 798 y FH(X)2667 783 y FK(.)605 900 y(If)34 b FF(X)43 b FK(is)35 b(spin,)f(then)g FF(Y)57 b FK(c)-5 b(an)34 b(b)-5 b(e)34 b(taken)h(to)g(b)-5 b(e)35 b(spin)f(as)g(wel)5 b(l.)605 1089 y FL(Pr)n(oof.)42 b FM(It)c(is)f(clear)h(that)f(w)m(e)i(can)f(alw)m(a)m(ys)h(increase)f FF(b)2725 1104 y FB(1)2803 1089 y FM(b)m(y)g(forming)e(the)456 1205 y(connected)g(sum)f(of)f FF(X)42 b FM(with)35 b(copies)g(of)f FF(S)2052 1169 y FB(1)2115 1205 y FG(\002)24 b FF(S)2282 1169 y FB(3)2321 1205 y FM(.)50 b(This)35 b(preserv)m(es)j(the)d(in)m (ter-)456 1321 y(section)d(form)g(and)g(the)h(prop)s(ert)m(y)h(of)e(b)s (eing)g(spin.)605 1437 y(W)-8 b(e)24 b(can)f(also)f(decrease)j FF(b)1536 1452 y FB(1)1599 1437 y FM(if)d(it)g(is)h(p)s(ositiv)m(e.)40 b(Let)23 b FF(S)2483 1401 y FB(1)2550 1437 y FG(\032)28 b FF(X)j FM(b)s(e)23 b(an)g(em)m(b)s(edded)456 1553 y(circle)34 b(whic)m(h)i(is)f(a)g(basis)g(elemen)m(t)g(of)g(the)h(free)f(part)g(of) g FF(H)2668 1568 y FB(1)2708 1553 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\).)49 b(By)35 b(doing)456 1670 y(elemen)m(tary)27 b(surgery)i(on)e FF(S)1485 1634 y FB(1)1552 1670 y FM(w)m(e)h(decrease) h FF(b)2111 1685 y FB(1)2179 1670 y FM(without)e(c)m(hanging)g FF(H)3022 1685 y FB(2)3061 1670 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\))25 b(or)456 1786 y FF(w)526 1801 y FB(2)565 1786 y FM(\()p FF(X)8 b FM(\).)p 3374 1786 4 66 v 3378 1723 59 4 v 3378 1786 V 3436 1786 4 66 v 456 1983 a(W)-8 b(e)41 b(in)m(vite)f(the)h(reader)g(to)f(c)m(hec)m(k)j(that)e(the)g (argumen)m(t)f(breaks)i(do)m(wn)f(if)f(the)456 2099 y(circle)31 b(has)i(\014nite)g(order)f(in)g FF(H)1583 2114 y FB(1)1622 2099 y FM(\()p FF(X)r(;)17 b FA(Z)p FM(\).)605 2215 y(Although)42 b(Lemma)f(4.7)h(sho)m(ws)j(that)d(the)h(\014rst)g(Betti)f(n)m(um)m(b)s (er)h(b)m(y)h(itself)456 2332 y(do)s(es)50 b(not)h(constrain)f(the)g (in)m(tersection)h(form,)i(one)e(should)f(k)m(eep)i(in)e(mind)456 2448 y(that)27 b(the)i(group)e(cohomology)f(of)i(the)g(fundamen)m(tal)f (group)h(do)s(es)g(imp)s(ose)f(con-)456 2564 y(strain)m(ts.)43 b(Namely)-8 b(,)32 b(if)1482 2706 y FF(c)11 b FM(:)33 b FF(X)j FG(\000)-16 b(!)27 b FF(K)7 b FM(\()p FF(\031)2083 2721 y FB(1)2123 2706 y FM(\()p FF(X)h FM(\))p FF(;)17 b FM(1\))456 2876 y(is)32 b(the)h(classifying)e(map)h(of)g(the)h(univ)m (ersal)f(co)m(v)m(ering)2507 2851 y(~)2479 2876 y FF(X)k FG(!)27 b FF(X)8 b FM(,)32 b(then)1388 3058 y FF(c)1430 3017 y FC(\003)1480 3058 y FM(:)i FF(H)1630 3017 y FC(\003)1668 3058 y FM(\()p FF(\031)1761 3073 y FB(1)1801 3058 y FM(\()p FF(X)8 b FM(\)\))27 b FG(\000)-16 b(!)28 b FF(H)2309 3017 y FC(\003)2348 3058 y FM(\()p FF(X)8 b FM(\))456 3241 y(is)27 b(an)h(algebra)e(homomorphism)f(whic)m(h)k(is)e(an)h (isomorphism)d(in)i(degrees)j(0)d(and)456 3357 y(1)g(and)h(is)f (injectiv)m(e)h(in)f(degree)h(2.)42 b(Th)m(us,)30 b FF(b)2029 3372 y FB(2)2069 3357 y FM(\()p FF(X)8 b FM(\))27 b(=)h FF(r)s(k)s FM(\()p FF(Q)2581 3372 y FH(X)2648 3357 y FM(\))g(is)f(b)s(ounded)i(b)s(elo)m(w)456 3473 y(b)m(y)j FF(b)631 3488 y FB(2)671 3473 y FM(\()p FF(\031)764 3488 y FB(1)803 3473 y FM(\()p FF(X)8 b FM(\)\).)43 b(Moreo)m(v)m(er,)33 b(the)f(signature)f(is)g(constrained)g(as)h(w)m(ell,)f(as)g(in)g(the) 456 3589 y(follo)m(wing)e(example:)605 3778 y FL(Example)37 b(4.8.)49 b FM(Let)22 b FF(X)29 b FM(b)s(e)22 b(a)g(closed)g(orien)m (ted)g(4{manifold)c(with)j FF(\031)3164 3793 y FB(1)3204 3778 y FM(\()p FF(X)8 b FM(\))27 b(=)456 3894 y FA(Z)525 3858 y FB(2)561 3894 y FM(.)42 b(Then)29 b FF(c)11 b FM(:)34 b FF(X)h FG(!)27 b FF(T)1308 3858 y FB(2)1348 3894 y FM(,)h(and)g(as)g FF(c)1745 3858 y FC(\003)1813 3894 y FM(preserv)m(es)i(the)f(cup)f(pro)s(duct)g(it)f(m)m(ust)h(map) 456 4011 y FF(H)545 3974 y FB(2)584 4011 y FM(\()p FF(T)693 3974 y FB(2)732 4011 y FM(\))36 b(to)f(an)h(isotropic)f(subspace)j(in)e FF(H)2085 3974 y FB(2)2123 4011 y FM(\()p FF(X)8 b FM(\).)54 b(Th)m(us)38 b FF(Q)2697 4026 y FH(X)2801 4011 y FM(cannot)e(b)s(e)g (de\014-)456 4127 y(nite.)605 4316 y(In)31 b(fact,)g(it)e(is)h(not)g (hard)h(to)f(see)i(that)e(for)g(no)g FF(k)h(>)d FM(1)i(can)h FA(Z)2804 4280 y FH(k)2874 4316 y FM(b)s(e)g(the)g(funda-)456 4432 y(men)m(tal)d(group)i(of)f(a)g(manifold)e(with)i(de\014nite)h(in)m (tersection)f(form,)g(although)g FA(Z)456 4548 y FM(is)j(the)h (fundamen)m(tal)e(group)i(of)f FF(S)1739 4512 y FB(1)1800 4548 y FG(\002)23 b FF(S)1966 4512 y FB(3)2005 4548 y FM(.)1351 4818 y FD(2.)56 b(Pro)s(of)37 b(of)h(Theorem)f(4.4)605 4993 y FM(Let)22 b FF(X)30 b FM(b)s(e)22 b(a)f(smo)s(oth)g(4{manifold)e (with)j(de\014nite)g(in)m(tersection)g(form.)38 b(After)456 5109 y(c)m(hanging)46 b(the)i(orien)m(tation)d(if)h(necessary)-8 b(,)53 b(w)m(e)48 b(ma)m(y)f(assume)g FF(b)2921 5068 y FB(+)2921 5133 y(2)3033 5109 y FM(=)52 b(0.)87 b(By)456 5225 y(Lemma)31 b(4.7)h(w)m(e)i(ma)m(y)e(also)g(assume)h FF(b)1890 5240 y FB(1)1957 5225 y FM(=)28 b(0.)p eop 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b(+)g(8)p FF(k)s FM(,)36 b(so)g(dim)15 b FF(M)2734 1548 y FH(!)2818 1533 y FM(=)33 b(2)p FF(k)27 b FG(\000)e FM(1)35 b(is)g(an)456 1649 y(o)s(dd)d(n)m(um)m(b)s(er.)44 b(W)-8 b(e)33 b(no)m(w)g(mak)m(e)g(the)g(follo)m(wing:)605 1824 y FL(Assumption)k(4.9.)49 b FM(dim)15 b FF(M)1715 1839 y FH(!)1794 1824 y FF(>)27 b FM(0,)33 b(equiv)-5 b(alen)m(tly)32 b FF(k)e(>)e FM(0.)605 2000 y(Then)45 b(for)d(generic)i FF(!)i FM(w)m(e)f(ha)m(v)m(e)g(a)e(compact)g (non{empt)m(y)h(mo)s(duli)c(space)456 2116 y(whic)m(h)33 b(is)f(smo)s(oth)g(a)m(w)m(a)m(y)i(from)d(the)i(unique)g(reducible)g (solution.)42 b(W)-8 b(e)33 b(need)h(to)456 2232 y(understand)g(the)f (neigh)m(b)s(ourho)s(o)s(d)e(of)h(this)g(reducible)h(solution.)605 2348 y(The)g(linearised)e(equation)h(at)g(the)g(reducible)g(uncouples)h (in)m(to)e(the)i(half)e(de)456 2465 y(Rham)42 b(complex)h(\(1.2\))h (and)f(the)h(Dirac)f(op)s(erator.)76 b(As)44 b FF(b)2708 2480 y FB(1)2795 2465 y FM(=)i FF(b)2958 2423 y FB(+)2958 2489 y(2)3064 2465 y 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%%Page: 53 7 53 52 bop 1429 236 a FE(2.)35 b(PR)n(OOF)d(OF)i(THEOREM)f Fg(??)871 b FE(53)456 435 y FM(fact)36 b(that)g FF(M)961 450 y FH(!)1048 435 y FM(giv)m(es)h(a)f(zero{b)s(ordism)f(of)h FF(\014)6 b FM(,)37 b(as)f(a)g(bundle)h(o)m(v)m(er)g FF(N)10 b FM(,)38 b(b)s(ecause)456 551 y(zero{b)s(ordan)m(t)32 b(bundles)i(ha)m(v)m(e)f(v)-5 b(anishing)32 b(Stiefel{Whitney)f(n)m(um) m(b)s(ers.)605 667 y(W)-8 b(e)29 b(ha)m(v)m(e)h(sho)m(wn)h(that)d (Assumption)h(4.9)f(leads)h(to)f(a)h(con)m(tradiction.)41 b(The-)456 783 y(orem)46 b(4.4)h(no)m(w)h(follo)m(ws)e(from)g(the)i (next)g(Prop)s(osition)e(whic)m(h)i(asserts)h(that)456 900 y(Assumption)34 b(4.9)g(can)g(b)s(e)h(satis\014ed)g(if)e(and)h (only)g(if)g(the)g(in)m(tersection)h(form)e(of)456 1016 y FF(X)40 b FM(is)32 b(not)g(diagonalisable)e(o)m(v)m(er)j(the)g(in)m (tegers.)605 1184 y FL(Pr)n(oposition)38 b(4.10.)48 b FM(\(Elkies\))37 b FK(A)h(ne)-5 b(gative)36 b(de\014nite)g(unimo)-5 b(dular)37 b(sym-)456 1300 y(metric)51 b(biline)-5 b(ar)51 b(form)g FF(Q)h FK(admits)f(a)g(char)-5 b(acteristic)51 b(ve)-5 b(ctor)52 b FF(v)j FK(with)d FF(v)3271 1264 y FB(2)3369 1300 y FF(>)456 1416 y FG(\000)17 b FM(rk\()p FF(Q)p FM(\))35 b FK(if)g(and)f(only)h(if)f FF(Q)i FK(is)e(not)h (diagonalisable)e(over)h FA(Z)p FK(.)605 1584 y FM(One)42 b(direction)e(is)h(ob)m(vious:)62 b(if)40 b FF(Q)i FM(is)f (diagonalisable,)f(then)i(the)g(sum)g(of)456 1700 y(the)29 b(basis)g(v)m(ectors)i(with)e(resp)s(ect)h(to)f(whic)m(h)h FF(Q)f FM(is)g(diagonal)e(is)i(a)g(v)m(ector)h FF(v)j FM(with)456 1816 y FF(v)507 1780 y FB(2)584 1816 y FM(maximal)h(among)j (all)f(c)m(haracteristic)i(v)m(ectors,)j(and)d FF(v)2664 1780 y FB(2)2740 1816 y FM(=)f FG(\000)17 b FM(rk)q(\()p FF(Q)p FM(\).)59 b(The)456 1932 y(con)m(v)m(erse)39 b(w)m(as)f(pro)m(v) m(ed)g(b)m(y)g(Elkies)f(in)f([)p FD(9)p FM(].)57 b(Note)37 b(that)f(for)h(sp)s(eci\014c)g(examples)456 2049 y(of)31 b(non{diagonalisable)e(de\014nite)k(forms,)e(sa)m(y)j(ev)m(en)g(ones,)f (one)f(do)s(es)h(not)f(need)456 2165 y(to)i(app)s(eal)f(to)h(Elkies's)h (prop)s(osition,)e(as)i(the)g(required)g(c)m(haracteristic)f(v)m(ector) 456 2281 y(is)e(easily)f(found)i(explicitely)-8 b(.)605 2397 y(There)40 b(are)f(a)f(n)m(um)m(b)s(er)i(of)e(p)s(ossible)g(v)-5 b(ariations)37 b(of)h(the)h(argumen)m(t)g(in)f(the)456 2514 y(pro)s(of)31 b(of)h(Theorem)h(4.4:)605 2681 y FL(Remark)38 b(4.11.)48 b FM(Instead)23 b(of)e(arguing)g(with)g(the)h(Kuranishi)f (map)g(\011)11 b(:)33 b FA(C)3317 2645 y FH(k)r FB(+)p FH(r)3482 2681 y FG(!)456 2798 y FA(C)522 2761 y FH(r)603 2798 y FM(for)k(an)h(unk)m(o)m(wn)h FF(r)s FM(,)f(one)g(could)f(reduce) i(to)f(a)f(situation)f(where)j FF(r)f FM(=)f(0,)h(so)456 2914 y(that)32 b FF(N)39 b FM(=)29 b FA(C)20 b FF(P)1032 2878 y FH(k)r FC(\000)p FB(1)1170 2914 y FM(.)45 b(This)33 b(can)g(b)s(e)h(done)f(b)m(y)h(sho)m(wing)f(that)g(for)g(generic)g FF(!)j FM(the)456 3030 y(Dirac)31 b(op)s(erator)h FF(D)1197 2989 y FB(+)1194 3057 y FH(A)1247 3065 y Fx(!)1328 3030 y FM(is)g(surjectiv)m(e.)605 3198 y FL(Remark)38 b(4.12.)48 b FM(Using)f(the)g(orien)m(tabilit)m(y)d(of)i(the)h(mo)s(duli)d(space)k FF(M)3377 3162 y FC(\003)3367 3222 y FH(!)3417 3198 y FM(,)456 3314 y(the)d(\014nal)f(con)m(tradiction)g(is)h(a)f (consequence)k(of)d(Stok)m(es's)i(theorem)d(applied)456 3430 y(to)37 b(the)h(top)f(p)s(o)m(w)m(er)i(of)e(the)h(Euler)f(class)h (of)f(the)h(basep)s(oin)m(t{\014bration.)57 b(Th)m(us,)456 3546 y(at)35 b(the)i(exp)s(ense)h(of)e(in)m(v)m(oking)g(orien)m (tabilit)m(y)-8 b(,)34 b(one)j(do)s(es)g(not)f(need)h(to)f(use)h(the) 456 3663 y(v)-5 b(anishing)31 b(of)h(the)h(Stiefel{Whitney)f(n)m(um)m (b)s(ers)h(of)f(zero{b)s(ordan)m(t)h(bundles.)605 3830 y FD(2.1.)56 b(Comparison)29 b(with)g(Donaldson's)h(pro)s(of.)49 b FM(The)27 b(ab)s(o)m(v)m(e)f(pro)s(of)f(of)456 3947 y(Theorem)34 b(4.4)f(is)h(v)m(ery)h(indirect,)f(and)g(the)g(reader)h (migh)m(t)d(b)s(e)j(w)m(ondering)f(ho)m(w)456 4063 y(Donaldson)24 b(ev)m(er)j(though)m(t)f(of)g(the)g(statemen)m(t.)41 b(He)27 b(found)e(the)i(statemen)m(t,)g(and)456 4179 y(the)41 b(original)d(pro)s(of,)k(using)e(Y)-8 b(ang{Mills)39 b(theory)i(whic)m(h)h(giv)m(es)f(an)g(approac)m(h)456 4295 y(whic)m(h,)33 b(though)f(more)g(direct,)h(is)f(tec)m(hnically)g (harder)g(to)h(carry)g(out.)605 4411 y(The)25 b(mo)s(del)e(situation)g (is)h(that)g(of)g(the)h(4{sphere)h(app)s(earing)d(as)i(the)g(b)s(ound-) 456 4528 y(ary)33 b(of)f(the)h(so{called)f(one{instan)m(ton)h(mo)s (duli)d(space,)k(whic)m(h)g(happ)s(ens)g(to)e(b)s(e)456 4644 y(the)j(5{ball.)47 b(Donaldson)34 b(generalised)g(this)g(to)g(pro) m(v)m(e)i(that)f(if)e FF(X)43 b FM(is)34 b(a)g(smo)s(oth)456 4760 y(closed)25 b(orien)m(ted)h(4{manifold)c(with)j(negativ)m(e)h (de\014nite)g(in)m(tersection)f(form)g FF(Q)3350 4775 y FH(X)3417 4760 y FM(,)456 4876 y(then)34 b(the)g(mo)s(duli)d(space)j (of)f(ASD)h(connections)g(on)f(the)h FF(S)6 b(U)k FM(\(2\){bundle)34 b(with)456 4993 y(second)e(Chern)h(class)e(1)h(o)m(v)m(er)g FF(X)8 b FM(,)32 b(though)f(not)g(compact,)h(could)f(b)s(e)h(truncated) 456 5109 y(to)j(giv)m(e)h(a)g(compact)f(5{dimensional)e(space)k(with)f FF(X)44 b FM(as)36 b(its)f(b)s(oundary)-8 b(.)54 b(This)456 5225 y(space)29 b(is)e(a)h(manifold)d(a)m(w)m(a)m(y)k(from)e(some)g (singularities)f(arising)g(from)h(reducible)p eop %%Page: 54 8 54 53 bop 456 235 a FE(54)860 b(4.)34 b(NON{EXISTENCE)f(RESUL)-6 b(TS)456 435 y FM(connections,)50 b(and)d(thinking)e(of)h(it)g(as)g(a)g (cob)s(ordism)g(b)s(et)m(w)m(een)i FF(X)54 b FM(and)47 b(the)456 551 y(union)32 b(of)h(the)h(links)f(of)f(the)i(singularities) d(giv)m(es)j(the)f(desired)h(diagonalisation)456 667 y(of)e FF(Q)644 682 y FH(X)711 667 y FM(.)605 783 y(This)22 b(is)f(a)h(v)m(ery)h(natural,)g(direct,)h(argumen)m(t,)f(and)f(\014ts)g (in)f(with)h(the)g(cob)s(ordism{)456 900 y(in)m(v)-5 b(ariance)26 b(of)h(the)h(signature.)41 b(It)28 b(do)s(es)g(not)f (require)h(an)m(y)g(algebraic)e(result)h(lik)m(e)456 1016 y(Prop)s(osition)21 b(4.10.)40 b(Ho)m(w)m(ev)m(er,)27 b(the)d(argumen)m(t)f(do)s(es)h(require)f(the)h(cob)s(ordism)e(to)456 1132 y(b)s(e)33 b(orien)m(ted,)i(whereas)g(the)f(indirect)f(approac)m (h)h(with)f(the)h(monop)s(ole)e(mo)s(duli)456 1248 y(spaces)i(do)s(es)f (not)f(need)i(an)m(y)f(orien)m(tations.)1351 1516 y FD(3.)56 b(Pro)s(of)37 b(of)h(Theorem)f(4.5)605 1690 y FM(Theorem)c(4.5)f(can)h (b)s(e)g(form)m(ulated)e(in)h(the)h(follo)m(wing)d(equiv)-5 b(alen)m(t)32 b(w)m(a)m(y:)605 1879 y FL(Theorem)38 b(4.13.)49 b FK(L)-5 b(et)25 b FF(X)33 b FK(b)-5 b(e)24 b(a)h(smo)-5 b(oth)24 b(close)-5 b(d)24 b(oriente)-5 b(d)24 b(spin)h FM(4)p FK({manifold.)456 1995 y(If)34 b FM(min)n FG(f)p FF(b)811 1954 y FB(+)811 2019 y(2)871 1995 y FM(\()p FF(X)8 b FM(\))p FF(;)17 b(b)1121 1954 y FC(\000)1121 2019 y FB(2)1180 1995 y FM(\()p FF(X)8 b FM(\))p FG(g)27 b(\024)h FM(2)p FK(,)35 b(then)f FF(\033)t FM(\()p FF(X)8 b FM(\))28 b(=)f(0)p FK(.)605 2183 y FM(It)e(is)g(clear)f(that)h(this)g (statemen)m(t)g(w)m(ould)g(b)s(e)g(false)g(in)f(the)h(case)h(of)f (manifolds)456 2300 y(with)32 b FF(min)p FG(f)p FF(b)945 2258 y FB(+)945 2324 y(2)1005 2300 y FF(;)17 b(b)1090 2258 y FC(\000)1090 2324 y FB(2)1149 2300 y FG(g)28 b FM(=)f(3,)33 b(b)s(ecause)h(of)e(the)h(existence)h(of)e(the)h FF(K)7 b FM(3)32 b(surface.)605 2416 y(In)40 b(outline,)h(the)f(pro)s (of)f(of)h(Theorem)g(4.5)g(follo)m(ws)e(that)i(of)f(Theorem)i(4.4)456 2532 y(giv)m(en)g(in)f(the)h(previous)g(section.)69 b(Ho)m(w)m(ev)m (er,)45 b(there)d(is)e(one)h(new)h(ingredien)m(t,)456 2648 y(whic)m(h)e(is)f(the)i(symmetry)f FF(\034)51 b FM(of)40 b(the)g(monop)s(ole)e(equations)i(for)f(Spin-)g(rather)456 2765 y(than)33 b(Spin)874 2723 y FH(c)909 2765 y FM({structures.)48 b(A)34 b(v)-5 b(arian)m(t)32 b(of)i(the)g(pro)s(of)e(will)g(b)s(e)i (used)g(in)f(the)h(next)456 2881 y(Chapter)f(to)f(calculate)g(the)h (monop)s(ole)d(in)m(v)-5 b(arian)m(ts)32 b(of)g(the)h FF(K)7 b FM(3)33 b(surface.)-3231 b Fm(Homotop)n(y)-40 2980 y(K3?)139 b(Men-)-40 3080 y(tion)27 b Fh(\034)9 b Fm(?)605 2997 y FM(F)-8 b(or)39 b(the)g(pro)s(of)g(of)g(the)g (Theorem)h(w)m(e)g(ma)m(y)f(assume)h FF(b)2687 3012 y FB(1)2766 2997 y FM(=)f(0,)i(b)s(ecause)g(of)456 3113 y(Lemma)k(4.7.)87 b(Cho)s(osing)46 b(the)i(appropriate)e(orien)m (tation)f(and)i(forming)e(the)456 3229 y(connected)c(sum)f(with)f FF(S)1427 3193 y FB(2)1494 3229 y FG(\002)27 b FF(S)1664 3193 y FB(2)1743 3229 y FM(if)39 b(necessary)-8 b(,)44 b(w)m(e)d(can)f(assume)h FF(b)3023 3188 y FB(+)3023 3254 y(2)3122 3229 y FM(=)f(2.)65 b(If)456 3346 y(the)28 b(signature)g(is)g (not)g(zero,)h(then)g(b)m(y)g(Rokhlin's)e(theorem)h(1.15)g(it)f(is)h FG(\000)p FM(16)p FF(k)j FM(for)456 3462 y(some)h FF(k)f(>)c FM(0.)44 b(W)-8 b(e)33 b(will)d(sho)m(w)j(that)g FF(k)e(>)c FM(0)32 b(leads)h(to)f(a)g(con)m(tradiction.)605 3578 y(W)-8 b(e)33 b(consider)f(the)g(monop)s(ole)f(equations)h(for)g(a)f (connection)i(and)f(a)g(spinor)456 3694 y(for)j(a)h(Spin)883 3653 y FH(c)918 3694 y FM({structure)h(coming)e(from)g(an)h(honest)h (spin)f(structure)i(on)e FF(X)8 b FM(.)54 b(In)456 3811 y(this)43 b(case)j FF(c)918 3826 y FB(1)957 3811 y FM(\()p FF(V)1052 3826 y FC(\006)1111 3811 y FM(\))h(=)g(0,)g(and)e(there)g(is) e(a)h(reducible)g(solution)f(\(0)p FF(;)17 b FM(0\))43 b(to)h(the)456 3927 y(unp)s(erturb)s(ed)49 b(monop)s(ole)d(equations)i (\(2.1\))f(and)h(\(2.3\),)k(meaning)46 b(that)i(the)456 4054 y(connection)967 4029 y(^)941 4054 y FF(A)30 b FM(on)h FF(det)1309 4069 y Fv(C)1361 4054 y FF(V)1418 4069 y FC(\006)1507 4054 y FM(is)f(a)g(pro)s(duct)g(connection)h(and)f(the)h (spinor)f FF(\036)g FM(v)-5 b(an-)456 4171 y(ishes)39 b(iden)m(tically)-8 b(.)59 b(This)39 b(solution)e(is)h(unique)h(up)g (to)f(gauge)g(equiv)-5 b(alence.)62 b(It)456 4287 y(giv)m(es)33 b(a)f(p)s(oin)m(t)g(in)f(a)i(mo)s(duli)d(space)j FF(M)44 b FM(of)32 b(dimension)1090 4511 y(dim)15 b FF(M)39 b FM(=)27 b FG(\000)1592 4444 y FM(1)p 1592 4488 49 4 v 1592 4580 a(4)1651 4511 y(\(2)p FF(\037)p FM(\()p FF(X)8 b FM(\))22 b(+)g FF(\033)t FM(\()p FF(X)8 b FM(\)\))27 b(=)h(4)p FF(k)d FG(\000)e FM(3)32 b FF(:)605 4722 y FM(The)c(linearised)d(equation)i(at)g(the)g(reducible)g(solution)e (uncouples)j(in)m(to)e(the)456 4839 y(half)f(de)i(Rham)e(complex)h (\(1.2\))g(and)h(the)g(Dirac)e(op)s(erator.)41 b(The)27 b(half)e(de)i(Rham)456 4955 y(complex)36 b(do)s(es)h(not)f(con)m (tribute)h(to)f FF(H)1928 4919 y FB(1)1920 4984 y(\()p FH(A)2000 4992 y Fx(!)2045 4984 y FH(;)p FB(0\))2131 4955 y FM(,)i(and)f(con)m(tributes)g(a)f(cop)m(y)i(of)e FA(R)3399 4919 y FB(2)456 5088 y FM(to)30 b FF(H)662 5052 y FB(2)654 5118 y(\()p FH(A)734 5126 y Fx(!)779 5118 y FH(;)p FB(0\))866 5088 y FM(.)43 b(The)32 b(complex)f(index)g (of)g(the)g(Dirac)f(op)s(erator)g FF(D)2788 5103 y FH(A)2877 5088 y FM(is)g(2)p FF(k)s FM(,)i(so)f(if)f(its)456 5225 y(cok)m(ernel)j(is)f FA(C)998 5189 y FH(r)1075 5225 y FM(for)f(some)i FF(r)d FG(\025)e FM(0,)k(its)g(k)m(ernel)h(is)f FA(C)2342 5189 y FB(2)p FH(k)r FB(+)p FH(r)2514 5225 y FM(.)44 b(Th)m(us,)34 b(the)f(Kuranishi)p eop %%Page: 55 9 55 54 bop 1429 236 a FE(3.)35 b(PR)n(OOF)d(OF)i(THEOREM)f Fg(??)871 b FE(55)456 435 y FM(mo)s(del)23 b(of)h(a)g(neigh)m(b)s (ourho)s(o)s(d)g(of)g([0)p FF(;)17 b FM(0])25 b(in)f FF(M)35 b FM(is)24 b(giv)m(en)h(b)m(y)h(\011)2673 399 y FC(\000)p FB(1)2767 435 y FM(\(0\))p FF(=U)10 b FM(\(1\),)26 b(where)456 551 y FF(U)10 b FM(\(1\))32 b(is)h(the)g(stabiliser)d(of)j ([0)p FF(;)17 b FM(0])32 b(in)f(the)i(gauge)g(group,)f(and)1478 750 y(\011)11 b(:)33 b FA(C)1691 709 y FB(2)p FH(k)r FB(+)p FH(r)1891 750 y FG(\000)-16 b(!)27 b FA(C)2145 709 y FH(r)2211 750 y FG(\010)c FA(R)2377 709 y FB(2)456 945 y FM(is)38 b(a)h FF(U)10 b FM(\(1\){equiv)-5 b(arian)m(t)39 b(map)f(with)h(\011\(0\))f(=)h(0)g(whic)m(h)g(is)g(transv)m(erse)i(to)e (0)g FG(2)456 1061 y FA(C)522 1025 y FH(r)572 1061 y FG(\010)6 b FA(R)721 1025 y FB(2)792 1061 y FM(a)m(w)m(a)m(y)26 b(from)d(the)j(origin)c(in)i FA(C)1851 1025 y FB(2)p FH(k)r FB(+)p FH(r)2024 1061 y FM(.)41 b(The)25 b FF(U)10 b FM(\(1\){action)24 b(is)h(the)g(standard)456 1177 y(one)34 b(on)h(the)g(spaces)h FA(C)1311 1141 y FB(2)p FH(k)r FB(+)p FH(r)1518 1177 y FM(and)e FA(C)1775 1141 y FH(r)1854 1177 y FM(of)g(harmonic)f(spinors,)i(and)g(is)f(trivial)d(on)456 1293 y(the)i(space)g FA(R)950 1257 y FB(2)1028 1293 y FM(of)f(self{dual)g(harmonic)f(t)m(w)m(o{forms.)605 1410 y(The)25 b(mo)s(duli)c(space)k FF(M)1472 1374 y FC(\003)1536 1410 y FM(of)f(irreducible)e(solutions)h(to)h(the)g(monop)s(ole)e (equa-)456 1526 y(tions)32 b(has)i(some)f(n)m(um)m(b)s(er)g(of)g(ends,) h(all)d(con)m(v)m(erging)j(to)f(the)g(unique)h(reducible)456 1642 y(solution)j(b)s(ecause)j FF(M)49 b FM(is)38 b(compact.)62 b(These)40 b(ends)g(are)f(describ)s(ed)g(b)m(y)h(the)f(re-)456 1758 y(striction)1327 1912 y(\011)11 b(:)33 b FA(C)1540 1871 y FB(2)p FH(k)r FB(+)p FH(r)1735 1912 y FG(n)21 b(f)p FM(0)p FG(g)28 b(\000)-17 b(!)28 b FA(C)2237 1871 y FH(r)2303 1912 y FG(\010)22 b FA(R)2468 1871 y FB(2)2546 1912 y FF(;)456 2087 y FM(whic)m(h)28 b(descends)i(to)d(a)h(section)f (of)g(the)h(bundle)g FF(H)2311 2051 y FC(\010)p FH(r)2416 2087 y FG(\010)12 b FA(R)2571 2051 y FB(2)2644 2087 y FM(o)m(v)m(er)29 b(the)f(punctured)456 2203 y(cone)33 b FF(c)p FM(\()p FA(C)20 b FF(P)901 2167 y FB(2)p FH(k)r FB(+)p FH(r)r FC(\000)p FB(1)1163 2203 y FM(\))32 b(asso)s(ciated)h (with)f(the)h FF(U)10 b FM(\(1\){action.)605 2319 y(Let)46 b FF(e)51 b FG(2)g FF(H)1095 2283 y FB(2)1134 2319 y FM(\()p FG(B)1240 2283 y FC(\003)1280 2319 y FF(;)17 b FA(Z)p FM(\))43 b(b)s(e)k(the)f(Euler)g(class)g(of)g(the)h(basep)s (oin)m(t{\014bration)456 2436 y FF(\014)6 b FM(.)82 b(W)-8 b(e)46 b(c)m(ho)s(ose)h(a)f(smo)s(oth)f(submanifold)e FG(V)54 b FM(of)46 b FG(B)2454 2399 y FC(\003)2539 2436 y FM(of)g(co)s(dimension)e(4)p FF(k)34 b FG(\000)456 2552 y FM(4)45 b(represen)m(ting)i FF(e)1159 2516 y FB(2)p FH(k)r FC(\000)p FB(2)1328 2552 y FM(,)i(suc)m(h)e(that)f FG(V)54 b FM(in)m(tersects)47 b(the)g(link)d(of)i FG(B)35 b(n)c(B)3183 2516 y FC(\003)3268 2552 y FM(in)46 b(a)456 2668 y(standard)32 b FA(C)20 b FF(P)1005 2632 y FB(2)p FH(k)r FC(\000)p FB(2)1207 2668 y FG(\032)28 b FA(C)20 b FF(P)1455 2632 y FC(1)1535 2668 y FM(.)43 b(If)33 b FG(V)40 b FM(is)32 b(generic)h(enough,)g(its)e(in)m(tersection)i(with) 456 2784 y FF(M)560 2748 y FC(\003)632 2784 y FM(is)e(a)g(smo)s(oth)g (1{dimensional)e(manifold)g(with)i(some)h(n)m(um)m(b)s(er)g(of)f(ends)i (all)456 2900 y(aproac)m(hing)40 b(the)i(reducible)f(solution.)69 b(The)42 b(n)m(um)m(b)s(er)g(of)f(ends)i(is)e(the)g(same,)456 3017 y(mo)s(dulo)32 b(2,)i(as)g(the)g(top)g(Stiefel{Whitney)f(n)m(um)m (b)s(er)h(of)g FF(H)2654 2981 y FC(\010)p FH(r)2769 3017 y FG(\010)24 b FA(R)2936 2981 y FB(2)3011 3017 y FG(!)30 b FA(C)20 b FF(P)3284 2981 y FH(r)r FB(+1)3417 3017 y FM(,)456 3133 y(whic)m(h)33 b(is)f(=)27 b(0)33 b(b)s(ecause)h(of)e(the) h(trivial)d(summand.)605 3249 y(So)h(far,)h(this)f(argumen)m(t)g(has)h (not)f(pro)s(duced)h(an)m(y)h(con)m(tradictions.)42 b(A)32 b(con-)456 3365 y(tradiction)27 b(could)h(only)h(ha)m(v)m(e)h(o)s (ccurred)g(if)e(the)h(Euler)g(n)m(um)m(b)s(er)h(had)f(b)s(een)g(o)s (dd.)456 3482 y(Then)e FF(M)808 3445 y FC(\003)874 3482 y FM(w)m(ould)f(ha)m(v)m(e)h(b)s(een)g(a)f(1{dimensional)d(manifold)g (with)j(an)g(o)s(dd)g(n)m(um-)456 3598 y(b)s(er)32 b(of)h(ends.)44 b(Ho)m(w)m(ev)m(er,)36 b(w)m(e)d(ha)m(v)m(e)h(not)f(really)e(used)j (the)f(fact)f(that)h FF(X)40 b FM(is)32 b(spin,)456 3714 y(w)m(e)f(could)f(ha)m(v)m(e)h(made)f(all)e(these)k(argumen)m(ts)e (with)g(a)g(Spin)2677 3672 y FH(c)2711 3714 y FM({structure)i(on)e(the) 456 3830 y(connected)48 b(sum)f(of)f(t)m(w)m(o)h(copies)g(of)g(the)g (Enriques)h(surface,)j(whic)m(h)c(clearly)456 3947 y(exists!)605 4063 y(W)-8 b(e)37 b(use)g(the)g(assumption)f(that)g(w)m(e)h(are)g (dealing)e(with)h(a)g(Spin)o({structure)456 4179 y(b)m(y)c(using)g(the) g(symmetry)h FF(\034)43 b FM(in)m(tro)s(duced)32 b(in)f(Section)h(2)g (of)f(Chapter)i(2.)43 b(Recall)456 4295 y(that)28 b FF(\034)40 b FM(in)m(terc)m(hanges)30 b(a)f(Spin)1568 4253 y FH(c)1602 4295 y FM({structure)h(with)f(its)f(conjugate,)i(and)f(therefore)456 4411 y(preserv)m(es)48 b(the)f(Spin)1259 4370 y FH(c)1293 4411 y FM({structures)h(induced)e(from)f(honest)i(Spin{structures.)456 4528 y(F)-8 b(or)42 b(suc)m(h)j(a)e(Spin)1154 4486 y FH(c)1188 4528 y FM({structure)i FF(\034)55 b FM(acts)43 b(as)h(an)f(in)m(v)m(olution)f(on)h FG(B)s FM(,)k(preserving)456 4644 y(the)30 b(mo)s(duli)d(space)k FF(M)10 b FM(.)44 b(This)30 b(action)f(\014xes)i(the)g(gauge)f(equiv)-5 b(alence)30 b(classes)h(of)456 4760 y(reducible)h(con\014gurations)g (and)h(is)f(free)h(on)f FG(B)2180 4724 y FC(\003)2220 4760 y FM(.)605 4876 y(Th)m(us)i(w)m(e)f(can)g(consider)f FF(M)1657 4840 y FC(\003)1697 4876 y FF(=\034)44 b FM(instead)32 b(of)g FF(M)2383 4840 y FC(\003)2423 4876 y FM(,)g(and)h(in)m(tersect)g (this)f(with)456 4993 y FG(V)8 b FF(=\034)j FM(,)33 b(b)m(y)h (arranging)e FG(V)41 b FM(to)32 b(b)s(e)i FF(\034)11 b FM({in)m(v)-5 b(arian)m(t.)43 b(The)34 b(n)m(um)m(b)s(er)g(of)e(ends) i(in)e(\()p FF(M)3315 4956 y FC(\003)3378 4993 y FG(\\)456 5109 y(V)8 b FM(\))p FF(=\034)56 b FM(is)45 b(the)g(same,)j(mo)s(dulo) 43 b(2,)48 b(as)e(the)f(top)g(Stiefel{Whitney)f(n)m(um)m(b)s(er)i(of) 456 5225 y(\()p FF(H)r(=\034)11 b FM(\))717 5189 y FC(\010)p FH(r)832 5225 y FG(\010)23 b FF(\017)971 5189 y FC(\010)p FB(2)1095 5225 y FG(!)28 b FA(C)20 b FF(P)1366 5189 y FH(r)r FB(+1)1500 5225 y FF(=\034)11 b FM(,)34 b(where)g FF(\017)g FM(is)f(the)g(non{trivial)e(real)h(line)g(bundle)p eop %%Page: 56 10 56 55 bop 456 235 a FE(56)860 b(4.)34 b(NON{EXISTENCE)f(RESUL)-6 b(TS)456 435 y FM(asso)s(ciated)38 b(with)g FF(\034)11 b FM(.)60 b(After)39 b(cutting)e(do)m(wn)i(w)m(e)h(are)e(lo)s(oking)e (at)i FF(\017)2988 399 y FC(\010)p FB(2)3120 435 y FG(!)f FA(R)5 b FF(P)3400 399 y FB(2)456 551 y FM(for)32 b(whic)m(h)h FF(w)957 515 y FB(2)954 576 y(1)1023 551 y FG(6)p FM(=)28 b(0.)605 667 y(W)-8 b(e)31 b(ha)m(v)m(e)h(no)m(w)g(reac)m(hed)g(the)f (desired)g(con)m(tradiction,)f(b)s(ecause)i FF(M)3126 631 y FC(\003)3166 667 y FF(=\034)42 b FM(is)31 b(a)456 783 y(1{dimensional)g(manifold)h(with)j(an)f(o)s(dd)h(n)m(um)m(b)s(er)g (of)f(ends,)j(whic)m(h)e(is)f(imp)s(os-)456 900 y(sible.)605 1016 y(This)f(completes)f(the)h(pro)s(of)f(of)g(Theorem)h(4.5.)605 1190 y FD(3.1.)56 b(Comparison)34 b(with)f(Donaldson's)i(pro)s(of.)49 b FM(Donaldson's)29 b([)p FD(6,)34 b(7)p FM(])456 1306 y(approac)m(h)k(to)f(Theorem)h(4.5)f(w)m(as)i(a)e(natural)f (generalisation)g(of)h(his)g(pro)s(of)g(of)456 1423 y(Theorem)31 b(4.4)g({)h(just)g(as)f(the)h(pro)s(of)f(w)m(e)h(ha)m(v)m(e)h(giv)m(en) f(here)g(is)f(a)g(natural)g(exten-)456 1539 y(sion)40 b(of)g(the)h(pro)s(of)e(of)h(Theorem)h(4.4)f(using)g(the)h(monop)s(ole) e(mo)s(duli)f(spaces.)456 1655 y(Ho)m(w)m(ev)m(er,)33 b(the)f(parallels)c(go)j(further:)43 b(in)30 b(b)s(oth)h(approac)m (hes,)h(with)f(instan)m(tons)456 1771 y(and)39 b(with)g(monop)s(oles,)h (Theorem)g(4.4)f(is)g(pro)m(v)m(ed)i(using)f(the)g(rational)d(coho-)456 1888 y(mology)42 b(of)i(a)g(mo)s(duli)e(space)k(truncated)f(around)f (the)h(reducible)f(solutions.)456 2004 y(Also)34 b(in)h(b)s(oth)g (approac)m(hes,)i(Theorem)f(4.5)f(is)f(pro)m(v)m(ed)j(relying)d(on)h (more)g(sub-)456 2120 y(tle)k(torsion)g(information)e(on)j(the)g(mo)s (duli)e(spaces,)43 b(whic)m(h)e(is)e(a)m(v)-5 b(ailable)38 b(only)456 2236 y(when)d(the)f(base)h(manifold)c(is)j(spin,)g(viz.)h (our)e(use)i(of)f FF(\034)46 b FM(and)34 b(Donaldson's)f(use)456 2353 y(of)f(2{torsion)f(classes)i(on)g(mo)s(duli)d(spaces.)1351 2550 y FD(4.)56 b(Pro)s(of)37 b(of)h(Theorem)f(4.6)605 2724 y FM(!!!TO)c(BE)g(WRITTEN!!!)456 2914 y Fq(Notes:)456 3059 y Fp(1.)39 b(The)757 3024 y FB(11)p 757 3039 71 4 v 774 3091 a(8)837 3059 y Fp({Conjecture,)27 b(for)e(simply)e (connected)j(manifolds,)e(made)i(its)e(app)s(earance)456 3167 y(in)k(prin)m(t)g(in)g(Matsumoto)j([)p Fo(24)q Fp(],)f(although)f (other)h(w)m(ork)m(ers)g(in)e(the)i(\014eld)e(also)h(arriv)m(ed)456 3275 y(at)i(it)f(around)f(the)i(same)f(time.)456 3383 y(2.)56 b(Donaldson's)35 b(pro)s(of)g(of)h(Theorem)f(4.4)i(in)d([)p Fo(5)q(,)41 b(7)p Fp(])36 b(w)m(as)g(the)f(\014rst)g(application)f(of) 456 3491 y(gauge)c(theory)-8 b(,)30 b(in)e(the)h(guise)f(of)h(the)g(Y) -8 b(ang{Mills)29 b(equations,)g(to)g(four{dimensional)456 3599 y(top)s(ology)-8 b(.)41 b(In)28 b(Octob)s(er/No)m(v)m(em)m(b)s(er) i(1994,)i(man)m(y)d(p)s(eople)f(noticed)g(that)i(the)f(mono-)456 3707 y(p)s(ole)h(equations)g(giv)m(e)i(a)f(simple)e(pro)s(of)h(for)h (the)g(case)h(of)f(ev)m(en)h(forms,)f(and)f(that)h(this)456 3815 y(pro)s(of)h(w)m(ould)f(giv)m(e)i(the)g(full)e(result)g(if)h(the)h (algebraic)f(Prop)s(osition)f(4.10)j(w)m(ere)f(true.)456 3923 y(In)f(resp)s(onse)f(to)j(a)f(question)f(from)g(Mro)m(wk)-5 b(a,)34 b(Elkies)e([)p Fo(9)p Fp(])h(pro)m(v)m(ed)g(Prop)s(osition)e (4.10)456 4031 y(in)e(Jan)m(uary/F)-8 b(ebruary)30 b(1995.)456 4139 y(3.)50 b(Theorem)34 b(4.5)h(w)m(as)f(pro)m(v)m(ed)f(b)m(y)h (Donaldson)f([)p Fo(6)q(,)38 b(7)p Fp(],)d(using)d(Y)-8 b(ang{Mills)33 b(theory)-8 b(,)456 4247 y(with)20 b(the)j(additional)c (assumption)i(that)h Ff(H)1960 4261 y FB(1)1999 4247 y Fp(\()p Ff(X)r(;)15 b Fe(Z)p Fp(\))20 b(b)s(e)h(free)h(of)g (2{torsion.)38 b(E\013orts)23 b(to)456 4355 y(remo)m(v)m(e)32 b(this)e(assumption)f(from)h(his)g(argumen)m(t)h(w)m(ere)g (unsuccessful,)e(cf.)j([)p Fo(27)p Fp(].)43 b(The)456 4463 y(pro)s(of)28 b(giv)m(en)i(here)f(using)f(the)i(monop)s(ole)f(mo)s (duli)e(spaces)j(w)m(as)g(found)e(in)g(No)m(v)m(em)m(b)s(er)456 4570 y(1994.)456 4678 y(4.)69 b(The)40 b(ideas)f(of)h(subsection)f (1.1,)44 b(esp)s(ecially)38 b(those)i(b)s(ehind)d(Example)i(4.8,)44 b(are)456 4786 y(pursued)28 b(systematically)i(in)f([)p Fo(14)q Fp(].)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF