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FD(;)1070 3314 y(e)1101 3324 y Fz(0)1142 3314 y FE(^)g FD(e)1233 3324 y Fz(3)1274 3314 y FI(+)h FD(e)1372 3324 y Fz(1)1413 3314 y FE(^)f FD(e)1504 3324 y Fz(2)1552 3314 y FD(:)409 3440 y FI(No)n(w)20 b(let)f(\()p FD(X)r(;)11 b(g)r FI(\))18 b(b)r(e)h(an)g(orien)n(ted)g(Riemannian)e (four-manifold.)28 b(Applying)308 3519 y(the)d(ab)r(o)n(v)n(e)g (construction)f(to)g(the)h(tangen)n(t)f(space)h(at)g(eac)n(h)g(p)r(oin) n(t,)g(w)n(e)g(obtain)1307 3620 y FC(5)p eop %%Page: 6 2 6 5 bop 308 159 a FC(6)110 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)308 294 y FI(a)k(decomp)r(osition)f (of)i(the)e(space)i(of)f(2-forms)f(on)h FD(X)32 b FI(in)n(to)27 b(the)g FJ(self-dual)33 b FI(and)308 373 y FJ(anti-self-dual)28 b FI(forms:)933 492 y(\012)981 464 y Fz(2)1007 492 y FI(\()p FD(X)5 b FI(\))18 b(=)h(\012)1255 464 y Fz(2)1255 509 y(+)1295 492 y FI(\()p FD(X)5 b FI(\))14 b FE(\010)i FI(\012)1537 464 y Fz(2)1537 509 y FA(\000)1576 492 y FI(\()p FD(X)5 b FI(\))p FD(:)308 611 y FI(Because)27 b(the)f(formal)g(adjoin)n(t)g FD(d)1157 587 y FA(\003)1210 611 y FI(of)h(the)f(exterior)g(di\013eren)n(tial)h FD(d)f FI(is)h FE(\006)19 b(\003)e FD(d)p FE(\003)p FI(,)308 690 y(the)22 b(space)g(of)h(closed)f(and)g(co-closed)h(2-forms)901 809 y FE(H)959 781 y Fz(2)1004 809 y FI(=)c FE(f)11 b FD(\013)19 b FE(j)g FD(d\013)g FI(=)g(0)p FD(;)34 b(d)1502 781 y FA(\003)1529 809 y FD(\013)19 b FI(=)g(0)11 b FE(g)308 928 y FI(is)21 b(mapp)r(ed)e(to)h(itself)h(b)n(y)f(the)h(star)e(op)r (erator,)h(and)g(w)n(e)h(ha)n(v)n(e)f(an)h(induced)f(split-)308 1007 y(ting)1086 1099 y FE(H)1144 1071 y Fz(2)1189 1099 y FI(=)f FE(H)1317 1071 y Fz(2)1316 1115 y(+)1371 1099 y FE(\010)c(H)1496 1071 y Fz(2)1495 1115 y FA(\000)1535 1099 y FD(:)308 1206 y FI(On)28 b(a)f(compact)f(manifold,)i FE(H)1109 1181 y Fz(2)1163 1206 y FI(is)f(the)g(space)h(of)g(harmonic)e (2-forms)g(and)g(is)308 1284 y(iden)n(ti\014ed)21 b(with)g FD(H)805 1260 y Fz(2)832 1284 y FI(\()p FD(X)r(;)11 b Fy(R)p FI(\))22 b(b)n(y)f(Ho)r(dge)h(theory)-6 b(,)21 b(so)g(w)n(e)h(can)f(regard)g(this)g(split-)308 1363 y(ting)27 b(as)f(a)h(metric-dep)r(enden)n(t)e(decomp)r(osition)h(of)h (the)f(second)h(cohomology)-6 b(.)308 1442 y(Seen)16 b(this)g(w)n(a)n(y)-6 b(,)17 b FE(H)778 1417 y Fz(+)834 1442 y FI(and)e FE(H)1014 1417 y FA(\000)1069 1442 y FI(are)h(maximal)e(p)r(ositiv)n(e)j(and)e(negativ)n(e)h(subspaces)308 1520 y(for)22 b(the)g(quadratic)f(form)g(on)h FD(H)1126 1496 y Fz(2)1152 1520 y FI(\()p FD(X)r(;)11 b Fy(R)p FI(\))23 b(de\014ned)f(b)n(y)f(the)h(cup)g(pro)r(duct:)762 1640 y FD(H)822 1612 y Fz(2)849 1640 y FI(\()p FD(X)r(;)11 b Fy(R)p FI(\))16 b FE(\002)f FD(H)1177 1612 y Fz(2)1204 1640 y FI(\()p FD(X)r(;)c Fy(R)p FI(\))19 b FE(\000)-10 b(!)19 b Fy(R)974 1743 y FI(\()p FD(a;)11 b(b)p FI(\))290 b FE(7\000)-10 b(!)19 b FI(\()p FD(a)f Fx(`)h FD(b)p FI(\)[)p FD(X)5 b FI(])23 b FD(:)308 1867 y FI(It)18 b(follo)n(ws)h(in)g(particular)f(that)f(the)i(dimensions)e(of)i FE(H)1678 1842 y Fz(+)1736 1867 y FI(and)f FE(H)1919 1842 y FA(\000)1977 1867 y FI(are)g(indep)r(en-)308 1945 y(den)n(t)d(of)h(the)f(metric.)27 b(These)16 b(w)n(e)f(denote)h(as)f (usual)g(b)n(y)h FD(b)1691 1917 y Fz(+)1691 1962 y(2)1731 1945 y FI(\()p FD(X)5 b FI(\))15 b(and)g FD(b)2008 1917 y FA(\000)2008 1962 y Fz(2)2048 1945 y FI(\()p FD(X)5 b FI(\);)17 b(their)308 2024 y(sum)25 b(is)h(the)f(second)h(Betti)f(n)n (um)n(b)r(er,)h(and)f(their)g(di\013erence)h(is)g(the)f FJ(signatur)m(e)308 2103 y FI(of)d FD(X)5 b FI(:)962 2222 y FD(b)990 2232 y Fz(2)1017 2222 y FI(\()p FD(X)g FI(\))18 b(=)h FD(b)1245 2194 y Fz(+)1245 2239 y(2)1285 2222 y FI(\()p FD(X)5 b FI(\))15 b(+)g FD(b)1506 2194 y FA(\000)1506 2239 y Fz(2)1546 2222 y FI(\()p FD(X)5 b FI(\))p FD(;)977 2329 y(\033)r FI(\()p FD(X)g FI(\))18 b(=)h FD(b)1245 2301 y Fz(+)1245 2345 y(2)1285 2329 y FI(\()p FD(X)5 b FI(\))15 b FE(\000)g FD(b)1507 2301 y FA(\000)1507 2345 y Fz(2)1547 2329 y FI(\()p FD(X)5 b FI(\))p FD(:)409 2448 y FI(W)-6 b(e)31 b(will)g(denote)f(the)g (self-dual)g(and)g(an)n(ti-self-dual)g(parts)f(of)i(a)f(2-form)308 2527 y(using)22 b FE(\006)g FI(subscripts.)29 b(If)22 b FD(\013)d FE(2)g FI(\012)1129 2502 y Fz(1)1155 2527 y FI(,)k(w)n(e)f(write)g FD(d)1496 2502 y FA(\006)1536 2527 y FD(\013)g FI(for)g(\()p FD(d\013)p FI(\))1829 2537 y FA(\006)1869 2527 y FI(.)409 2626 y FK(Pr)o(oposition)j FI(1.1)p FK(.)j FJ(If)c FD(X)29 b FJ(is)24 b(close)m(d,)f(then)308 2767 y FI(\(1.2\))611 b(\012)1103 2739 y Fz(0)1190 2729 y FF(d)1148 2767 y FE(\000)-10 b(!)19 b FI(\012)1324 2739 y Fz(1)1394 2729 y FF(d)1419 2713 y Fw(+)1369 2767 y FE(\000)-11 b(!)19 b FI(\012)1544 2739 y Fz(2)1544 2784 y(+)308 2888 y FJ(is)e(a)g(c)m(omplex)g(whose)f(thr)m(e)m(e)g(c)m (ohomolo)m(gy)f(gr)m(oups)i(ar)m(e)f(isomorphic)g(to)h FD(H)2118 2864 y Fz(0)2144 2888 y FI(\()p FD(X)r(;)11 b Fy(R)p FI(\))p FJ(,)308 2967 y FD(H)368 2942 y Fz(1)395 2967 y FI(\()p FD(X)r(;)g Fy(R)p FI(\))24 b FJ(and)g FE(H)791 2942 y Fz(2)790 2983 y(+)829 2967 y FJ(.)31 b(In)24 b(p)m(articular,)f(the)h(index)g(of)38 b FI(\(1.2\))23 b FJ(is)675 3123 y FD(b)703 3133 y Fz(0)730 3123 y FI(\()p FD(X)5 b FI(\))14 b FE(\000)h FD(b)951 3133 y Fz(1)978 3123 y FI(\()p FD(X)5 b FI(\))14 b(+)i FD(b)1199 3096 y Fz(+)1199 3140 y(2)1239 3123 y FI(\()p FD(X)5 b FI(\))18 b(=)1446 3078 y(1)p 1446 3108 34 3 v 1446 3170 a(2)1486 3123 y(\()p FD(\037)p FI(\()p FD(X)5 b FI(\))14 b(+)h FD(\033)r FI(\()p FD(X)5 b FI(\)\))23 b FD(;)308 3263 y FJ(wher)m(e)h FD(\037)f FJ(is)h(the)g(Euler)g(char)m(acteristic)f (and)g FD(\033)j FJ(is)d(the)h(signatur)m(e.)409 3362 y FI(This,)d(together)e(with)g(the)g(Dirac)i(equation,)f(is)g(the)f(mo) r(del)g(for)h(the)f(elliptic)308 3440 y(complex)30 b(w)n(e)h(will)g (encoun)n(ter)f(when)g(linearizing)h(the)g(nonlinear)f(monop)r(ole)308 3519 y(equations.)p eop %%Page: 7 3 7 6 bop 1038 160 a FC(2.)24 b(Spin)1203 139 y Fv(c)1225 160 y FC(-STR)o(UCTURES)703 b(7)409 294 y FK(Pr)o(oof.)27 b FI(F)-6 b(or)27 b FD(H)853 270 y Fz(0)880 294 y FI(\()p FD(X)r(;)11 b Fy(R)p FI(\))28 b(the)f(claim)g(is)h(clear.)45 b(F)-6 b(or)28 b(the)f(cohomology)f(at)308 373 y(the)c(middle)f(term,)g (observ)n(e)h(that)f(for)h(all)h FD(\013)c FE(2)g FI(\012)1558 348 y Fz(1)1606 373 y FI(w)n(e)j(ha)n(v)n(e)459 455 y Fu(Z)496 607 y FF(X)542 546 y FI(\()p FE(j)p FD(d)621 519 y Fz(+)660 546 y FD(\013)p FE(j)721 519 y Fz(2)763 546 y FE(\000)16 b(j)p FD(d)884 519 y FA(\000)924 546 y FD(\013)p FE(j)985 519 y Fz(2)1011 546 y FI(\))p FD(dv)r(ol)1157 556 y FF(g)1203 546 y FI(=)1273 455 y Fu(Z)1311 607 y FF(X)1367 546 y FD(d\013)g FE(^)f FD(d\013)k FI(=)1684 455 y Fu(Z)1722 607 y FF(X)1779 546 y FD(d)p FI(\()p FD(\013)c FE(^)g FD(d\013)p FI(\))k(=)g(0)308 727 y(b)n(y)f(Stok)n (es's)f(theorem.)27 b(Th)n(us)17 b FD(d)1119 703 y Fz(+)1159 727 y FD(\013)i FI(=)h(0)d(if)h(and)f(only)h(if)g FD(d)1752 703 y FA(\000)1792 727 y FD(\013)h FI(=)g(0.)28 b(This)18 b(pro)n(v)n(es)308 806 y(that)j(the)h(\014rst)f(cohomology)g(of)h(the)g (complex)g(is)g FD(H)1636 782 y Fz(1)1663 806 y FI(\()p FD(X)r(;)11 b Fy(R)p FI(\).)409 885 y(F)-6 b(or)23 b(the)e(last)h (term,)f(note)h(that)f(for)h FD(h)d FE(2)f(H)1511 860 y Fz(2)1510 901 y(+)1572 885 y FI(w)n(e)k(ha)n(v)n(e)411 975 y Fu(Z)449 1127 y FF(X)494 1067 y FE(h)p FD(h;)11 b(d)621 1039 y Fz(+)662 1067 y FD(\013)p FE(i)p FD(dv)r(ol)850 1077 y FF(g)897 1067 y FI(=)967 975 y Fu(Z)1004 1127 y FF(X)1061 1067 y FD(h)k FE(^)g FD(d)1208 1039 y Fz(+)1248 1067 y FD(\013)20 b FI(=)1380 975 y Fu(Z)1417 1127 y FF(X)1474 1067 y FD(h)15 b FE(^)g FD(d\013)k FI(=)1752 975 y Fu(Z)1790 1127 y FF(X)1847 1067 y FD(d)p FI(\()p FD(h)c FE(^)f FD(\013)p FI(\))19 b(=)g(0)p FD(;)308 1249 y FI(again)24 b(b)n(y)f(Stok)n(es's)h(theorem.)33 b(Th)n(us)23 b FE(H)1345 1225 y Fz(2)1344 1266 y(+)1407 1249 y FI(is)h FD(L)1520 1225 y Fz(2)1547 1249 y FI(-orthogonal)f(to)g FD(I)5 b(m)p FI(\()p FD(d)2135 1225 y Fz(+)2175 1249 y FI(\))23 b(and)308 1330 y(injects)e(in)n(to)f(\012)694 1305 y Fz(2)694 1347 y(+)733 1330 y FD(=I)5 b(m)p FI(\()p FD(d)918 1305 y Fz(+)958 1330 y FI(\).)29 b(W)-6 b(e)21 b(w)n(an)n(t)e(to)h(sho)n(w)f(that)g(this)h(injection)g(is)g(surjec-) 308 1410 y(tiv)n(e)j(as)f(w)n(ell.)30 b(Let)22 b FD(!)g FE(2)c FI(\012)969 1386 y Fz(2)969 1427 y(+)1031 1410 y FI(ha)n(v)n(e)k(Ho)r(dge)h(decomp)r(osition)1040 1552 y FD(!)e FI(=)e FD(h)c FI(+)h FD(d\013)f FI(+)h FE(\003)p FD(d\014)26 b(;)308 1694 y FI(with)c FD(h)c FE(2)h(H)637 1669 y Fz(2)663 1694 y FI(.)30 b(Applying)22 b(the)g(star)f(op)r (erator,)g(w)n(e)h(obtain)1023 1835 y FD(!)f FI(=)e FE(\003)p FD(h)c FI(+)g FD(d\014)20 b FI(+)15 b FE(\003)p FD(d\013)23 b(:)308 1977 y FI(The)i(uniqueness)f(of)g(the)h(Ho)r(dge)g(decomp)r (osition)e(sho)n(ws)h(that)g FD(h)g FI(is)h(self-dual)308 2056 y(and)e FD(d\013)f FI(=)g FD(d\014)t FI(.)34 b(Th)n(us)22 b FD(!)i FI(=)e FD(h)16 b FI(+)g FD(d)1197 2031 y Fz(+)1237 2056 y FI(\(2)p FD(\013)p FI(\),)24 b(and)f FD(h)e FE(2)g(H)1719 2031 y Fz(2)1718 2072 y(+)1782 2056 y FI(represen)n(ts)i FD(!)j FI(in)d(the)308 2136 y(quotien)n(t)f(\012)617 2112 y Fz(2)617 2153 y(+)656 2136 y FD(=I)5 b(m)p FI(\()p FD(d)841 2112 y Fz(+)881 2136 y FI(\).)409 2215 y(The)19 b(index)f(of)h(the)g(complex)f(\(the)g(alternating)g(sum)f(of)i(the)g (dimensions)f(of)308 2293 y(the)h(cohomology)e(groups\))h(can)g(b)r(e)g (expressed)h(in)f(terms)g(of)g FD(\037)h FI(and)f FD(\033)j FI(as)d(stated)308 2372 y(in)k(the)g(prop)r(osition)f(b)r(ecause)h FD(\033)f FI(=)e FD(b)1247 2344 y Fz(+)1247 2388 y(2)1287 2372 y FI(\()p FD(X)5 b FI(\))14 b FE(\000)i FD(b)1509 2344 y FA(\000)1509 2388 y Fz(2)1549 2372 y FI(\()p FD(X)5 b FI(\))21 b(and)954 2514 y FD(\037)e FI(=)g FD(b)1112 2524 y Fz(0)1154 2514 y FE(\000)c FD(b)1249 2524 y Fz(1)1291 2514 y FI(+)g FD(b)1385 2524 y Fz(2)1427 2514 y FE(\000)g FD(b)1522 2524 y Fz(3)1564 2514 y FI(+)g FD(b)1658 2524 y Fz(4)1014 2612 y FI(=)k(2)p FD(b)1145 2622 y Fz(0)1187 2612 y FE(\000)c FI(2)p FD(b)1315 2622 y Fz(1)1357 2612 y FI(+)g FD(b)1451 2622 y Fz(2)1478 2612 y FD(;)308 2754 y FI(b)n(y)22 b(P)n(oincar)n(\023)-31 b(e)22 b(dualit)n(y)-6 b(.)1394 b Fx(\003)1016 2989 y FB(2.)38 b FI(Spin)1241 2961 y FF(c)1265 2989 y FB(-structures)409 3107 y(2.1.)g(Cli\013ord)27 b(mo)r(dules.)34 b FI(The)23 b(standard)e(Dirac)k(op)r(erator)d(on)h Fy(R)2173 3083 y FF(n)2230 3107 y FI(is)h(a)308 3186 y(particular)h(self-adjoin)n(t)h(op)r(erator)e(whose)g(square)h(is)h (the)e(Laplace)j(op)r(erator.)308 3264 y(Consider)22 b(a)g(\014rst-order)e(di\013eren)n(tial)i(op)r(erator)f FD(D)j FI(of)e(the)g(form)308 3457 y(\(1.3\))660 b FD(D)20 b FI(=)1284 3372 y FF(n)1249 3392 y Fu(X)1259 3535 y FF(i)p Fz(=1)1358 3457 y FD(A)1407 3467 y FF(i)1462 3411 y FD(@)p 1433 3441 96 3 v 1433 3503 a(@)t(x)1510 3483 y FF(i)p eop %%Page: 8 4 8 7 bop 308 159 a FC(8)110 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)308 294 y FI(acting)c(on)g Fy(C)637 270 y FF(N)686 294 y FI(-v)l(alued)g(functions)g(on)g Fy(R)1330 270 y FF(n)1363 294 y FI(,)h(where)f(the)f FD(A)1748 304 y FF(i)1787 294 y FI(are)h(constan)n(t)f FD(N)7 b FI(-b)n(y-)308 373 y FD(N)32 b FI(matrices.)37 b(This)24 b(op)r(erator)g(is)h(formally)f(self-adjoin)n(t)h(if)g(the)g FD(A)2016 383 y FF(i)2060 373 y FI(are)g(sk)n(ew-)308 451 y(adjoin)n(t.)42 b(The)26 b(Laplacian)g(acting)h(on)e Fy(C)1358 427 y FF(N)1407 451 y FI(-v)l(alued)i(functions)e(is)i(the)f (second-)308 530 y(order)c(op)r(erator)1022 660 y(\001)c(=)h FE(\000)1263 575 y FF(n)1229 596 y Fu(X)1239 738 y FF(i)p Fz(=1)1338 660 y FD(I)1367 670 y FF(N)1412 660 y FI(\()1473 614 y FD(@)p 1445 644 96 3 v 1445 706 a(@)t(x)1522 686 y FF(i)1547 660 y FI(\))1573 632 y Fz(2)1599 660 y FD(;)308 818 y FI(where)j FD(I)528 828 y FF(N)596 818 y FI(is)g(the)g(iden)n (tit)n(y)g(matrix,)e(so)i(the)g(condition)g FD(D)1794 793 y Fz(2)1839 818 y FI(=)d(\001)j(is)308 925 y(\(1.4\))357 b FD(A)850 897 y Fz(2)850 941 y FF(i)896 925 y FI(=)19 b FE(\000)p FD(I)1047 935 y FF(N)1159 925 y FI(and)65 b FD(A)1380 935 y FF(i)1400 925 y FD(A)1449 935 y FF(j)1489 925 y FI(+)15 b FD(A)1604 935 y FF(j)1629 925 y FD(A)1678 935 y FF(i)1717 925 y FI(=)k(0)p FD(:)308 1031 y FI(In)g(the)g (presence)h(of)g(the)f(\014rst)f(of)i(these)f(t)n(w)n(o)g(relations,)i (the)e(condition)g(that)f FD(A)2311 1041 y FF(i)308 1110 y FI(is)23 b(sk)n(ew-adjoin)n(t)e(is)i(equiv)l(alen)n(t)f(to)g(it)g(b)r (eing)g(unitary)-6 b(.)409 1188 y(This)32 b(situation)f(can)g(b)r(e)h (abstracted)e(as)i(follo)n(ws.)59 b(Let)32 b FD(H)k FI(b)r(e)31 b(a)h(\014nite-)308 1267 y(dimensional)21 b(real)h(v)n(ector)g(space)g (with)f(inner)g(pro)r(duct)f FD(g)r FI(.)29 b(A)22 b FJ(Cli\013or)m(d)h(mo)m(dule)308 1346 y FI(for)h FD(H)29 b FI(is)24 b(a)g(complex)g(v)n(ector)g(space)g FD(V)39 b FI(with)24 b(Hermitian)f(inner)h(pro)r(duct)e(and)308 1424 y(a)g(linear)h(map)1073 1504 y FD(\015)11 b FI(:)24 b FD(H)f FE(!)c FI(End\()p FD(V)c FI(\))p FD(;)308 1597 y FI(satisfying)22 b(the)g(relations)308 1750 y(\(1.5\))1231 1694 y FD(\015)t FI(\()p FD(v)r FI(\))1355 1666 y FA(\003)1400 1694 y FI(=)d FE(\000)p FD(\015)t FI(\()p FD(v)r FI(\))777 1796 y FD(\015)t FI(\()p FD(v)r FI(\))p FD(\015)t FI(\()p FD(w)r FI(\))13 b(+)j FD(\015)t FI(\()p FD(w)r FI(\))p FD(\015)t FI(\()p FD(v)r FI(\))h(=)i FE(\000)p FI(2)p FD(g)r FI(\()p FD(v)r(;)11 b(w)r FI(\))p FB(1)1791 1806 y FF(V)1855 1796 y FD(:)308 1903 y FI(The)22 b(elemen)n(ts)g(of)g FD(V)37 b FI(are)22 b(called)h FJ(spinors)p FI(.)409 1982 y(The)e(ab)r(o)n(v)n(e)h(relations)f(ensure)g(that)g(the)g(linear) g(map)f FD(\015)25 b FI(extends)c(to)g(an)g(al-)308 2060 y(gebra)f(homomorphism)d(of)j(the)g(Cli\013ord)g(algebra)g(of)g(\()p FD(H)r(;)11 b(g)r FI(\),)20 b(denoted)g(Cl\()p FD(H)5 b FI(\),)308 2139 y(whic)n(h)25 b(is)g(de\014ned)f(to)g(b)r(e)g(the)h (quotien)n(t)f(of)h(the)f(tensor)g(algebra)g(of)h FD(H)30 b FI(b)n(y)24 b(the)308 2218 y(relation)836 2297 y FD(v)18 b FE(\012)d FD(w)i FI(+)e FD(w)i FE(\012)f FD(v)h FI(+)e(2)p FD(g)r FI(\()p FD(v)r(;)c(w)r FI(\))p FB(1)1599 2307 y FF(V)1659 2297 y FI(=)19 b(0)j FD(:)308 2390 y FI(As)j(a)g(v)n(ector) f(space)h(the)g(Cli\013ord)e(algebra)i(Cl\()p FD(H)5 b FI(\))24 b(is)h(isomorphic)e(to)h(the)h(ex-)308 2468 y(terior)j(algebra)f(\003)776 2444 y FA(\003)803 2468 y FI(\()p FD(H)5 b FI(\).)46 b(The)27 b(action)h(of)f FD(H)33 b FI(on)27 b FD(V)43 b FI(via)28 b FD(\015)j FI(is)d(called)h FJ(Cli\013or)m(d)308 2547 y(multiplic)m(ation)p FI(,)22 b(as)g(is)h(the)e(m)n(ultiplication)h(in)g(the)g(Cli\013ord)f (algebra)h(itself.)409 2626 y(Although)16 b(w)n(e)g(will)h(not)e(need)i (to)e(kno)n(w)h(the)f(structure)g(of)i(Cli\013ord)e(mo)r(dules)308 2704 y(in)20 b(general)g(for)g(this)g(b)r(o)r(ok,)g(w)n(e)g(state)f (here,)i(without)e(pro)r(of,)h(this)f(basic)i(result.)409 2797 y FK(Pr)o(oposition)h FI(1.2)p FK(.)j FJ(If)d FD(n)d FI(=)g(2)p FD(m)h FJ(is)h(even)h(then)f(up)g(to)f(isomorphism)g(ther)m (e)308 2876 y(is)25 b(a)g(unique)h(irr)m(e)m(ducible)f(Cli\013or)m(d)g (mo)m(dule)f FI(\()p FD(V)t(;)11 b(\015)t FI(\))25 b FJ(for)f Fy(R)1779 2851 y FF(n)1813 2876 y FJ(.)34 b(Its)25 b(dimension)g(is)308 2954 y FI(2)341 2930 y FF(m)386 2954 y FJ(.)30 b(If)20 b FD(n)f FI(=)g(2)p FD(m)6 b FI(+)g(1)20 b FJ(is)g(o)m(dd)e(then)i(ther)m(e)f(ar)m(e)h(two)f(irr)m(e)m(ducible)h (Cli\013or)m(d)f(mo)m(dules)308 3033 y(for)26 b Fy(R)463 3008 y FF(n)497 3033 y FJ(.)38 b(Both)27 b(have)f(dimension)g FI(2)1224 3008 y FF(m)1270 3033 y FJ(,)h(and)f(one)g(is)g(obtaine)m(d)g (fr)m(om)g(the)g(other)308 3112 y(by)e(r)m(eplacing)g(the)f(Cli\013or)m (d)h(multiplic)m(ation)f FD(\015)k FJ(with)d FE(\000)p FD(\015)t FJ(.)522 b Fx(\003)409 3204 y FI(An)n(y)19 b(\014nite-dimensional)e(Cli\013ord)g(mo)r(dule)g(is)h(a)g(sum)f(of)h (irreducible)g(mo)r(d-)308 3283 y(ules:)34 b(since)25 b(eac)n(h)f FD(\015)t FI(\()p FD(v)r FI(\))g(is)g(sk)n(ew-adjoin)n(t,)h (the)e(orthogonal)g(complemen)n(t)g(of)h(a)308 3362 y(submo)r(dule)d (is)h(also)g(a)g(submo)r(dule.)409 3440 y(Th)n(us)f(in)g(the)f(ev)n(en) i(dimensional)e(case)i(w)n(e)f(will)h(call)g(an)n(y)f(irreducible)g (Clif-)308 3519 y(ford)h(mo)r(dule)f(a)h FJ(standar)m(d)f(Cli\013or)m (d)j(mo)m(dule)p FI(.)p eop %%Page: 9 5 9 8 bop 1038 160 a FC(2.)24 b(Spin)1203 139 y Fv(c)1225 160 y FC(-STR)o(UCTURES)703 b(9)409 294 y FI(In)21 b(the)g(o)r(dd)g (dimensional)f(case)i(w)n(e)g(can)f(use)h(an)e(orien)n(tation)h(of)h FD(H)k FI(distin-)308 373 y(guish)f(its)f(t)n(w)n(o)h(irreducible)g (Cli\013ord)f(mo)r(dules.)36 b(The)24 b(inner)h(pro)r(duct)e(and)h(an) 308 451 y(orien)n(tation)k(de\014ne)f(an)h(orien)n(ted)g(v)n(olume)f (form)g FE(\003)p FI(1)h(=)h FD(dv)r(ol)g FI(whic)n(h)f(acts)g(b)n(y) 308 530 y(Cli\013ord)23 b(m)n(ultiplication)h(as)f FE(\006)p FI(1.)36 b(W)-6 b(riting)24 b FD(n)e FI(=)g(2)p FD(m)16 b FI(+)g(1,)25 b(the)e(Cli\013ord)g(mo)r(d-)308 609 y(ules)i(can)g(b)r (e)f(written)g(\()p FD(V)t(;)11 b(\015)t FI(\))24 b(and)g(\()p FD(V)t(;)11 b FE(\000)p FD(\015)t FI(\),)26 b(with)e FD(V)38 b FI(=)24 b Fy(C)1815 584 y Fz(2)1839 568 y Fv(m)1885 609 y FI(.)37 b(F)-6 b(ollo)n(wing)26 b(the)308 687 y(usual)21 b(con)n(v)n(en)n(tion)f(w)n(e)h(will)g(call)h(an)e(irreducible)h (Cli\013ord)f(mo)r(dule)f(a)h FJ(standar)m(d)308 766 y(Cli\013or)m(d)j(mo)m(dule)f FI(if)h(the)e(v)n(olume)h(form)e(acts)i (as)h(\()p FE(\000)p FI(1\))1683 742 y FF(m)p Fz(+1)1788 766 y FI(.)409 845 y(2.1.1.)34 b FJ(A)29 b(standar)m(d)e(Cli\013or)m(d) h(mo)m(dule)g(for)h Fy(R)1580 820 y Fz(4)1608 845 y FJ(.)k FI(W)-6 b(e)29 b(de\014ne)e(an)g(action)h(of)308 923 y Fy(R)355 899 y Fz(4)405 923 y FI(on)22 b Fy(C)541 899 y Fz(4)594 923 y FI(as)g(follo)n(ws.)30 b(Cho)r(ose)21 b(an)h(orien)n(ted)f(orthonormal)f(basis,)i FD(e)2081 933 y Fz(0)2107 923 y FD(;)11 b(:)g(:)g(:)j(;)d(e)2286 933 y Fz(3)2312 923 y FI(,)308 1002 y(for)22 b Fy(R)456 978 y Fz(4)507 1002 y FI(and)f(de\014ne)940 1155 y FD(\015)t FI(\()p FD(e)1035 1165 y FF(j)1059 1155 y FI(\))d(=)h FD(A)1222 1165 y FF(j)1266 1155 y FI(=)1336 1060 y Fu(\022)1407 1114 y FI(0)77 b FE(\000)p FD(B)1622 1089 y FA(\003)1619 1131 y FF(j)1386 1194 y FD(B)1436 1204 y FF(j)1567 1194 y FI(0)1650 1060 y Fu(\023)308 1335 y FI(where)796 1489 y FD(B)846 1499 y Fz(0)892 1489 y FI(=)962 1394 y Fu(\022)1011 1448 y FE(\000)p FI(1)83 b(0)1038 1527 y(0)f FE(\000)p FI(1)1238 1394 y Fu(\023)1344 1489 y FD(;)11 b(B)1423 1499 y Fz(1)1469 1489 y FI(=)1539 1394 y Fu(\022)1594 1448 y FD(i)82 b FI(0)1589 1527 y(0)56 b FE(\000)p FD(i)1753 1394 y Fu(\023)1814 1489 y FD(;)796 1658 y(B)846 1668 y Fz(2)892 1658 y FI(=)962 1563 y Fu(\022)1011 1618 y FI(0)62 b FD(i)1017 1696 y(i)f FI(0)1134 1563 y Fu(\023)1344 1658 y FD(;)11 b(B)1423 1668 y Fz(3)1469 1658 y FI(=)1539 1563 y Fu(\022)1589 1618 y FI(0)56 b FE(\000)p FI(1)1589 1696 y(1)82 b(0)1764 1563 y Fu(\023)1825 1658 y FD(:)308 1845 y FI(One)28 b(can)f(c)n(hec)n(k)h(that)e FD(\015)31 b FI(extends)c(to)f(an)h(isomorphism)e FD(\015)31 b FI(:)d(Cl\()p Fy(R)2044 1820 y Fz(4)2072 1845 y FI(\))18 b FE(\012)h Fy(C)44 b FE(!)308 1923 y FI(End)427 1933 y Ft(C)462 1923 y FI(\()p Fy(C)532 1899 y Fz(4)563 1923 y FI(\).)27 b(Therefore)15 b(the)g(Cli\013ord)f(mo)r(dule)g(w)n(e)i(ha)n(v)n(e)f (de\014ned)f(is)i(irreducible.)308 2002 y(F)-6 b(rom)19 b(the)h(basic)g(theory)f(of)g(matrix)g(algebras)g(it)h(follo)n(ws)h (that)d(it)i(is)g(unique)f(up)308 2081 y(to)j(isomorphism.)409 2159 y(As)f(men)n(tioned)e(ab)r(o)n(v)n(e,)h(the)g(Cli\013ord)f (algebra)h(Cl\()p Fy(R)1745 2135 y Fz(4)1773 2159 y FI(\))g(is)g (linearly)h(isomor-)308 2238 y(phic)28 b(to)g(\003)590 2213 y FA(\003)616 2238 y FI(\()p Fy(R)689 2213 y Fz(4)717 2238 y FI(\).)47 b(The)28 b(standard)e(w)n(a)n(y)i(to)g(c)n(ho)r(ose)g (this)g(isomorphism)d(is)k(so)308 2317 y(that)839 2445 y FD(\015)t FI(\()p FD(e)934 2455 y FF(i)950 2461 y Fw(1)990 2445 y FE(^)15 b FD(:)c(:)g(:)17 b FE(^)d FD(e)1233 2455 y FF(i)1249 2463 y Fv(l)1268 2445 y FI(\))19 b(=)g FD(\015)t FI(\()p FD(e)1478 2455 y FF(i)1494 2461 y Fw(1)1520 2445 y FI(\))11 b FD(:)g(:)g(:)h(\015)t FI(\()p FD(e)1740 2455 y FF(i)1756 2463 y Fv(l)1775 2445 y FI(\))308 2587 y(whenev)n(er)24 b FD(i)623 2597 y Fz(1)672 2587 y FD(<)f(:)11 b(:)g(:)24 b(<)e(i)942 2597 y FF(l)959 2587 y FI(.)37 b(Ho)n(w)n(ev)n(er,)26 b(this)d(do)r(es)h FJ(not)31 b FI(mean)23 b(that)g FD(\015)28 b FI(is)d(a)f(ring)308 2666 y(homomorphism,)14 b(b)r(ecause)i(if)h(w)n(e)f(m)n(ultiply)f (forms)g(whic)n(h)h(are)g(not)f(orthogonal,)308 2745 y(w)n(e)22 b(obtain)g(di\013eren)n(t)f(answ)n(ers)h(in)g(the)f (exterior)h(and)g(Cli\013ord)f(algebras.)409 2823 y(F)-6 b(rom)23 b(the)f(de\014nition)h(of)g(the)f FD(A)1232 2833 y FF(j)1281 2823 y FI(it)h(is)g(natural)f(to)g(decomp)r(ose)h(the) f(space)308 2902 y(of)k(spinors)e Fy(C)659 2878 y Fz(4)713 2902 y FI(=)h Fy(C)833 2878 y Fz(2)833 2919 y(+)894 2902 y FE(\010)18 b Fy(C)1008 2878 y Fz(2)1008 2919 y FA(\000)1077 2902 y FI(so)26 b(that)e(the)g(v)n(olume)h(form)f(acts)h(as)g FE(\000)p FI(1)g(on)g Fy(C)2287 2878 y Fz(2)2287 2919 y(+)308 2982 y FI(and)20 b(1)g(on)g Fy(C)622 2958 y Fz(2)622 2999 y FA(\000)666 2982 y FI(.)29 b(The)20 b(elemen)n(ts)g(of)g Fy(C)1233 2958 y Fz(2)1233 2999 y(+)1297 2982 y FI(\(resp)r(ectiv)n (ely)h Fy(C)1725 2958 y Fz(2)1725 2999 y FA(\000)1769 2982 y FI(\))f(are)g(called)h(p)r(ositiv)n(e)308 3061 y(\(resp)r(ectiv)n(ely)i(negativ)n(e\))f(spinors.)409 3140 y(Under)g FD(\015)k FI(the)c(self-dual)g(t)n(w)n(o-forms)1070 3296 y FD(e)1101 3306 y Fz(0)1142 3296 y FE(^)15 b FD(e)1233 3306 y Fz(1)1274 3296 y FI(+)h FD(e)1372 3306 y Fz(2)1413 3296 y FE(^)f FD(e)1504 3306 y Fz(3)1552 3296 y FD(;)1069 3394 y(e)1100 3404 y Fz(0)1141 3394 y FE(^)g FD(e)1232 3404 y Fz(2)1273 3394 y FE(\000)h FD(e)1372 3404 y Fz(1)1413 3394 y FE(^)f FD(e)1504 3404 y Fz(3)1552 3394 y FD(;)1070 3492 y(e)1101 3502 y Fz(0)1142 3492 y FE(^)g FD(e)1233 3502 y Fz(3)1274 3492 y FI(+)h FD(e)1372 3502 y Fz(1)1413 3492 y FE(^)f FD(e)1504 3502 y Fz(2)1552 3492 y FD(;)p eop %%Page: 10 6 10 9 bop 308 159 a FC(10)84 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)308 294 y FI(act)g(non-trivially)f (only)g(on)g Fy(C)1079 270 y Fz(2)1079 311 y(+)1145 294 y FI(as)g(2)p FD(B)1309 304 y Fz(1)1336 294 y FD(;)11 b FI(2)p FD(B)1448 304 y Fz(2)1498 294 y FI(and)22 b(2)p FD(B)1710 304 y Fz(3)1759 294 y FI(resp)r(ectiv)n(ely)-6 b(.)32 b(In)22 b(fact)308 373 y FD(\015)k FI(induces)501 426 y Fu(\000)532 481 y FI(\003)578 453 y Fz(1)604 481 y FI(\()p Fy(R)677 453 y Fz(4)705 481 y FI(\))15 b FE(\010)g FI(\003)859 453 y Fz(3)886 481 y FI(\()p Fy(R)959 453 y Fz(4)987 481 y FI(\))1013 426 y Fu(\001)1058 481 y FE(\012)h Fy(C)1193 462 y FE(\030)1194 484 y FI(=)1264 481 y(Hom\()p Fy(C)1472 453 y Fz(2)1472 497 y(+)1516 481 y FD(;)11 b Fy(C)1590 453 y Fz(2)1590 497 y FA(\000)1634 481 y FI(\))k FE(\010)g FI(Hom\()p Fy(C)1950 453 y Fz(2)1950 497 y FA(\000)1994 481 y FD(;)c Fy(C)2069 453 y Fz(2)2069 497 y(+)2113 481 y FI(\))831 589 y(\003)877 561 y Fz(2)877 606 y FA(\006)917 589 y FI(\()p Fy(R)990 561 y Fz(4)1018 589 y FI(\))j FE(\012)i Fy(C)1193 570 y FE(\030)1194 592 y FI(=)1264 589 y(End)1383 599 y Fz(0)1409 589 y FI(\()p Fy(C)1480 561 y Fz(2)1480 606 y FA(\006)1524 589 y FI(\))p FD(:)308 697 y FI(Here)23 b(End)582 707 y Fz(0)631 697 y FI(denotes)f(the)f(trace-free)i(endomorphisms.)409 776 y(The)i(fact)h(that)e(the)h(space)h(of)f(complexi\014ed)g (self-dual)h(t)n(w)n(o-forms)e(is)h(iso-)308 855 y(morphic)20 b(to)g(the)g(traceless)i(endomorphisms)17 b(of)k(the)g(p)r(ositiv)n(e)f (spinors)h(will)g(b)r(e)308 933 y(crucial)c(in)f(the)f(form)n(ulation)g (of)h(the)f(monop)r(ole)g(equations.)27 b(See)17 b(the)e(b)r(eginning) 308 1012 y(of)22 b(Chapter)f(2.)409 1090 y(2.1.2.)34 b FJ(A)29 b(standar)m(d)e(Cli\013or)m(d)h(mo)m(dule)g(for)h Fy(R)1580 1066 y Fz(3)1608 1090 y FJ(.)k FI(W)-6 b(e)29 b(de\014ne)e(an)g(action)h(of)308 1169 y Fy(R)355 1145 y Fz(3)402 1169 y FI(on)19 b Fy(C)535 1145 y Fz(2)585 1169 y FI(as)f(follo)n(ws.)29 b(Cho)r(ose)19 b(an)f(orien)n(ted)g (orthonormal)f(basis)h FD(e)2033 1179 y Fz(1)2060 1169 y FD(;)11 b(e)2120 1179 y Fz(2)2146 1169 y FD(;)g(e)2206 1179 y Fz(3)2252 1169 y FI(for)308 1248 y Fy(R)355 1223 y Fz(3)1139 1328 y FD(\015)t FI(\()p FD(e)1234 1338 y FF(j)1258 1328 y FI(\))18 b(=)i FE(\000)p FD(B)1475 1338 y FF(j)308 1422 y FI(with)26 b FD(B)513 1432 y FF(j)565 1422 y FI(as)g(ab)r(o)n(v)n(e.)44 b(One)26 b(can)h(c)n(hec)n(k)g(that)f FD(\015)k FI(extends)c(to)h(an)f(isomorphism)308 1501 y FD(\015)11 b FI(:)35 b(Cl\()p Fy(R)545 1476 y Fz(3)573 1501 y FI(\))14 b FE(\012)i Fy(C)35 b FE(!)19 b FI(End)953 1511 y Ft(C)988 1501 y FI(\()p Fy(C)1058 1476 y Fz(2)1089 1501 y FI(\).)29 b(Note)23 b(that)815 1609 y FD(\015)t FI(\()p FD(e)910 1619 y Fz(1)951 1609 y FE(^)15 b FD(e)1042 1619 y Fz(2)1083 1609 y FE(^)g FD(e)1174 1619 y Fz(3)1201 1609 y FI(\))j(=)h FE(\000)p FD(B)1417 1619 y Fz(1)1444 1609 y FD(B)1494 1619 y Fz(2)1521 1609 y FD(B)1571 1619 y Fz(3)1617 1609 y FI(=)g FB(1)1725 1620 y Ft(C)1753 1607 y Fw(2)1805 1609 y FD(;)308 1717 y FI(so)j(this)g(is)g(a)g (standard)f(Cli\013ord)g(mo)r(dule.)409 1795 y(2.1.3.)34 b FJ(Conjugation)25 b(of)f(Cli\013or)m(d)f(mo)m(dules.)33 b FI(The)22 b(sets)g(of)h(complex)e(linear)308 1874 y(endomorphisms)26 b(of)i(a)h(complex)f(v)n(ector)g(space,)j FD(V)15 b FI(,)30 b(and)e(its)g(conjugate,)2269 1857 y(\026)2259 1874 y FD(V)15 b FI(,)308 1953 y(are)26 b(the)g(same.)42 b(Therefore)26 b(a)g(Cli\013ord)f(mo)r(dule)g FD(\015)11 b FI(:)25 b FD(H)30 b FE(!)c FI(End\()p FD(V)14 b FI(\))26 b(giv)n(es)h(us)308 2031 y(another)i(Cli\013ord)f(mo)r(dule)g FD(\015)11 b FI(:)26 b FD(H)35 b FE(!)d FI(End)o(\()1477 2014 y(\026)1467 2031 y FD(V)15 b FI(\).)51 b(In)29 b(particular)f(if)i FD(\015)11 b FI(:)25 b FD(H)36 b FE(!)308 2110 y FI(End\()p FD(V)14 b FI(\))27 b(is)h(standard)e(then)h(so)g(is)h FD(\015)11 b FI(:)25 b FD(H)33 b FE(!)28 b FI(End\()1651 2093 y(\026)1641 2110 y FD(V)14 b FI(\),)29 b(and)e(hence)h(they)f(are) 308 2188 y(isomorphic.)409 2310 y FB(2.2.)38 b(Spin)712 2285 y FF(c)736 2310 y FB(-structures)h(on)g(manifolds.)34 b FI(In)f(constructing)f(a)i(Dirac)308 2388 y(op)r(erator)22 b(on)g(a)h(general)g(Riemannian)f(manifold,)g(the)h(\014rst)e(problem)h (one)h(en-)308 2467 y(coun)n(ters)f(is)h(top)r(ological.)32 b(T)-6 b(o)23 b(b)r(egin)f(with,)g(let)h FD(X)28 b FI(b)r(e)22 b(a)h(paracompact)e(Haus-)308 2546 y(dor\013)h(space)i(and)e FD(H)k FE(!)21 b FD(X)28 b FI(an)23 b(orien)n(ted)g(real)g(v)n(ector)g (bundle)g(equipp)r(ed)f(with)308 2624 y(an)i(inner)g(pro)r(duct.)34 b(\(Later)24 b FD(X)29 b FI(will)24 b(b)r(e)g(an)g(orien)n(ted)g (Riemannian)f(manifold)308 2703 y(and)f FD(H)27 b FI(will)22 b(b)r(e)g(its)g(cotangen)n(t)g(bundle.\))409 2796 y FK(Definition)30 b FI(1.3)p FK(.)f FI(A)d(Spin)1132 2768 y FF(c)1155 2796 y FJ(-structur)m(e)p FI(,)g Fs(s)p FI(,)g(on)e FD(H)29 b FE(!)24 b FD(X)29 b FI(is)d(a)e(pair)h(\()p FD(V)t(;)11 b(\015)t FI(\))308 2875 y(where)29 b FD(V)46 b FE(!)31 b FD(X)j FI(is)c(a)f(complex)g(v)n(ector)g(bundle)f(with)h(a)g (Hermitian)f(metric)308 2954 y(and)22 b FD(\015)27 b FI(is)22 b(a)h(bundle)f(map)f FD(\015)j FI(:)c FD(H)k FE(!)c FI(End)o(\()p FD(V)15 b FI(\))22 b(whic)n(h)h(is)g(\014b)r (erwise)f(a)g(standard)308 3032 y(Cli\013ord)f(mo)r(dule.)308 3126 y(It)e(is)g(easy)g(to)g(see,)h(that)e(existence)i(or)e (non-existence)i(of)f(a)g(Spin)1936 3097 y FF(c)1959 3126 y FI(-structure)f(on)308 3204 y(a)j(giv)n(en)g FD(H)j FE(!)19 b FD(X)25 b FI(is)c(indeed)g(a)g(purely)f(top)r(ological)h (problem,)f(whic)n(h)h(w)n(e)f(shall)308 3283 y(no)n(w)i(discuss)g(in)g (the)g(framew)n(ork)e(of)1265 3266 y(\024)1257 3283 y(Cec)n(h)j (cohomology)-6 b(.)409 3362 y(F)g(or)23 b(a)f(Lie)h(group)e FD(G)i FI(w)n(e)f(shall)h(denote)f(b)n(y)f FE(S)1553 3372 y FF(G)1616 3362 y FI(the)g(sheaf)i(assigning)f(to)g(an)308 3440 y(op)r(en)j(subset)f(of)h FD(X)30 b FI(the)25 b(con)n(tin)n(uous)f (functions)h(on)f(that)g(set)h(with)f(v)l(alues)i(in)308 3519 y FD(G)p FI(.)p eop %%Page: 11 7 11 10 bop 1038 160 a FC(2.)24 b(Spin)1203 139 y Fv(c)1225 160 y FC(-STR)o(UCTURES)677 b(11)409 294 y FI(Let)23 b FE(f)p FD(U)607 304 y FF(a)635 294 y FE(g)f FI(b)r(e)h(a)f(co)n(v)n (ering)h(of)g FD(X)28 b FI(b)n(y)22 b(op)r(en)g(sets)h(for)f(whic)n(h)h FD(H)5 b FE(j)2007 304 y FF(U)2039 310 y Fv(a)2090 294 y FI(is)23 b(trivial)308 373 y(for)j(all)g FD(a)p FI(.)41 b(W)-6 b(e)26 b(can)g(then)f(\014nd)g(Spin)1263 345 y FF(c)1286 373 y FI(-structures)g Fs(s)1651 383 y FF(a)1704 373 y FI(for)h(the)f(restrictions)h(to)308 451 y(these)31 b(op)r(en)g(sets.)56 b(Moreo)n(v)n(er,)35 b(these)c(Spin)1462 423 y FF(c)1486 451 y FI(-structures)e(are)i(unique)g(up)f(to)308 530 y(isomorphism.)e(F)-6 b(or)22 b(eac)n(h)h(ordered)e(pair)h(\()p FD(a;)11 b(b)p FI(\),)22 b(c)n(ho)r(ose)g(an)g(isomorphism)308 655 y(\(1.6\))515 b FD(\036)998 665 y FF(ab)1054 655 y FI(:)23 b Fs(s)1125 665 y FF(a)1153 655 y FE(j)1172 665 y FF(U)1204 671 y Fv(a)1229 665 y FA(\\)p FF(U)1293 673 y Fv(b)1336 655 y FE(!)c Fs(s)1452 665 y FF(b)1475 655 y FE(j)1494 665 y FF(U)1526 671 y Fv(a)1552 665 y FA(\\)p FF(U)1616 673 y Fv(b)1662 655 y FD(;)308 783 y FI(arranging)i(that)g FD(\036)787 793 y FF(ba)855 783 y FI(=)e FD(\036)964 755 y FA(\000)p Fz(1)964 801 y FF(ab)1028 783 y FI(.)29 b(If)22 b(eac)n(h)h(triple-o)n(v)n(erlap)f(function)1090 908 y FD( )1133 918 y FF(abc)1221 908 y FI(=)d FD(\036)1330 918 y FF(ab)1378 908 y FD(\036)1417 918 y FF(bc)1461 908 y FD(\036)1500 918 y FF(ca)308 1033 y FI(is)j(the)f(iden)n(tit)n(y) h(on)f(its)h(domain)e(of)i(de\014nition,)f(then)g(one)g(ma)n(y)g(form)f (a)i(Spin)2285 1005 y FF(c)2309 1033 y FI(-)308 1112 y(structure)h Fs(s)h FI(on)f FD(H)k FE(!)22 b FD(X)29 b FI(from)23 b(the)h(giv)n(en)g(collection.)37 b(In)23 b(general)h(ho)n(w)n(ev)n(er,)308 1190 y FD( )351 1200 y FF(abc)452 1190 y FI(is)33 b(an)e(automorphism)e(of)j Fs(s)1192 1200 y FF(a)1220 1190 y FE(j)1239 1200 y FF(U)1271 1206 y Fv(a)1297 1200 y FA(\\)p FF(U)1361 1208 y Fv(b)1382 1200 y FA(\\)p FF(U)1446 1206 y Fv(c)1503 1190 y FI(.)59 b(Since)32 b(a)g(standard)f(Cli\013ord)308 1269 y(mo)r(dule)23 b(has)g(automorphism)d(group)j FD(U)7 b FI(\(1\))20 b(=)i FD(S)1564 1245 y Fz(1)1590 1269 y FI(,)i FD( )1675 1279 y FF(abc)1768 1269 y FI(m)n(ust)e(tak)n(e)i(v)l(alues)g(in)308 1348 y FD(S)353 1323 y Fz(1)379 1348 y FI(.)30 b(W)-6 b(e)23 b(therefore)f(ha)n(v)n(e)g(functions)308 1473 y(\(1.7\))507 b FD( )994 1483 y FF(abc)1071 1473 y FI(:)23 b FD(U)1157 1483 y FF(a)1200 1473 y FE(\\)15 b FD(U)1305 1483 y FF(b)1343 1473 y FE(\\)g FD(U)1448 1483 y FF(c)1490 1473 y FE(!)k FD(S)1621 1445 y Fz(1)1669 1473 y FD(:)308 1598 y FI(These)25 b(satisfy)f(the)g(co)r(cycle)i(condition)e(on)g (quadruple)g(o)n(v)n(erlaps,)h(and)f(there-)308 1677 y(fore)h(de\014ne)g(a)g(2-co)r(cycle)h(for)e(the)h(sheaf)g FE(S)1409 1688 y FF(S)1441 1675 y Fw(1)1467 1677 y FI(.)38 b(An)n(y)25 b(other)f(c)n(hoice)i FD(\036)2080 1652 y FA(0)2080 1694 y FF(ab)2153 1677 y FI(of)f(iso-)308 1755 y(morphisms)g(is)j(of)f(the)g(form)f FD(\036)1119 1731 y FA(0)1119 1773 y FF(ab)1196 1755 y FI(=)i FD(\031)1312 1765 y FF(ab)1361 1755 y FD(\036)1400 1765 y FF(ab)1449 1755 y FI(,)h(where)e FD(\031)1729 1765 y FF(ab)1805 1755 y FI(is)h FD(S)1922 1731 y Fz(1)1948 1755 y FI(-v)l(alued,)h(with) 308 1834 y FD(S)353 1810 y Fz(1)406 1834 y FI(again)e(acting)g(p)r(oin) n(t)n(wise)f(as)h(the)g(automorphisms)c(of)k Fs(s)1868 1844 y FF(a)1896 1834 y FI(.)44 b(The)27 b(e\013ect)g(of)308 1913 y(c)n(ho)r(osing)g FD(\036)620 1888 y FA(0)620 1930 y FF(ab)696 1913 y FI(rather)e(than)h FD(\036)1096 1923 y FF(ab)1172 1913 y FI(is)h(to)f(alter)h FE(f)p FD( )1564 1923 y FA(\003\003\003)1638 1913 y FE(g)f FI(b)n(y)h(the)f(cob)r (oundary)g(of)308 1995 y(the)c(1-co)r(c)n(hain)h FE(f)p FD(\031)787 2005 y FA(\003\003)838 1995 y FE(g)p FI(.)30 b(Th)n(us)21 b(the)1208 1978 y(\024)1201 1995 y(Cec)n(h)i(cohomology)e (class)i(of)f(the)g(2-co)r(cycle)308 2074 y(dep)r(ends)g(only)f(on)h FD(H)i FE(!)19 b FD(X)27 b FI(and)21 b(not)h(on)f(the)h(c)n(hoices)h (made;)f(w)n(e)g(write)308 2199 y(\(1.8\))478 b Fs(o)p FI(\()p FD(H)5 b FI(\))18 b(=)h([)p FD( )1216 2209 y FA(\003\003\003)1290 2199 y FI(])g FE(2)g FD(H)1451 2171 y Fz(2)1478 2199 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1634 2211 y FF(S)1666 2198 y Fw(1)1692 2199 y FI(\))308 2324 y(for)20 b(this)f(class.)30 b(It)19 b(is)g(the)h(obstruction)e(to)h (the)g(existence)i(of)f(a)f(Spin)2024 2296 y FF(c)2048 2324 y FI(-structure)308 2403 y(for)28 b FD(H)33 b FE(!)28 b FD(X)5 b FI(.)47 b(Its)27 b(construction)f(sho)n(ws)h(that)g(it)g(b)r (eha)n(v)n(es)h(naturally)f(under)308 2482 y(pull-bac)n(k,)c(and)e(is)h (th)n(us)g(a)g(c)n(haracteristic)g(class)h(of)f(SO\()p FD(n)p FI(\)-bundles.)409 2560 y(If)e(a)g(Spin)655 2532 y FF(c)678 2560 y FI(-structure)f(on)g FD(H)24 b FE(!)19 b FD(X)25 b FI(exists,)c(one)f(can)g(similarly)g(discuss)g(the)308 2639 y(question)26 b(of)g(uniqueness.)41 b(If)26 b Fs(s)f FI(and)h Fs(s)1312 2615 y FA(0)1353 2639 y FI(are)g(t)n(w)n(o)f(Spin) 1724 2611 y FF(c)1747 2639 y FI(-structures,)h(then)g(w)n(e)308 2718 y(can)i(\014nd)e(an)h(op)r(en)g(co)n(v)n(er)h FE(f)p FD(U)1089 2728 y FF(a)1117 2718 y FE(g)f FI(whic)n(h)g(trivializes)i(b) r(oth,)f(and)f(w)n(e)g(can)h(\014nd)308 2796 y(isomorphisms)19 b FD(\034)748 2806 y FF(a)795 2796 y FI(:)g Fs(s)p FE(j)881 2806 y FF(U)913 2812 y Fv(a)960 2796 y FE(!)g Fs(s)1076 2772 y FA(0)1092 2796 y FE(j)1111 2806 y FF(U)1143 2812 y Fv(a)1171 2796 y FI(.)29 b(The)21 b(collection)i(of)e(functions)g FD(\033)2042 2806 y FF(ab)2109 2796 y FI(=)e FD(\034)2216 2772 y FA(\000)p Fz(1)2208 2813 y FF(a)2279 2796 y FD(\034)2308 2806 y FF(b)308 2875 y FI(is)27 b(a)g(1-co)r(cycle)h(with)e(v)l(alues)h (in)g FE(S)1205 2886 y FF(S)1237 2874 y Fw(1)1263 2875 y FI(.)43 b(An)n(y)27 b(c)n(hange)g(in)g(the)f FD(\034)1922 2885 y FA(\003)1975 2875 y FI(c)n(hanges)h(this)308 2954 y(co)r(cycle)d(b)n(y)d(a)h(cob)r(oundary)-6 b(.)29 b(So)22 b(the)f(cohomology)g(class)308 3079 y(\(1.9\))470 b FD(\016)s FI(\()p Fs(s)p FD(;)11 b Fs(s)1061 3051 y FA(0)1077 3079 y FI(\))18 b(=)h([)p FD(\033)1247 3089 y FA(\003\003)1298 3079 y FI(])g FE(2)f FD(H)1458 3051 y Fz(1)1485 3079 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1641 3090 y FF(S)1673 3077 y Fw(1)1699 3079 y FI(\))308 3204 y(is)23 b(indep)r(enden)n(t)d (of)j(the)e(c)n(hoices)i(made.)29 b(It)21 b(is)h(the)g(obstruction)f (to)g(\014nding)g(an)308 3283 y(isomorphism)f(b)r(et)n(w)n(een)i(the)g (Spin)1191 3255 y FF(c)1214 3283 y FI(-structures.)409 3362 y(Con)n(v)n(ersely)-6 b(,)19 b(giv)n(en)f(an)f(elemen)n(t)g FD(\016)k FE(2)e FD(H)1420 3337 y Fz(1)1446 3362 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1602 3373 y FF(S)1634 3360 y Fw(1)1660 3362 y FI(\),)19 b(and)d(a)h(Spin)2024 3333 y FF(c)2048 3362 y FI(-structure)308 3440 y Fs(s)p FI(,)k(one)g(can)f (construct)g Fs(s)935 3416 y FA(0)971 3440 y FI(with)g FD(\016)s FI(\()p Fs(s)p FD(;)11 b Fs(s)1267 3416 y FA(0)1282 3440 y FI(\))19 b(=)g FD(\016)s FI(.)29 b(This)20 b(ma)n(y)f(b)r(e)i (done)f(in)g(the)g(st)n(yle)308 3519 y(of)i(the)g(ab)r(o)n(v)n(e)g (discussion,)h(using)e(transition)g(functions.)p eop %%Page: 12 8 12 11 bop 308 159 a FC(12)84 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)409 294 y FI(Alternativley)-6 b(,)32 b(note)d(that)f(the)h(group)f FD(H)1511 270 y Fz(1)1537 294 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1693 305 y FF(S)1725 293 y Fw(1)1751 294 y FI(\))29 b(classi\014es)h (principal)308 373 y(bundles)21 b(on)g FD(X)27 b FI(with)20 b(structure)h(group)f(U\(1\))h(up)g(to)g(isomorphism,)f(or)h(equiv-)308 451 y(alen)n(tly)26 b(complex)f(line-bundles)h(with)e(Hermitian)h (metric.)40 b(T)-6 b(o)25 b(construct)g Fs(s)2297 427 y FA(0)2312 451 y FI(,)308 530 y(de\014ne)d FD(V)537 540 y Fr(s)558 527 y Fq(0)598 530 y FI(as)308 668 y(\(1.10\))632 b FD(V)1147 678 y Fr(s)1168 665 y Fq(0)1205 668 y FI(=)19 b FD(V)1313 678 y Fr(s)1352 668 y FE(\012)c FD(L)1464 678 y FF(\016)1512 668 y FD(;)308 806 y FI(where)21 b FD(L)543 816 y FF(\016)590 806 y FI(is)h(a)f(Hermitian)g(line-bundle)h (corresp)r(onding)e(to)h FD(\016)s FI(.)29 b(There)22 b(is)f(then)308 885 y(a)27 b(canonical)g(isomorphism)d FD(\023)7 b FI(:)36 b(End)o(\()p FD(V)1318 895 y Fr(s)1342 885 y FI(\))25 b FE(!)i FI(End)o(\()p FD(V)1669 895 y Fr(s)1690 882 y Fq(0)1708 885 y FI(\),)g(and)f(one)g(de\014nes)g FD(\015)2292 895 y Fr(s)2313 882 y Fq(0)308 963 y FI(as)c(the)g(comp)r (osite)f(represen)n(tation)h FD(\023)p FE(\016)p FD(\015)1336 973 y Fr(s)1359 963 y FI(.)409 1042 y(The)27 b(classes)g Fs(o)p FI(\()p FD(H)5 b FI(\))26 b(and)g FD(\016)s FI(\()p Fs(s)p FD(;)11 b Fs(s)1215 1018 y FA(0)1231 1042 y FI(\))26 b(b)r(ecome)g(computable)f(if)i(w)n(e)g(relate)g(the)308 1121 y(ab)r(o)n(v)n(e)c(discussion)g(to)g(in)n(tegral)g(cohomology)-6 b(.)32 b(Consider)22 b(the)h(exact)g(sequence)308 1199 y(of)f(shea)n(v)n(es)1029 1310 y FE(S)1070 1320 y Ft(Z)1121 1310 y FE(\000)-10 b(!)19 b(S)1290 1320 y Ft(R)1361 1269 y Fz(exp)1344 1310 y FE(\000)-11 b(!)19 b(S)1512 1321 y FF(S)1544 1308 y Fw(1)1592 1310 y FD(:)308 1434 y FI(It)34 b(giv)n(es)i(rise)f(to)f(a)h(long)f(exact)h(sequence)g(in)g(cohomology) -6 b(,)38 b(in)d(whic)n(h)f(all)308 1513 y(the)29 b(groups)g FD(H)710 1488 y FF(i)729 1513 y FI(\()p FD(X)5 b FI(;)11 b FE(S)885 1523 y Ft(R)920 1513 y FI(\))29 b(with)g FD(i)h(>)i FI(0)d(v)l(anish)g(b)r(ecause)h FE(S)1835 1523 y Ft(R)1899 1513 y FI(is)g(a)f(\014ne)g(sheaf.)308 1591 y(So)d FD(H)464 1567 y Fz(1)491 1591 y FI(\()p FD(X)5 b FI(;)11 b FE(S)647 1603 y FF(S)679 1590 y Fw(1)705 1591 y FI(\))25 b(is)i(isomorphic)e(to) h FD(H)1310 1567 y Fz(2)1336 1591 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-5 b FI(\))o(:)34 b(this)26 b(is)g(the)g(statemen)n(t)f (that)308 1670 y(Hermitian)g(line-bundles)h(are)f(classi\014ed)h(up)f (to)g(isomorphism)e(b)n(y)j(their)f(\014rst)308 1749 y(Chern)d(classes.)33 b Fp(Mutatis)22 b(m)n(utandis)p FI(,)g(the)g(group)g FD(H)1663 1724 y Fz(2)1689 1749 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1845 1760 y FF(S)1877 1747 y Fw(1)1903 1749 y FI(\))23 b(is)g(isomorphic)308 1827 y(to)f FD(H)449 1803 y Fz(3)476 1827 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))o(.)409 1906 y(Our)28 b(discussion)h(of)f (Spin)1077 1878 y FF(c)1100 1906 y FI(-structures)f(can)h(b)r(e)h (summarized)d(in)i(the)g(fol-)308 1984 y(lo)n(wing)22 b(prop)r(osition.)409 2093 y FK(Pr)o(oposition)17 b FI(1.4)p FK(.)22 b FJ(A)n(n)c FI(SO\()p FD(n)p FI(\))p FJ(-bund)s(le)f FD(H)24 b FE(!)19 b FD(X)j FJ(admits)16 b(a)g FI(Spin)2099 2065 y FF(c)2122 2093 y FJ(-structur)m(e)308 2171 y(if)24 b(and)f(only)h(if)g(the)f(char)m(acteristic)h(class)f Fs(o)p FI(\()p FD(H)5 b FI(\))18 b FE(2)h FD(H)1672 2147 y Fz(3)1698 2171 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-5 b FI(\))19 b FJ(vanishes.)409 2250 y(If)29 b FD(H)j FE(!)27 b FD(X)33 b FJ(admits)28 b(a)f FI(Spin)1161 2222 y FF(c)1184 2250 y FJ(-structur)m(e,)j(then)e(the)g(set)g(of)g(isomorphism)308 2329 y(classes)e(of)f FI(Spin)734 2300 y FF(c)757 2329 y FJ(-structur)m(es)h(is)g(an)f(a\016ne)h(sp)m(ac)m(e)f(for)g FD(H)1783 2304 y Fz(2)1810 2329 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))o FJ(;)23 b(the)j(action)308 2407 y(of)h(an)g (element)h(of)f FD(H)876 2383 y Fz(2)903 2407 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))23 b FJ(is)k(de\014ne)m(d)g(thr)m (ough)f(the)h(tensor)g(pr)m(o)m(duct)f(with)308 2486 y(the)e(c)m(orr)m(esp)m(onding)e(line)i(bund)s(le)h(as)e(in)29 b FI(\(1.10\))p FJ(.)409 2594 y FI(2.2.1.)34 b FJ(Conjugation)26 b(of)f FI(Spin)1179 2566 y FF(c)1203 2594 y FJ(-structur)m(es.)33 b FI(The)23 b(conjugation)h(op)r(eration)308 2673 y(in)n(tro)r(duced)15 b(in)h(section)g(2.1.3)g(extends)g(to)f(an)h(in)n(v)n(olution)f(on)h (Spin)1980 2645 y FF(c)2004 2673 y FI(-structures.)308 2752 y(W)-6 b(e)22 b(denote)f(this)g(map)e(b)n(y)i Fs(s)e FE(7!)p 1132 2712 30 3 v 19 w Fs(s)p FI(.)29 b(It)21 b(ma)n(y)f(happ)r(en)g(that)p 1796 2712 V 20 w Fs(s)h FI(is)g(in)g(fact)h(isomor-)308 2830 y(phic)g(to)g Fs(s)p FI(.)30 b(If)21 b(so,)i(w)n(e)f(mak)n(e)f(the)h(follo)n(wing)h (de\014nition.)409 2939 y FK(Definition)j FI(1.5)p FK(.)h FI(A)21 b(Spin)1122 2910 y FF(c)1145 2939 y FI(-structure)f Fs(s)h FI(together)f(with)g(a)h(c)n(hoice)h(of)f(iso-)308 3017 y(morphism,)f FD(J)14 b FI(:)p 726 2977 V 23 w Fs(s)19 b FE(!)g Fs(s)p FI(,)j(is)g(called)h(a)f FJ(spin)i(structur)m(e)p FI(.)409 3126 y(W)-6 b(e)21 b(can)e(think)g(of)g FD(J)27 b FI(as)19 b(a)h Fy(C)12 b FI(-an)n(ti-li)q(near)23 b(isomorphism)17 b FD(V)34 b FE(!)19 b FD(V)35 b FI(comm)n(ut-)308 3204 y(ing)22 b(with)g FD(\015)t FI(.)409 3283 y(In)k(an)n(y)f(case)i(there) e(is)h(a)g(unique)g(complex)f(line)h(bundle)f FD(L)1943 3293 y Fr(s)1993 3283 y FI(so)h(that)e Fs(s)h FI(=)p 308 3321 V 308 3362 a Fs(s)19 b FE(\012)g FD(L)473 3372 y Fr(s)496 3362 y FI(.)45 b(This)27 b FD(L)760 3372 y Fr(s)811 3362 y FI(is)g(called)h(the)f FJ(char)m(acteristic)g FI(line)h(bundle)f(of)g(the)g(Spin)2285 3333 y FF(c)2309 3362 y FI(-)308 3440 y(structure)19 b Fs(s)p FI(.)29 b(The)20 b(\014rst)f(Chern)h(class)h(of)f FD(L)1402 3450 y Fr(s)1446 3440 y FI(is)g(an)g(in)n(tegral)h(lift)g(of)f FD(w)2062 3450 y Fz(2)2089 3440 y FI(\()p FD(H)5 b FI(\),)20 b(the)308 3519 y(second)i(Stiefel-Whitney)h(class)f(of)g FD(H)5 b FI(.)30 b(This)22 b(will)g(b)r(e)g(de\014ned)g(b)r(elo)n(w.)p eop %%Page: 13 9 13 12 bop 1038 160 a FC(2.)24 b(Spin)1203 139 y Fv(c)1225 160 y FC(-STR)o(UCTURES)677 b(13)409 294 y FB(2.3.)38 b FI(Spin)694 266 y FF(c)717 294 y FB(-structures)18 b(via)e(principal)h(bundles)g(and)g(Stiefel-Whitney)308 373 y(classes.)32 b FI(The)20 b(de\014nition)g(of)h(a)f(Spin)1260 345 y FF(c)1283 373 y FI(-structure)f(can)i(b)r(e)f(rephrased)g(in)g (terms)308 451 y(of)26 b(principal)g(bundles.)39 b(Let)26 b FD(\015)1100 461 y Fz(0)1152 451 y FI(:)f Fy(R)1242 427 y FF(n)1300 451 y FE(!)g FI(End\()p Fy(C)1581 427 y FF(N)1630 451 y FI(\))h(b)r(e)f(a)h(standard)e(Cli\013ord)308 530 y(mo)r(dule.)29 b(One)22 b(can)g(de\014ne)g(a)g(group)f(Spin)1388 502 y FF(c)1412 530 y FI(\()p FD(n)p FI(\))h(as)g(the)f(set)h(of)g (pairs)991 653 y(\()p FD(\034)t(;)11 b(\033)r FI(\))18 b FE(2)g FI(SO\()p FD(n)p FI(\))d FE(\002)h FI(U\()p FD(N)7 b FI(\))308 776 y(for)22 b(whic)n(h)g(the)g(follo)n(wing)g (diagram)f(comm)n(utes:)1010 904 y Fy(R)1057 879 y FF(n)1286 866 y(\034)1215 904 y FE(\000)-20 b(\000)-14 b(\000)-21 b(!)145 b Fy(R)1575 879 y FF(n)980 1017 y(\015)1005 1023 y Fw(0)1028 951 y Fu(?)1028 992 y(?)1028 1032 y(y)1498 1017 y FF(\015)1523 1023 y Fw(0)1546 951 y Fu(?)1546 992 y(?)1546 1032 y(y)918 1176 y FI(End\()p Fy(C)1107 1151 y FF(N)1156 1176 y FI(\))1236 1132 y Fz(A)p FF(d)o Fz(\()p FF(\033)r Fz(\))1215 1176 y FE(\000)-20 b(\000)-14 b(\000)-21 b(!)34 b FI(End)o(\()p Fy(C)1605 1151 y FF(N)1655 1176 y FI(\))21 b FD(;)308 1298 y FI(where)h(A)p FD(d)q FI(\()p FD(\033)r FI(\))g(denotes)g(the)g(adjoin)n(t)g(action)g(of)h FD(\033)h FI(on)e(End\()p Fy(C)1891 1273 y FF(N)1940 1298 y FI(\).)31 b(The)22 b(homo-)308 1376 y(morphism)j(Spin)751 1348 y FF(c)774 1376 y FI(\()p FD(n)p FI(\))j FE(!)h FI(SO\()p FD(n)p FI(\))f(giv)n(en)g(b)n(y)f FD(\034)35 b FI(is)28 b(surjectiv)n(e)g(b)r(ecause)g(of)g(the)308 1455 y(uniqueness)h(of)h(standard)e(Cli\013ord)h(mo)r(dules,)h(and)f (its)h(k)n(ernel)f(is)h(the)f(auto-)308 1534 y(morphism)20 b(group)i(of)h(the)f(mo)r(dule,)g(whic)n(h)h(is)g FD(S)1549 1509 y Fz(1)1575 1534 y FI(.)31 b(So)23 b(there)f(is)h(a)g(short)f (exact)308 1612 y(sequence)308 1753 y(\(1.11\))471 b FD(S)993 1725 y Fz(1)1061 1712 y FF(j)1038 1753 y FE(!)19 b FI(Spin)1253 1724 y FF(c)1276 1753 y FI(\()p FD(n)p FI(\))g FE(!)g FI(SO\()p FD(n)p FI(\))i FD(:)308 1876 y FI(T)-6 b(o)26 b(giv)n(e)f(a)g(Spin)734 1848 y FF(c)757 1876 y FI(-structure)f(for)h(an)g(orien)n(ted)g(v)n(ector)g(bundle)f FD(H)29 b FE(!)24 b FD(X)30 b FI(with)308 1954 y(a)f(metric)e(is)i (equiv)l(alen)n(t)f(to)g(giving)h(a)f(principal)g(Spin)1734 1926 y FF(c)1757 1954 y FI(\()p FD(n)p FI(\)-bundle)f FD(Q)j FE(!)g FD(X)308 2033 y FI(together)e(with)f(an)h(isomorphism)e (of)j FD(Q=S)1433 2009 y Fz(1)1489 2033 y FE(!)g FD(X)34 b FI(with)27 b(F)-6 b(r)q(\()p FD(H)5 b FI(\))28 b FE(!)i FD(X)5 b FI(,)30 b(the)308 2112 y(orien)n(ted)22 b(orthonormal)e(frame) h(bundle.)409 2190 y(In)26 b(one)h(direction,)h(giv)n(en)f(a)f(Spin) 1285 2162 y FF(c)1309 2190 y FI(-structure)f Fs(s)h FI(=)h(\()p FD(V)t(;)11 b(\015)t FI(\),)28 b(one)e(can)h(con-)308 2269 y(struct)14 b FD(Q)19 b FE(!)g FD(X)h FI(b)n(y)14 b(de\014ning)g(its)h(\014b)r(ers)f FD(Q)1358 2279 y FF(x)1403 2269 y FI(as)g(sets)h(of)g(pairs)f(\()p FD(t;)d(s)p FI(\))19 b FE(2)f FI(Hom\()p Fy(R)2249 2245 y FF(n)2282 2269 y FD(;)11 b(H)2366 2279 y FF(x)2396 2269 y FI(\))p FE(\002)308 2348 y FI(Hom\()p Fy(C)516 2323 y FF(N)566 2348 y FD(;)g(V)633 2358 y FF(x)663 2348 y FI(\))19 b(whic)n(h)f(preserv)n(e)h(b)r(oth)f (inner)g(pro)r(ducts)f(and)i(mak)n(e)f(the)g(follo)n(w-)308 2426 y(ing)k(diagram)f(comm)n(utativ)n(e:)1008 2536 y FD(H)1063 2546 y FF(x)1202 2536 y FE( )-20 b(\000)-14 b(\000)-21 b(\000)1278 2576 y FF(t)1515 2536 y Fy(R)1562 2512 y FF(n)977 2668 y(\015)1002 2674 y Fv(x)1028 2602 y Fu(?)1028 2642 y(?)1028 2683 y(y)1485 2668 y FF(\015)1510 2674 y Fw(0)1533 2602 y Fu(?)1533 2642 y(?)1533 2683 y(y)931 2805 y FI(End\()p FD(V)1114 2815 y FF(x)1143 2805 y FI(\))33 b FE( )-20 b(\000)-14 b(\000)-21 b(\000)1227 2850 y Fz(A)p FF(d)o Fz(\()p FF(s)p Fz(\))1404 2805 y FI(End)o(\()p Fy(C)1592 2780 y FF(N)1642 2805 y FI(\))21 b FD(:)308 2968 y FI(The)h(map)f(from)g FD(Q)h FI(to)f(F)-6 b(r)q(\()p FD(H)5 b FI(\))21 b(is)h(giv)n(en)h(b)n(y)e(\()p FD(t;)11 b(s)p FI(\))19 b FE(7!)g FD(t)p FI(.)409 3047 y(In)30 b(the)f(other)g(direction,)j(giv)n(en)e FD(Q)p FI(,)j(one)d(can)f(asso)r(ciate)i(to)e(it)h(a)g(v)n(ector)308 3126 y(bundle)g FD(V)45 b FI(via)30 b(the)g(homomorphism)c(Spin)1456 3097 y FF(c)1480 3126 y FI(\()p FD(n)p FI(\))32 b FE(!)h FI(U\()p FD(N)7 b FI(\))29 b(pro)n(vided)h(b)n(y)f FD(\033)r FI(;)308 3204 y(and)22 b FD(\015)k FI(is)c(then)f(obtained)h(from)f FD(\015)1175 3214 y Fz(0)1201 3204 y FI(.)409 3283 y(The)26 b(pro)r(of)g(of)g(Prop)r(osition)f(1.4)i(can)f(no)n(w)g(b)r(e)g (rephrased)f(as)h(follo)n(ws.)43 b(A)308 3362 y(principal)26 b FD(G)p FI(-bundle)f(is)h(a)g(1-co)r(c)n(hain)g(with)f(v)l(alues)i(in) e FE(S)1783 3372 y FF(G)1823 3362 y FI(,)i(and)e(the)h(isomor-)308 3440 y(phism)e(class)h(of)g(the)f(bundle)g(corresp)r(onds)g(to)g(the)h (cohomology)f(class)h(repre-)308 3519 y(sen)n(ted)d(b)n(y)g(this)g(co)r (c)n(hain.)p eop %%Page: 14 10 14 13 bop 308 159 a FC(14)84 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)409 294 y FI(The)f(short)f(exact)h (sequence)h(of)f(shea)n(v)n(es)967 406 y FE(S)1008 418 y FF(S)1040 405 y Fw(1)1084 406 y FE(!)d(S)1211 417 y Fz(Spin)1304 400 y Fv(c)1325 417 y Fz(\()p FF(n)p Fz(\))1413 406 y FE(!)g(S)1540 417 y Fz(SO\()p FF(n)p Fz(\))308 518 y FI(giv)n(es)k(rise)f(to)g(an)g(exact)g(sequence)h(in)f (cohomology)-6 b(,)22 b(whic)n(h)f(reads)h(in)g(part:)308 630 y(\(1.12\))212 b FD(H)749 603 y Fz(1)776 630 y FI(\()p FD(X)5 b FI(;)11 b FE(S)932 641 y Fz(Spin)1025 624 y Fv(c)1047 641 y Fz(\()p FF(n)p Fz(\))1115 630 y FI(\))19 b FE(!)g FD(H)1306 603 y Fz(1)1333 630 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1489 641 y Fz(SO)o(\()p FF(n)p Fz(\))1621 630 y FI(\))19 b FE(!)g FD(H)1812 603 y Fz(2)1838 630 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1994 642 y FF(S)2026 629 y Fw(1)2052 630 y FI(\))22 b FD(:)308 742 y FI(This)f(is)f(an)h (exact)f(sequence)h(of)g(p)r(oin)n(ted)f(sets,)h(the)f(base)h(p)r(oin)n (t)f(b)r(eing)g(the)g(iso-)308 821 y(morphism)d(class)k(of)e(the)h (trivial)g(bundle.)28 b(Due)20 b(to)f(the)h(nonab)r(elian)f(co)r (e\016cien)n(t)308 900 y(groups,)j(the)f(sequence)i(do)r(es)f(not)f (extend)h(in)g(a)g(naiv)n(e)g(w)n(a)n(y)-6 b(.)409 978 y(The)22 b(isomorphism)d(class)j(of)g(a)f(principal)g(SO\()p FD(n)p FI(\)-bundle)g(F)-6 b(r\()p FD(H)5 b FI(\))21 b(giv)n(es)h(an)308 1057 y(elemen)n(t)k(of)g FD(H)695 1033 y Fz(1)721 1057 y FI(\()p FD(X)5 b FI(;)11 b FE(S)877 1067 y Fz(SO\()p FF(n)p Fz(\))1010 1057 y FI(\).)40 b(A)26 b(Spin)1298 1029 y FF(c)1321 1057 y FI(-structure)e(exists)i(if)g(and)f (only)h(if)g(this)308 1138 y(elemen)n(t)c(is)f(in)h(the)f(image)g(of)h FD(H)1132 1113 y Fz(1)1159 1138 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1315 1148 y Fz(Spin)1408 1132 y Fv(c)1429 1148 y Fz(\()p FF(n)p Fz(\))1498 1138 y FI(\).)29 b(If)22 b(the)f FE(f)p FD(\036)1823 1148 y FA(\003\003)1873 1138 y FE(g)g FI(in)h(\(1.6\))f(de\014ne)308 1216 y(a)27 b(co)r(cycle,)j (then)c(this)g(co)r(cycle)j(represen)n(ts)d(suc)n(h)h(a)f(lift.)45 b(By)26 b(exactness,)j(the)308 1295 y(existence)23 b(of)e(the)h(lift)g (is)g(equiv)l(alen)n(t)g(to)f(the)g(v)l(anishing)h(of)f(the)h(image)f Fs(o)p FI(\()p FD(H)5 b FI(\))18 b FE(2)308 1374 y FD(H)368 1349 y Fz(2)395 1374 y FI(\()p FD(X)r(;)11 b FE(S)548 1385 y FF(S)580 1372 y Fw(1)605 1374 y FI(\))23 b(of)g(the)g (isomorphism)e(class)j(of)f(F)-6 b(r)q(\()p FD(H)5 b FI(\).)32 b(This)23 b(is)g(represen)n(ted)g(b)n(y)308 1452 y(the)f(co)r(c)n(hain)g FE(f)p FD( )737 1462 y FA(\003\003\003)811 1452 y FE(g)g FI(in)g(\(1.7\).)409 1531 y(Spin)538 1503 y FF(c)561 1531 y FI(-structures)15 b(on)h(F)-6 b(r\()p FD(H)5 b FI(\))15 b(are)h(co)r(cycles)i(with)d(v)l(alues)i(in)f FE(S)1961 1541 y Fz(Spin)2054 1525 y Fv(c)2075 1541 y Fz(\()p FF(n)p Fz(\))2160 1531 y FI(lifting)308 1610 y(the)21 b FE(S)462 1620 y FF(S)s(O)q Fz(\()p FF(n)p Fz(\))600 1610 y FI(-co)r(cycle)h(giv)n(en)g(b)n(y)e(F)-6 b(r\()p FD(H)5 b FI(\).)29 b(An)n(y)21 b(t)n(w)n(o)g(di\013er)f(b)n(y)h (a)g(1-co)r(c)n(hain)g(with)308 1688 y(v)l(alues)27 b(in)f FE(S)631 1700 y FF(S)663 1687 y Fw(1)689 1688 y FI(.)42 b(The)26 b(cohomology)g(class)h(of)f(this)g(co)r(c)n(hain)h(is)g(the)f (di\013erence)308 1767 y(elemen)n(t)c FD(\016)s FI(\()p Fs(s)p FD(;)11 b Fs(s)699 1742 y FA(0)715 1767 y FI(\))21 b(in)h(\(1.9\).)409 1846 y(2.3.1.)34 b FJ(Spin)26 b(structur)m(es)g (and)g(princip)m(al)f(bund)s(les.)34 b FI(A)25 b(Spin)1933 1817 y FF(c)1957 1846 y FI(-structure,)f Fs(s)p FI(,)308 1924 y(whic)n(h)16 b(admits)e(an)i(isomorphism)d FD(J)h FI(:)p 1257 1884 30 3 v 23 w Fs(s)19 b FE(!)g Fs(s)c FI(giv)n(es)i(rise)f(to)f(a)h(spin)f(structure.)26 b(In)308 2003 y(terms)20 b(of)i(principal)f(bundles)g(the)g(c)n(hoice)h(of)g FD(J)28 b FI(de\014nes)21 b(a)g(reduction)g FD(P)30 b FI(of)21 b(the)308 2082 y(structure)15 b(group)h(of)g(the)g(principal)h (Spin)1342 2053 y FF(c)1365 2082 y FI(\()p FD(n)p FI(\))f(bundle,)i FD(Q)p FI(,)g(de\014ned)d(b)n(y)h Fs(s)p FI(,)i(to)e(the)308 2160 y(subgroup)22 b(Spin)o(\()p FD(n)p FI(\))e FE(\032)g FI(Spin)1036 2132 y FF(c)1059 2160 y FI(\()p FD(n)p FI(\))j(preserving) f FD(J)30 b FI(\014b)r(erwise.)h(In)23 b(a)g(single)g(\014b)r(er,)308 2239 y(the)h(automorphisms)c(of)k(a)g(standard)e(Cli\013ord)g(mo)r (dule)h(preserving)g(complex)308 2318 y(conjugation)i(consist)g(of)g FE(f\006)p FI(1)p FE(g)e(\032)i FD(S)1265 2293 y Fz(1)1291 2318 y FI(.)38 b(Th)n(us)24 b(Spin\()p FD(n)p FI(\))g(is)i(a)e(double)h (co)n(v)n(er)h(of)308 2396 y(SO\()p FD(n)p FI(\):)308 2508 y(\(1.13\))483 b Fy(Z)1009 2518 y Fz(2)1050 2508 y FE(!)19 b FI(Spin\()p FD(n)p FI(\))f FE(!)h FI(SO\()p FD(n)p FI(\))j FD(:)308 2620 y FI(W)-6 b(e)19 b(lea)n(v)n(e)h(to)e(the) g(reader)g(to)g(c)n(hec)n(k)h(that)e(this)h(is)h(the)f(connected)g (double)g(co)n(v)n(er.)308 2699 y(Th)n(us)25 b FD(P)34 b FI(is)25 b(a)g(connected)g(double)g(co)n(v)n(ering)h(of)f(the)g (frame)f(bundle)h(of)g FD(H)5 b FI(.)39 b(In)308 2778 y(this)22 b(w)n(a)n(y)g(w)n(e)g(reco)n(v)n(er)h(the)e(usual)h (de\014nition)f(of)i(a)f(spin)f(structure.)409 2856 y(Applying)33 b(the)g(sheaf)g(cohomology)f(argumen)n(t)f(ab)r(o)n(v)n(e)i(sho)n(ws)f (that)g(the)308 2935 y(obstruction)22 b(to)h(the)f(existence)i(of)f(a)g (spin)g(structure)f(on)g FD(H)28 b FI(is)c(a)e(w)n(ell-de\014ned)308 3014 y(elemen)n(t)g(of)g FD(H)687 2989 y Fz(2)714 3014 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)878 3024 y Fz(2)901 3014 y FI(\).)409 3109 y FK(Definition)32 b FI(1.6)p FK(.)e FI(F)-6 b(or)27 b(an)g(orien)n(ted)f Fy(R)1458 3084 y FF(n)1519 3109 y FI(bundle)g FD(H)31 b FE(!)d FD(X)5 b FI(,)28 b(the)e FJ(se)m(c)m(ond)308 3188 y(Stiefel-Whitney)34 b(class)j FD(w)1002 3198 y Fz(2)1029 3188 y FI(\()p FD(H)5 b FI(\))36 b FE(2)g FD(H)1318 3163 y Fz(2)1344 3188 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)1508 3198 y Fz(2)1531 3188 y FI(\))32 b(is)g(the)g(obstruction)f(to)h(the)308 3266 y(existence)23 b(of)f(a)g(spin)g(structure.)409 3362 y(The)c(reader)f(familiar)g(with)g(other)g(de\014nitions)f(of)i (Stiefel-Whitney)g(classes)308 3440 y(should)i(ha)n(v)n(e)g(no)g (trouble)g(v)n(erifying)h(the)f(equiv)l(alence)h(of)g(this)f (de\014nition)f(with)308 3519 y(their)j(fa)n(v)n(orite)g(one.)p eop %%Page: 15 11 15 14 bop 1038 160 a FC(2.)24 b(Spin)1203 139 y Fv(c)1225 160 y FC(-STR)o(UCTURES)677 b(15)409 294 y FI(F)-6 b(rom)26 b(the)f(inclusion)i(of)f(the)g(sequence)g(\(1.13\))g(in)g(\(1.11\),)h (one)f(sees)h(that)308 373 y Fs(o)p FI(\()p FD(H)5 b FI(\))22 b(is)h(the)f(image)h(of)f FD(w)970 383 y Fz(2)997 373 y FI(\()p FD(H)5 b FI(\))22 b(under)g(the)g(map)f FD(H)1640 348 y Fz(2)1667 373 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)1831 383 y Fz(2)1854 373 y FI(\))19 b FE(!)h FD(H)2046 348 y Fz(2)2073 373 y FI(\()p FD(X)5 b FI(;)11 b FE(S)2229 384 y FF(S)2261 371 y Fw(1)2287 373 y FI(\).)308 451 y(Alternativ)n(ely)-6 b(,)23 b(if)e(w)n(e)g(iden)n(tify)g FD(H)1174 427 y Fz(2)1200 451 y FI(\()p FD(X)5 b FI(;)11 b FE(S)1356 463 y FF(S)1388 450 y Fw(1)1414 451 y FI(\))20 b(with)g FD(H)1669 427 y Fz(3)1696 451 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))17 b(as)j(ab)r(o)n(v)n(e,)i Fs(o)p FI(\()p FD(H)5 b FI(\))308 530 y(is)24 b(the)f(image)g(of)h FD(w)807 540 y Fz(2)833 530 y FI(\()p FD(H)5 b FI(\))23 b(under)f(a)i(map)e FD(H)1421 506 y Fz(2)1447 530 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)1612 540 y Fz(2)1634 530 y FI(\))21 b FE(!)g FD(H)1829 506 y Fz(3)1856 530 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))19 b(whic)n(h)24 b(w)n(e)308 609 y(can)e(iden)n(tify)h(as)f(the)f(connecting)h(homomorphism)c(for)k (the)g(exact)g(sequence)308 773 y(\(1.14\))637 b Fy(Z)1181 734 y FA(\002)p Fz(2)1177 773 y FE(!)20 b Fy(Z)8 b FE(!)19 b Fy(Z)1462 783 y Fz(2)1507 773 y FD(:)308 923 y FI(F)-6 b(rom)22 b(the)f(long)h(exact)g(sequence)h(in)f(cohomology)f(w)n(e)h (deduce:)409 1037 y FK(Pr)o(oposition)17 b FI(1.7)p FK(.)22 b FJ(The)17 b(obstruction)h Fs(o)p FI(\()p FD(H)5 b FI(\))16 b FJ(to)g(the)h(existenc)m(e)h(of)f(a)g FI(Spin)2285 1009 y FF(c)2309 1037 y FJ(-)308 1116 y(structur)m(e)36 b(for)f FD(H)46 b FE(!)41 b FD(X)f FJ(vanishes)c(if)g(and)f(only)g(if)h FD(w)1788 1126 y Fz(2)1815 1116 y FI(\()p FD(H)5 b FI(\))35 b FJ(has)f(a)h(lift)h(to)308 1194 y FD(H)368 1170 y Fz(2)395 1194 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))p FJ(.)409 1308 y FI(By)33 b(a)g(Spin)718 1280 y FF(c)742 1308 y FI(-structure)e(on)i(a)g(Riemannian)e(manifold)i FD(X)5 b FI(,)36 b(w)n(e)d(mean)f(a)308 1387 y(Spin)437 1359 y FF(c)460 1387 y FI(-structure)c(for)h(its)g(cotangen)n(t)f(bundle.)50 b(Although)28 b(not)h(ev)n(ery)g(v)n(ector)308 1466 y(bundle)c(of)g (rank)f(4)h(on)f(a)h(4-manifold)f(has)h(a)g(Spin)1609 1437 y FF(c)1633 1466 y FI(-stucture,)g(the)f(cotangen)n(t)308 1544 y(bundle)e(alw)n(a)n(ys)g(has)g(one.)409 1659 y FK(Theorem)29 b FI(1.8)h(\(Whitney\))p FK(.)e FJ(Every)f(oriente)m(d)g (4-manifold)f FD(X)32 b FJ(admits)26 b(a)308 1737 y FI(Spin)437 1709 y FF(c)460 1737 y FJ(-structur)m(e.)31 b(That)23 b(is,)h FD(w)1087 1747 y Fz(2)1114 1737 y FI(\()p FD(X)5 b FI(\))23 b FJ(is)h(the)f(r)m(e)m(duction)g(of)h(an)f(inte)m(ger)i (class.)409 1903 y FK(Pr)o(oof.)i FI(W)-6 b(e)23 b(consider)f(the)f (comm)n(utativ)n(e)g(diagram)322 2072 y FD(E)t(xt)p FI(\()p FD(H)518 2082 y Fz(1)543 2072 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\);)11 b Fy(Z)-10 b FI(\))946 2031 y FF(f)875 2072 y FE(\000)-19 b(\000)k(\000)-21 b(!)47 b FD(H)1150 2048 y Fz(2)1177 2072 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))1481 2031 y FF(g)1409 2072 y FE(\000)-20 b(\000)-14 b(\000)-21 b(!)47 b FD(H)5 b(om)p FI(\()p FD(H)1855 2082 y Fz(2)1880 2072 y FI(\()p FD(X)g FI(;)11 b Fy(Z)-5 b FI(\))o(;)11 b Fy(Z)-9 b FI(\))42 b FE(\000)-20 b(\000)-14 b(\000)-21 b(!)54 b FI(0)530 2190 y FF(r)553 2120 y Fu(?)553 2160 y(?)553 2201 y(y)1181 2190 y FF(r)1204 2120 y Fu(?)1204 2160 y(?)1204 2201 y(y)1850 2190 y FF(r)1873 2120 y Fu(?)1873 2160 y(?)1873 2201 y(y)308 2339 y FD(E)t(xt)p FI(\()p FD(H)504 2349 y Fz(1)530 2339 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))o(;)11 b Fy(Z)794 2349 y Fz(2)817 2339 y FI(\))946 2298 y FF(f)875 2339 y FE(\000)-19 b(\000)k(\000)-21 b(!)34 b FD(H)1137 2314 y Fz(2)1163 2339 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)1328 2349 y Fz(2)1350 2339 y FI(\))1481 2298 y FF(g)1409 2339 y FE(\000)-20 b(\000)-14 b(\000)-21 b(!)34 b FD(H)5 b(om)p FI(\()p FD(H)1842 2349 y Fz(2)1867 2339 y FI(\()p FD(X)g FI(;)11 b Fy(Z)-6 b FI(\);)11 b Fy(Z)2131 2349 y Fz(2)2154 2339 y FI(\))33 b FE(\000)-20 b(\000)-14 b(\000)-21 b(!)33 b FI(0)22 b FD(;)308 2480 y FI(where)29 b(the)f(v)n(ertical)h(maps)e(are)i(giv)n(en)g(b)n(y)f (reduction)g(mo)r(dulo)f(2)h(of)h(the)f(co-)308 2558 y(e\016cien)n(ts,)36 b(and)c(the)g(ro)n(ws)f(arise)i(from)e(the)h(univ) n(ersal)g(co)r(e\016cien)n(t)h(theorem)308 2637 y(and)27 b(are)g(exact.)44 b(W)-6 b(e)28 b(need)f(to)g(pro)n(v)n(e)g(that)f (there)g(is)i(a)f(class)g FD(c)h FE(2)f FD(H)2118 2612 y Fz(2)2145 2637 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Fy(Z)-6 b FI(\).)24 b(W)-6 b(e)17 b(claim)g(that)f(this)h(is)h(the)e(case)i(for)f FD(w)r FI(.)28 b(Indeed,)18 b(if)f FD(x)i FE(2)f FD(H)2118 2961 y Fz(2)2145 2951 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))308 3030 y(is)23 b(a)f(torsion)f(class,)i(then)f FD(x)14 b FE(\001)h FD(x)j FI(=)h(0,)j(hence)h FD(w)r FI(\()p FD(x)p FI(\))17 b FE(\021)i FD(x)c FE(\001)g FD(x)j FE(\021)h FI(0)45 b(mo)r(d)21 b(2.)409 3109 y(Th)n(us)28 b(w)n(e)h(can)f(\014nd)g(an)g(elemen)n(t)g FD(\013)i FE(2)g FD(H)1508 3084 y Fz(2)1534 3109 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-5 b FI(\))24 b(suc)n(h)k(that)g FD(\014)33 b FI(=)d FD(r)r FI(\()p FD(\013)p FI(\))308 3187 y(has)j(the)f(same)h (image)f(under)g FD(g)j FI(as)e FD(w)1343 3197 y Fz(2)1369 3187 y FI(.)63 b(By)32 b(exactness,)k(there)d(is)g(a)g FD(\015)41 b FE(2)308 3266 y FD(E)t(xt)p FI(\()p FD(H)504 3276 y Fz(1)530 3266 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\))o(;)11 b Fy(Z)794 3276 y Fz(2)817 3266 y FI(\))20 b(with)h FD(f)7 b FI(\()p FD(\015)t FI(\))18 b(=)h FD(\014)e FE(\000)c FD(w)1397 3276 y Fz(2)1424 3266 y FI(,)22 b(and)e(since)i (the)f(restriction)g FD(r)h FI(is)g(a)308 3345 y(surjection)16 b(on)g FD(E)t(xt)p FI(\()p FD(H)887 3355 y Fz(1)912 3345 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-6 b FI(\);)11 b Fy(Z)-10 b FI(\),)13 b(w)n(e)j(can)g(\014nd)f FD(\016)22 b FE(2)c FD(E)t(xt)p FI(\()p FD(H)1875 3355 y Fz(1)1900 3345 y FI(\()p FD(X)5 b FI(;)11 b Fy(Z)-5 b FI(\))o(;)11 b Fy(Z)-9 b FI(\))11 b(with)308 3423 y(\()p FD(r)i FE(\016)f FD(f)7 b FI(\)\()p FD(\016)s FI(\))17 b(=)i FD(\014)d FE(\000)c FD(w)824 3433 y Fz(2)851 3423 y FI(.)29 b(Let)20 b FD(c)g FI(=)f FD(\013)12 b FE(\000)g FD(f)7 b FI(\()p FD(\016)s FI(\).)28 b(Then)20 b FD(r)r FI(\()p FD(c)p FI(\))e(=)h FD(\014)d FE(\000)c FI(\()p FD(\014)j FE(\000)d FD(w)2097 3433 y Fz(2)2124 3423 y FI(\))18 b(=)i FD(w)2286 3433 y Fz(2)2312 3423 y FI(,)308 3502 y(hence)j FD(c)f FI(is)g(an)g(in)n (tegral)g(class)h(lifting)g FD(w)1335 3512 y Fz(2)1361 3502 y FI(\()p FD(X)5 b FI(\).)787 b Fx(\003)p eop %%Page: 16 12 16 15 bop 308 159 a FC(16)84 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)1012 294 y FB(3.)37 b(Dirac)26 b(op)r(erators)409 412 y FI(Let)32 b FD(X)k FI(b)r(e)31 b(a)g(smo)r(oth)e(manifold)h(with)h(a)g(Riemannian)f (metric)g FD(g)r FI(.)57 b(W)-6 b(e)308 491 y(no)n(w)29 b(seek)h(to)f(globalize)i(the)e(righ)n(t-hand-side)f(of)35 b(\(1.3\))o(.)52 b(F)-6 b(or)30 b(that)e(w)n(e)i(will)308 569 y(need)23 b(a)f(Spin)646 541 y FF(c)670 569 y FI(-structure)f Fs(s)e FI(=)g(\()p FD(V)t(;)11 b(\015)t FI(\))23 b(for)f(the)g (cotangen)n(t)g(bundle)f FD(T)2060 545 y FA(\003)2087 569 y FD(X)27 b FI(and)22 b(a)308 648 y(connection)k(in)f FD(V)15 b FI(.)40 b(The)26 b(latter)f(is)g(used)h(to)f(replace)h(the)f (partial)g(deriv)l(ativ)n(es)308 727 y(in)d(\(1.3\))g(with)f(co)n(v)l (arian)n(t)i(deriv)l(ativ)n(es.)30 b(The)21 b(global)i(v)n(ersion)f(of) 27 b(\(1.3\))21 b(is)308 897 y(\(1.15\))516 b FD(D)1040 907 y FF(A)1097 897 y FI(=)1202 813 y FF(n)1167 833 y Fu(X)1177 975 y FF(i)p Fz(=1)1276 897 y FD(\015)t FI(\()p FD(dx)1412 869 y FF(i)1430 897 y FI(\))p FE(r)1496 922 y FF(A;)1573 904 y Fv(@)p 1552 912 66 3 v 1552 946 a(@)r(x)1600 936 y(i)1627 897 y FD(;)308 1072 y FI(where)26 b FE(r)543 1089 y FF(A)608 1072 y FI(denotes)f(the)h(co)n(v)l(arian)n(t)g(deriv)l (ativ)n(e)g(of)h(the)e(connection)h FD(A)p FI(.)42 b(This)308 1151 y(expression)22 b(can)h(easily)g(b)r(e)f(seen)g(to)g(b)r(e)g (indep)r(enden)n(t)f(of)h(the)g(co)r(ordinate)g(sys-)308 1229 y(tem)g(since)i(it)f(is)g(the)g(comp)r(osition)f(of)h(t)n(w)n(o)f (in)n(trinsically)i(de\014ned)e(op)r(erations.)308 1308 y(First)g(w)n(e)g(write)g(the)g(co)n(v)l(arian)n(t)g(deriv)l(ativ)n(e)g (of)h FD(A)f FI(as)g(a)g(map)929 1424 y FE(r)969 1441 y FF(A)1015 1424 y FI(:)h(\000\()p FD(V)15 b FI(\))k FE(!)g FI(\000\()p FD(T)1422 1397 y FA(\003)1448 1424 y FD(X)h FE(\012)c FD(V)f FI(\))22 b FD(:)308 1541 y FI(Second,)g(w)n(e)h(ha)n(v)n(e)f(the)f(linear)i(map)d(induced)i(from)f (Cli\013ord)g(m)n(ultiplication)993 1657 y(\000\()p FD(T)1108 1629 y FA(\003)1134 1657 y FD(X)f FE(\012)c FD(V)f FI(\))j FE(!)h FI(\000\()p FD(V)c FI(\))22 b FD(:)308 1773 y FI(Their)32 b(comp)r(osition)f(is)i(a)f(\014rst)f(order)h(di\013eren)n (tial)g(op)r(erator)f FD(D)2011 1783 y FF(A)2082 1773 y FI(.)60 b(While)308 1852 y(this)28 b(op)r(erator)g(can)g(b)r(e)g (considered)h(for)f(an)g(arbitrary)f(connection)i FD(A)g FI(in)f FD(V)15 b FI(,)308 1931 y(in)31 b(practice)h(w)n(e)f(shall)g (restrict)g(ourselv)n(es)g(to)g(so)g(called)h(Spin)1951 1902 y FF(c)1975 1931 y FI(-connections)308 2009 y(satisfying)27 b(additional)f(prop)r(erties)f(giv)n(en)i(in)g(follo)n(wing)g (subsection.)42 b(In)26 b(this)308 2088 y(case)17 b FD(D)489 2105 y FF(A)543 2088 y FI(will)f(b)r(e)g(called)g(the)g FJ(Dir)m(ac)h(op)m(er)m(ator)k FI(asso)r(ciated)16 b(to)f(the)h (connection)f FD(A)308 2167 y FI(for)k(the)h(Spin)646 2138 y FF(c)669 2167 y FI(-structure)e Fs(s)p FI(.)29 b(The)19 b(Spin)1309 2138 y FF(c)1333 2167 y FI(-structure)f(is)i (clear)g(from)e(the)h(con)n(text)308 2245 y(and)j(is)g(not)f(made)g (explicit)i(in)f(the)g(notation.)409 2383 y FB(3.1.)38 b FI(Spin)694 2355 y FF(c)717 2383 y FB(-connections.)33 b FI(Consider)14 b(again)h(an)f(arbitrary)g(orien)n(ted)g(v)n(ec-)308 2462 y(tor)29 b(bundle)g(with)f(inner)i(pro)r(duct)d FD(H)35 b FI(with)28 b(a)i(Spin)1672 2434 y FF(c)1696 2462 y FI(-structure)e Fs(s)j FI(=)g(\()p FD(V)t(;)11 b(\015)t FI(\).)308 2541 y(Giv)n(en)29 b(an)f(orthogonal)f(connection)i FD(B)i FI(in)e FD(H)5 b FI(,)30 b(there)e(are)g(t)n(w)n(o)g(natural)g (con-)308 2619 y(strain)n(ts)c(to)h(imp)r(ose)f(on)h(a)g(connection)g FD(A)g FI(in)g FD(V)15 b FI(.)38 b(W)-6 b(e)26 b(can)f(require)g(that)e (it)i(is)308 2698 y(compatible)f(with)h(the)f(Hermitian)g(metric)g(on)h FD(V)15 b FI(.)38 b(In)24 b(addition,)h(w)n(e)g(can)g(ask)308 2777 y(that)c(Cli\013ord)g(m)n(ultiplication)g(is)g(parallel)i(in)e (the)g(sense)h(that)f(for)g(all)h(sections)308 2855 y(\010)g(of)g FD(V)37 b FI(and)22 b(sections)g FD(h)g FI(of)g FD(H)27 b FI(w)n(e)22 b(ha)n(v)n(e)308 2972 y(\(1.16\))330 b FE(r)847 2988 y FF(A)886 2972 y FI(\()p FD(\015)t FI(\()p FD(h)p FI(\)\010\))17 b(=)i FD(\015)t FI(\()p FE(r)1305 2988 y FF(B)1346 2972 y FD(h)p FI(\)\010)14 b(+)h FD(\015)t FI(\()p FD(h)p FI(\))p FE(r)1706 2988 y FF(A)1744 2972 y FI(\010)22 b FD(:)308 3088 y FI(Th)n(us,)f(if)h FD(h)f FI(is)h(parallel,)h(then)d(Cli\013ord)h(m)n(ultiplication)g(b)n(y)g FD(h)g FI(comm)n(utes)e(with)308 3167 y FE(r)348 3184 y FF(A)387 3167 y FI(.)409 3264 y FK(Definition)g FI(1.9)p FK(.)j FI(A)16 b(Hermitian)e(connection)h FD(A)g FI(satisfying)f (equation)h(\(1.16\))308 3343 y(is)23 b(called)g(a)f(Spin)745 3314 y FF(c)769 3343 y FI(-)p FJ(c)m(onne)m(ction)p FI(.)409 3440 y(The)29 b(basic)g(issues)g(to)f(address)g(are)g(existence)i(and)e (uniqueness)g(of)h(suc)n(h)308 3519 y(connections.)p eop %%Page: 17 13 17 16 bop 1026 159 a FC(3.)24 b(DIRA)o(C)d(OPERA)l(TORS)667 b(17)409 294 y FK(Lemma)19 b FI(1.10)p FK(.)j FJ(F)-5 b(or)16 b(every)i FI(Spin)1226 266 y FF(c)1250 294 y FJ(-structur)m(e)f Fs(s)g FJ(ther)m(e)f(ar)m(e)g FI(Spin)1983 266 y FF(c)2006 294 y FJ(-c)m(onne)m(ctions)308 373 y(on)23 b FD(V)16 b FJ(.)30 b(Given)24 b(any)f(two)g(such)h(c)m(onne)m(ctions)f FD(A)1501 383 y Fz(1)1552 373 y FJ(and)f FD(A)1728 383 y Fz(2)1779 373 y FJ(ther)m(e)h(is)g(a)g(one-form)308 451 y FD(\022)i FJ(so)f(that)985 537 y FE(r)1025 554 y FF(A)1061 560 y Fw(2)1103 537 y FE(\000)15 b(r)1210 554 y FF(A)1246 560 y Fw(1)1291 537 y FI(=)k FD(i\022)d FE(\012)g FI(Id)1559 547 y FF(V)1635 537 y FD(:)409 658 y FK(Pr)o(oof.)27 b FI(T)-6 b(o)27 b(pro)n(v)n(e)g(existence)h(w)n(e)f (can)g(argue)g(lo)r(cally)-6 b(,)30 b(and)d(then)f(patc)n(h)308 737 y(together)c(using)f(a)h(partition)f(of)i(unit)n(y)-6 b(.)409 816 y(First)22 b(notice)g(that)e(w)n(e)i(can)f(lo)r(cally)i (giv)n(e)f FD(H)k FI(a)c(\015at)f(connection)g FD(\034)29 b FI(induced)308 894 y(from)d(an)g(orien)n(ted)g(orthogonal)g (trivialization.)43 b(W)-6 b(e)27 b(then)f(c)n(ho)r(ose)h(a)g(trivial-) 308 973 y(ization)d(for)f FD(V)38 b FI(so)23 b(that)f(Cli\013ord)h(m)n (ultiplication)f(b)n(y)h(a)g(parallel)h(section)f(with)308 1052 y(resp)r(ect)g(to)g FD(\034)30 b FI(is)24 b(giv)n(en)f(b)n(y)g(a)g (constan)n(t)g(matrix.)31 b(Then)23 b(the)g(pro)r(duct)e(connec-)308 1130 y(tion,)j FD(A)513 1140 y Fz(0)540 1130 y FI(,)g(with)f(resp)r (ect)g(to)g(this)g(trivialization)h(of)g FD(V)38 b FI(is)24 b(compatible)e(with)h FD(\034)8 b FI(.)308 1209 y(Let)22 b(\002)d(=)g FE(r)606 1226 y FF(B)661 1209 y FE(\000)c(r)768 1225 y FF(\034)797 1209 y FI(.)30 b(W)-6 b(e)22 b(seek)g(a)f(sk)n (ew-Hermitian)g(End\()p FD(V)14 b FI(\)-v)l(alued)22 b(one-form)317 1276 y(~)308 1293 y(\002)h(so)g(that)f FD(A)657 1303 y Fz(0)700 1293 y FI(+)776 1276 y(~)767 1293 y(\002)h(is)g(compatible)f(with)g FE(r)1437 1310 y FF(B)1501 1293 y FI(in)h(the)f(sense)i(of)j(\(1.16\).)32 b(Ev)n(ery)308 1378 y(suc)n(h)466 1361 y(~)457 1378 y(\002)23 b(giv)n(es)f(us)g(a)g(lo)r(cal)h(Spin)1116 1349 y FF(c)1140 1378 y FI(-connection.)409 1456 y(T)-6 b(aking)15 b(the)f(di\013erence) h(of)g(the)g(corresp)r(onding)e(v)n(ersions)i(of)g(equation)f (\(1.16\),)308 1535 y(this)22 b(implies)g(that)308 1650 y(\(1.17\))929 1633 y(~)920 1650 y(\002)q FD(\015)t FI(\()p FD(h)p FI(\))14 b FE(\000)h FD(\015)t FI(\()p FD(h)p FI(\))1318 1633 y(~)1309 1650 y(\002)j(=)i FD(\015)t FI(\(\002\()p FD(h)p FI(\)\))g FD(:)409 1763 y FI(The)i(isomorphism)f FD(\015)11 b FI(:)23 b(\003)1064 1739 y FA(\003)1090 1763 y FI(\()p FD(H)5 b FI(\))15 b FE(\012)1269 1773 y Ft(R)1320 1763 y Fy(C)36 b FE(!)20 b FI(End)o(\()p FD(V)15 b FI(\))22 b(mak)n(es)f(End\()p FD(V)14 b FI(\))22 b(in)n(to)h(a)308 1842 y Fy(Z)-6 b FI(-graded)18 b(v)n(ector)k(space.) 30 b(Giv)n(en)22 b FD(M)j FE(2)19 b FI(End)o(\()p FD(V)c FI(\))22 b(w)n(e)g(write)1139 2010 y FD(M)j FI(=)1333 1925 y FF(n)1299 1945 y Fu(X)1303 2087 y FF(\027)s Fz(=0)1407 2010 y FD(M)1471 2020 y FF(\027)308 2181 y FI(for)15 b(the)f(decomp)r(osition)g(according)g(to)h(the)f(grading.)27 b(Notice)16 b(that)d(if)i FD(h)2093 2191 y Fz(1)2120 2181 y FD(;)c(:)g(:)g(:)i(;)e(h)2305 2191 y FF(n)308 2259 y FI(is)23 b(an)e(orthonormal)f(basis)i(for)g FD(H)5 b FI(,)22 b(then)692 2339 y FF(n)658 2359 y Fu(X)668 2501 y FF(i)p Fz(=1)766 2423 y FD(\015)t FI(\()p FD(h)868 2433 y FF(i)887 2423 y FI(\))p FD(M)7 b(\015)t FI(\()p FD(h)1086 2433 y FF(i)1104 2423 y FI(\))19 b(=)1253 2339 y FF(n)1219 2359 y Fu(X)1224 2501 y FF(\027)s Fz(=0)1316 2369 y Fu(\000)1347 2423 y FI(\()p FE(\000)p FI(1\))1484 2395 y FF(\027)s Fz(+1)1574 2423 y FI(\()p FD(n)c FE(\000)g FI(2)p FD(\027)t FI(\))1817 2369 y Fu(\001)1848 2423 y FD(M)1912 2433 y FF(\027)1963 2423 y FD(:)308 2599 y FI(Th)n(us)22 b(equation)f(\(1.17\))h(implies)627 2682 y FF(n)593 2702 y Fu(X)603 2844 y FF(i)p Fz(=1)702 2766 y FD(\015)t FI(\()p FD(h)804 2776 y FF(i)822 2766 y FI(\))p FD(\015)t FI(\(\002\()p FD(h)1027 2776 y FF(i)1046 2766 y FI(\)\))c(=)1220 2682 y FF(n)1186 2702 y Fu(X)1196 2844 y FF(i)p Fz(=1)1283 2766 y FI(\()p FD(\015)t FI(\()p FD(h)1411 2776 y FF(i)1429 2766 y FI(\))1464 2749 y(~)1455 2766 y(\002)p FD(\015)t FI(\()p FD(h)1608 2776 y FF(i)1627 2766 y FI(\))d(+)1743 2749 y(~)1734 2766 y(\002\))1116 2987 y(=)1220 2902 y FF(n)1186 2923 y Fu(X)1191 3064 y FF(\027)s Fz(=0)1283 2987 y FI(\()p FD(n)g FI(+)g(\()p FE(\000)p FI(1\))1566 2959 y FF(\027)s Fz(+1)1656 2987 y FI(\()p FD(n)g FE(\000)h FI(2)p FD(\027)t FI(\)\))1935 2970 y(~)1926 2987 y(\002)1977 2997 y FF(\027)2028 2987 y FD(:)308 3165 y FI(The)22 b(left-hand)g(side)g(of)g(the)g(equation)f (lies)i(in)f FD(\015)t FI(\(\003)1616 3140 y Fz(2)1642 3165 y FI(\))g(hence)964 3315 y(~)955 3332 y(\002)1006 3342 y Fz(2)1052 3332 y FI(=)1129 3286 y(1)p 1129 3317 34 3 v 1129 3378 a(4)1214 3248 y FF(n)1180 3268 y Fu(X)1190 3410 y FF(i)p Fz(=1)1288 3332 y FD(\015)t FI(\()p FD(h)1390 3342 y FF(i)1409 3332 y FI(\))p FD(\015)t FI(\(\002\()p FD(h)1614 3342 y FF(i)1632 3332 y FI(\)\))308 3519 y(while)490 3502 y(~)481 3519 y(\002)532 3529 y FF(\027)580 3519 y FI(=)d(0)j(for)g FD(\027)g FE(6)p FI(=)e(0)p FD(;)11 b FI(2)p FD(;)g(n)p FI(.)p eop %%Page: 18 14 18 17 bop 308 159 a FC(18)84 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)409 300 y FI(Th)n(us,)g(the)f (equation)g(for)1091 283 y(~)1081 300 y(\002)h(has)f(solutions.)32 b(The)22 b(most)f(general)i(solution)308 378 y(is)308 541 y(\(1.18\))842 524 y(~)833 541 y(\002)c(=)g FD(i\022)13 b FI(Id)1100 551 y FF(V)1153 541 y FI(+)1211 496 y(1)p 1211 526 34 3 v 1211 587 a(4)1296 457 y FF(n)1262 477 y Fu(X)1272 619 y FF(i)p Fz(=1)1370 541 y FD(\015)t FI(\()p FD(h)1472 551 y FF(i)1491 541 y FI(\))p FD(\015)t FI(\(\002\()p FD(h)1696 551 y FF(i)1714 541 y FI(\)\))22 b FD(;)308 725 y FI(where)g FD(\022)i FI(is)e(an)g(arbitrary)e(real)j(one-form)d (on)i FD(X)5 b FI(.)409 806 y(Ha)n(ving)22 b(found)f(the)g(most)g (general)1330 789 y(~)1320 806 y(\002)h(also)g(settles)g(the)f (\(non-\)uniqueness)308 884 y(question.)1711 b Fx(\003)409 1020 y FI(3.1.1.)34 b FJ(Conjugation)19 b(and)e(Spin)h(c)m(onne)m (ctions.)33 b FI(Supp)r(ose)15 b(w)n(e)h(ha)n(v)n(e)g(a)g(Spin)2285 992 y FF(c)2309 1020 y FI(-)308 1099 y(structure)22 b Fs(s)g FI(whic)n(h)h(comes)f(from)g(a)g(spin)h(structure,)f(so)g(that)g (a)h Fy(C)12 b FI(-)q(an)n(ti-linear)308 1178 y(isomorphism)20 b FD(J)14 b FI(:)23 b FD(V)34 b FE(!)19 b FD(V)37 b FI(is)23 b(\014xed.)409 1280 y FK(Definition)39 b FI(1.11)p FK(.)33 b FI(A)g FJ(spin)h(c)m(onne)m(ction)k FI(is)33 b(a)g(Spin)1823 1251 y FF(c)1847 1280 y FI(-connection)g(that)308 1358 y(comm)n(utes)21 b(with)g FD(J)6 b FI(.)409 1460 y(Unlik)n(e)23 b(Spin)744 1432 y FF(c)768 1460 y FI(-connections,)f(spin)f (connections)h(are)f(unique.)29 b(The)22 b(analog)308 1539 y(of)g(Lemma)f(1.10)h(is)h(simply:)409 1641 y FK(Lemma)29 b FI(1.12)p FK(.)f FJ(F)-5 b(or)25 b(every)h(spin)g(structur)m(e)f (ther)m(e)h(is)f(a)g(unique)i(spin)e(c)m(on-)308 1720 y(ne)m(ction)f(on)g FD(V)15 b FJ(.)409 1856 y FK(Pr)o(oof.)27 b FI(By)34 b(Lemma)g(1.10,)k FD(V)50 b FI(admits)33 b(Spin)1654 1827 y FF(c)1677 1856 y FI(-connections)i FD(A)p FI(.)68 b(But,)308 1934 y(giv)n(en)29 b FD(J)6 b FI(,)32 b(w)n(e)c(can)h(c)n (ho)r(ose)g(the)f FD(A)1195 1944 y Fz(0)1251 1934 y FI(in)h(the)f(pro)r (of)g(of)h(Lemma)e(1.10)i(to)f(com-)308 2015 y(m)n(ute)23 b(with)h FD(J)6 b FI(.)37 b(Then)911 1998 y(~)902 2015 y(\002)953 2025 y Fz(2)1004 2015 y FI(also)25 b(comm)n(utes)d(with)i FD(J)6 b FI(.)37 b(Ho)n(w)n(ev)n(er,)1997 1998 y(~)1988 2015 y(\002)2039 2025 y Fz(0)2089 2015 y FI(=)23 b FD(i\022)12 b FI(Id)2289 2025 y FF(V)308 2093 y FI(only)22 b(comm)n(utes)e(with)i FD(J)28 b FI(if)23 b FD(\022)d FI(=)f(0.)1067 b Fx(\003)409 2233 y FI(A)32 b(Spin)618 2205 y FF(c)641 2233 y FI(-connection)f FD(A)g FI(in)g FD(V)46 b FI(induces)30 b(a)h(connection)1915 2216 y(^)1898 2233 y FD(A)g FI(in)g(the)f(c)n(har-)308 2312 y(acteristic)i(line)f(bundle)f FD(L)1003 2322 y Fr(s)1026 2312 y FI(.)55 b(The)31 b(trivialization)g(of)f FD(V)46 b FI(from)29 b(the)i(pro)r(of)f(of)308 2390 y(Lemma)18 b(1.10)i(induces)f(a)h(trivialization)g(of)f FD(L)1468 2400 y Fr(s)1511 2390 y FI(in)g(whic)n(h)g(the)g(connection)h(one-)308 2470 y(form)h(of)557 2453 y(^)539 2470 y FD(A)i FI(is)f(2)p FD(i\022)r FI(.)409 2627 y FB(3.2.)38 b(The)e(square)g(of)h(a)f(Dirac)g (op)r(erator.)d FI(W)-6 b(e)33 b(will)f(compare)f(the)308 2705 y(square)22 b(of)h(the)f(Dirac)h(op)r(erator)f(with)f(the)i(co)n (v)l(arian)n(t)f(Laplacian,)i FE(r)2078 2681 y FA(\003)2062 2722 y FF(A)2105 2705 y FE(r)2145 2722 y FF(A)2183 2705 y FI(,)f(and)308 2792 y(the)h(curv)l(ature)f(of)815 2775 y(^)798 2792 y FD(A)p FI(.)35 b(The)23 b(main)g(result)h(is)g(the)f(so) h(called)h(W)-6 b(eitzen)n(b\177)-33 b(oc)n(k)25 b(for-)308 2870 y(m)n(ula)c(for)h(the)g(Dirac)h(op)r(erator.)409 2972 y FK(Theorem)18 b FI(1.13)p FK(.)23 b FJ(L)m(et)18 b Fs(s)f FJ(b)m(e)h(a)g FI(Spin)1303 2944 y FF(c)1327 2972 y FJ(-structur)m(e)f(on)h(an)f(oriente)m(d)g(R)n(iemann-)308 3051 y(ian)31 b(manifold)f FD(X)5 b FJ(,)32 b(and)e FD(A)h FJ(a)g FI(Spin)1230 3023 y FF(c)1253 3051 y FJ(-c)m(onne)m(ction.)51 b(Then)31 b(the)g(squar)m(e)f(of)h(the)308 3130 y(asso)m(ciate)m(d)23 b(Dir)m(ac)g(op)m(er)m(ator)e(satis\014es)j(the)g(identity)308 3284 y FI(\(1.19\))411 b FD(D)945 3256 y Fz(2)935 3301 y FF(A)993 3284 y FI(=)19 b FE(r)1119 3256 y FA(\003)1103 3301 y FF(A)1145 3284 y FE(r)1185 3301 y FF(A)1239 3284 y FI(+)1312 3239 y(1)p 1312 3269 V 1312 3330 a(4)1352 3284 y FD(s)1383 3294 y FF(g)1425 3284 y FI(+)1498 3239 y(1)p 1498 3269 V 1498 3330 a(2)1538 3284 y FD(\015)t FI(\()p FD(F)1657 3292 y Fz(^)1645 3304 y FF(A)1683 3284 y FI(\))k FD(:)409 3440 y FK(Pr)o(oof.)k FI(In)j(the)g(ab)r(o)n(v)n(e)h (form)n(ula)f FD(F)1385 3448 y Fz(^)1373 3460 y FF(A)1441 3440 y FI(is)h(an)g(imaginary)e(t)n(w)n(o-form)g(and)308 3519 y FD(\015)t FI(\()p FD(F)427 3527 y Fz(^)415 3538 y FF(A)453 3519 y FI(\))22 b(denotes)g(the)f(the)h(extension)g(of)g (Cli\013ord)f(m)n(ultiplication)h(to)f(forms.)p eop %%Page: 19 15 19 18 bop 1026 159 a FC(3.)24 b(DIRA)o(C)d(OPERA)l(TORS)667 b(19)409 294 y FI(W)-6 b(e)21 b(will)g(do)f(the)g(calculation)h(in)f(t) n(w)n(o)g(steps.)29 b(First)20 b(recall)h(that)e(the)h(curv)l(a-)308 373 y(ture)g(of)g FD(A)g FI(is)h(a)f(t)n(w)n(o-form)e(with)h(v)l(alues) i(in)f(End)o(\()p FD(V)15 b FI(\))20 b(giv)n(en)g(b)n(y)g(the)f (expression:)807 527 y FD(F)850 544 y FF(A)888 527 y FI(\()p FD(X)r(;)11 b(Y)k FI(\))j(=)h([)p FE(r)1225 544 y FF(A;X)1321 527 y FD(;)11 b FE(r)1390 544 y FF(A;Y)1481 527 y FI(])k FE(\000)g(r)1621 544 y FF(A;)p Fz([)p FF(X)q(;Y)d Fz(])1814 527 y FD(:)308 684 y FI(Lo)r(cally)992 861 y FD(F)1035 878 y FF(A)1092 861 y FI(=)1169 815 y(1)p 1169 846 34 3 v 1169 907 a(2)1261 777 y FF(n)1227 797 y Fu(X)1220 939 y FF(i;j)s Fz(=1)1343 861 y FD(F)1386 878 y FF(ij)1427 861 y FD(e)1458 871 y FF(i)1492 861 y FE(^)j FD(e)1583 871 y FF(j)1629 861 y FD(;)308 1073 y FI(where)25 b FE(f)p FD(e)567 1083 y Fz(1)593 1073 y FD(;)11 b(:)g(:)g(:)i(;)e(e)771 1083 y FF(n)803 1073 y FE(g)25 b FI(is)g(a)g(lo)r(cal)h(orthonormal)d(frame)h(for)g FD(T)1868 1048 y FA(\003)1895 1073 y FD(X)5 b FI(,)26 b(and)e(the)h FD(F)2290 1090 y FF(ij)308 1154 y FI(are)d(lo)r(cal)h (sections)g(of)f(End)o(\()p FD(V)15 b FI(\))22 b(with)f FD(F)1334 1171 y FF(ij)1393 1154 y FI(=)e FE(\000)p FD(F)1558 1171 y FF(j)s(i)1599 1154 y FI(.)409 1236 y(Using)k(Cli\013ord)f(m)n (ultiplication)g(w)n(e)g(de\014ne)g(an)g(endomorphism)e FD(\015)t FI(\()p FD(F)2192 1253 y FF(A)2230 1236 y FI(\))i(of)308 1315 y FD(V)37 b FI(as)22 b(the)g(image)g(of)g FD(F)887 1332 y FF(A)947 1315 y FI(under)f(the)h(comp)r(osition)589 1492 y(\003)635 1464 y Fz(2)677 1492 y FE(\012)15 b FI(End)o(\()p FD(V)g FI(\))986 1451 y FF(\015)s FA(\012)p Fz(Id)985 1492 y FE(\000)-10 b(!)19 b FI(End)o(\()p FD(V)c FI(\))g FE(\012)g FI(End)o(\()p FD(V)g FI(\))j FE(\000)-10 b(!)19 b FI(End)o(\()p FD(V)c FI(\))22 b FD(:)308 1647 y FI(The)32 b(second)g(map)f(is)i(giv)n(en)f(b)n(y)g(comp)r(osition)f(of)h (endomorphisms.)57 b(Th)n(us)308 1725 y(lo)r(cally)23 b(w)n(e)f(ha)n(v)n(e:)785 1909 y FD(\015)t FI(\()p FD(F)892 1926 y FF(A)930 1909 y FI(\)\(\010\))c(=)1151 1863 y(1)p 1151 1894 V 1151 1955 a(2)1201 1845 y Fu(X)1224 1987 y FF(i;j)1310 1909 y FD(\015)t FI(\()p FD(e)1405 1919 y FF(i)1438 1909 y FE(^)d FD(e)1529 1919 y FF(j)1554 1909 y FI(\)\()p FD(F)1649 1926 y FF(ij)1689 1909 y FI(\(\010\)\))20 b FD(:)409 2131 y FI(Using)j(the)f(metrics)f(on)h FD(V)37 b FI(and)21 b FD(T)1283 2106 y FA(\003)1310 2131 y FD(X)5 b FI(,)22 b(the)g(co)n(v)l(arian)n(t)g(deriv)l(ativ)n(e)929 2285 y FE(r)969 2302 y FF(A)1015 2285 y FI(:)h(\000\()p FD(V)15 b FI(\))j FE(\000)-11 b(!)20 b FI(\000\()p FD(V)30 b FE(\012)15 b FD(T)1598 2258 y FA(\003)1625 2285 y FD(X)5 b FI(\))308 2440 y(has)22 b(a)g(formal)f(adjoin)n(t)927 2595 y FE(r)983 2567 y FA(\003)967 2611 y FF(A)1017 2595 y FI(:)i(\000\()p FD(V)30 b FE(\012)16 b FD(T)1309 2567 y FA(\003)1335 2595 y FD(X)5 b FI(\))19 b FE(!)g FI(\000\()p FD(V)c FI(\))22 b FD(:)308 2749 y FI(Here)j(is)f(the)g(W)-6 b(eitzen)n(b\177)-33 b(oc)n(k)25 b(decomp)r(osition)e(of)h FD(D)1606 2725 y Fz(2)1604 2766 y FF(A)1642 2749 y FI(,)h(expressing)f (it)g(as)g(a)g(sum)308 2828 y(of)e(the)g(connection)g(Laplacian)h(and)e (a)h(curv)l(ature)f(term.)409 2999 y FK(Lemma)27 b FI(1.14)p FK(.)308 3170 y FI(\(1.20\))433 b FD(D)967 3142 y Fz(2)957 3187 y FF(A)995 3170 y FI(\010)19 b(=)g FE(r)1188 3142 y FA(\003)1172 3187 y FF(A)1215 3170 y FE(r)1255 3187 y FF(A)1293 3170 y FI(\010)c(+)1429 3124 y(1)p 1429 3154 V 1429 3216 a(2)1469 3170 y FD(\015)t FI(\()p FD(F)1576 3187 y FF(A)1614 3170 y FI(\)\010)23 b FD(:)409 3362 y FK(Pr)o(oof.)k FI(Fix)35 b(a)f(p)r(oin)n(t)g FD(x)40 b FE(2)g FD(X)f FI(and)34 b(c)n(ho)r(ose)h(an)f(orthonormal)f(frame)308 3440 y FE(f)p FD(e)373 3450 y Fz(1)399 3440 y FD(;)11 b(:)g(:)g(:)j(;)d(e)578 3450 y FF(n)610 3440 y FE(g)27 b FI(for)h FD(T)826 3416 y FA(\003)853 3440 y FD(X)33 b FI(with)27 b(\()p FE(r)1179 3416 y Fz(LC)1246 3440 y FD(e)1277 3450 y FF(i)1296 3440 y FI(\))p FE(j)1341 3450 y FF(x)1399 3440 y FI(=)i(0.)48 b(W)-6 b(riting)28 b FE(r)1866 3457 y FF(i)1913 3440 y FI(for)g(the)g(co)n(v)l(ari-)308 3519 y(an)n(t)f(deriv)l(ativ)n(e)h(with)f(resp)r(ect)g(to)g FD(A)h FI(in)f(the)g FD(e)1519 3529 y FF(i)1538 3519 y FI(-direction,)i(the)e(de\014nition)g(of)p eop %%Page: 20 16 20 19 bop 308 159 a FC(20)84 b(1.)23 b(F)o(OUR-DIMENSIONAL)g(LINEAR)g (ALGEBRA)g(AND)f(BUNDLE)h(THEOR)l(Y)308 294 y FI(the)f(Dirac)h(op)r (erator)e(giv)n(es:)429 423 y(\()p FD(D)512 395 y Fz(2)510 439 y FF(A)548 423 y FI(\010\))p FE(j)641 433 y FF(x)689 423 y FI(=)740 368 y Fu(\000)771 358 y(X)794 501 y FF(i;j)880 423 y FD(\015)t FI(\()p FD(e)975 433 y FF(i)993 423 y FI(\)\()p FE(r)1085 439 y FF(i)1104 423 y FI(\()p FD(\015)t FI(\()p FD(e)1225 433 y FF(j)1248 423 y FI(\)\()p FE(r)1340 439 y FF(j)1365 423 y FI(\010\)\)\))1491 368 y Fu(\001)1520 423 y FE(j)1539 433 y FF(x)689 612 y FI(=)740 557 y Fu(\000)771 548 y(X)812 690 y FF(i)869 612 y FI(\()p FE(\000r)987 629 y FF(i)1006 612 y FE(r)1046 629 y FF(i)1065 612 y FI(\)\(\010\))14 b(+)1271 548 y Fu(X)1282 690 y FF(i