%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: crrev.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips crrev %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.04.21:1807 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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 0]N/nn 0 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257 a Ff(Example)37 b(2)49 b(On)36 b Fi(R)644 269 y Fc(1)717 257 y Fk(=)f Fi(D)888 227 y Fc(1)948 257 y Fj(\002)23 b Fi(D)1107 227 y Fc(2)1168 257 y Fj(\002)g Fi(S)1312 227 y Fc(1)1385 257 y Ff(we)37 b(use)f(the)h(c)l(o)l(or)l (dinates)g Fi(x)g Ff(on)g Fi(D)2532 227 y Fc(1)2569 257 y Ff(,)g Fi(y)2672 269 y Fc(1)2709 257 y Fi(;)14 b(y)2787 269 y Fc(2)2860 257 y Ff(on)37 b Fi(D)3057 227 y Fc(2)3131 257 y Ff(and)g Fi(t)f Ff(on)h Fi(S)3547 227 y Fc(1)3584 257 y Ff(.)g(The)0 357 y(distribution)30 b(sp)l(anne)l(d)g(by)939 529 y Fi(W)35 b Fk(=)1164 473 y Fi(@)p 1149 510 79 4 v 1149 586 a(@)5 b(t)1257 529 y Fk(+)1350 473 y Fi(y)1391 485 y Fc(1)p 1350 510 78 4 v 1368 586 a Fk(2)1500 473 y Fi(@)p 1461 510 127 4 v 1461 586 a(@)g(y)1551 598 y Fc(1)1616 529 y Fk(+)18 b Fi(y)1740 541 y Fc(2)1840 473 y Fi(@)p 1801 510 V 1801 586 a(@)5 b(y)1891 598 y Fc(2)1956 529 y Fj(\000)2049 473 y Fi(x)p 2049 510 48 4 v 2052 586 a Fk(2)2154 473 y Fi(@)p 2130 510 97 4 v 2130 586 a(@)g(x)953 748 y(X)30 b Fk(=)22 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(\014x)g(a)g(round)f(handle)h(decomp)r(osition)g(b)n(y)g(Theorem)f (3.1.)g(W)-7 b(e)27 b(ma)n(y)e(assume)h(that)g(there)0 1225 y(is)k(only)f(one)h(round)f(handle)h(of)f(index)h(0)g(resp)r (ectiv)n(ely)f(3)g(and)h(that)g(the)g(round)f(handles)h(are)e(ordered)h (according)f(to)0 1325 y(their)34 b(index.)g(W)-7 b(e)34 b(shall)g(construct)f(an)h(Engel)f(structure)h(on)f Fi(M)43 b Fk(b)n(y)34 b(successiv)n(ely)e(attac)n(hing)h(round)g(handles)h(and) 0 1425 y(extending)28 b(an)f(Engel)g(structure)g(to)g(the)h(new)g (round)f(handleb)r(o)r(dy)-7 b(.)83 1524 y(Supp)r(ose)22 b(that)h(w)n(e)e(ha)n(v)n(e)g(constructed)h(an)g(orien)n(ted)f(Engel)g (structure)h(on)f(the)i(union)f Fi(M)2887 1494 y Fd(0)2933 1524 y Fj(\032)h Fi(M)30 b Fk(of)22 b(the)h(\014rst)f Fi(l)h Fk(round)0 1624 y(handles)29 b(suc)n(h)f(that)i(the)f(c)n (haracteristic)e(foliation)i(is)g(transv)n(erse)e(to)i Fi(@)5 b(M)2377 1594 y Fd(0)2428 1624 y Fk(and)29 b(the)h(induced)f (con)n(tact)g(structure)f(on)0 1724 y Fi(@)5 b(M)139 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(remains)f(to)h(ensure)f(that)h(the)g(con)n(tact)f(structure)h(on)f Fi(M)3464 3586 y Fd(0)3511 3616 y Fj([)24 b Fi(R)3653 3628 y Fe(k)3730 3616 y Fk(is)0 3716 y(o)n(v)n(ert)n(wisted)38 b(and)h(that)g(the)h(Engel)f(framing)f(extends)i(from)f Fi(M)2145 3686 y Fd(0)2194 3716 y Fj([)26 b Fi(R)2338 3728 y Fe(k)2419 3716 y Fk(to)39 b Fi(M)9 b Fk(.)39 b(The)g(pro)r(of)g (in)h([8)o(,9])f(of)g(the)h(last)0 3816 y(assertion)27 b(relies)g(on)h(the)h(fact)f(that)h(there)f(is)g(only)g(one)g(round)g (3{handle)f(and)h(it)h(requires)e(that)h(the)h(Engel)f(framing)0 3915 y(on)33 b(the)i(union)e Fi(M)588 3927 y Fc(1)659 3915 y Fk(of)h(all)f(round)h(handles)f(of)h(index)g(0)f(and)g(1)h(is)g (homotopic)f(to)g(the)h(restriction)f(of)h(an)f(auxiliary)0 4015 y(framing)h(of)h Fi(M)9 b Fk(.)34 b(It)h(is)g(p)r(ossible)g(to)f (c)n(ho)r(ose)g(the)h(mo)r(del)g(Engel)f(structures)g(on)g Fi(R)2674 4027 y Fc(0)2746 4015 y Fk(and)h Fi(R)2978 4027 y Fc(1)3050 4015 y Fk(suc)n(h)g(that)g(the)g(Engel)0 4115 y(structure)27 b(on)g Fi(M)552 4127 y Fc(1)617 4115 y Fk(satis\014es)g(this)h(condition.)0 4421 y Fl(4.)47 b(Connected)32 b(sums)d(of)j(Engel)f(manifolds)83 4641 y Fk(W)-7 b(e)34 b(\014nish)h(this)f(note)g(with)g(the)g(discussion)g (of)f(another)g(theorem)h(from)g([8)o(])g(whic)n(h)g(can)g(also)e(b)r (e)j(pro)n(v)n(ed)d(using)0 4741 y(round)27 b(handles)g(and)h(mo)r(del) g(Engel)e(structures.)h(A)h(detailed)g(exp)r(osition)f(will)h(app)r (ear)e(in)i([9].)83 4840 y(Notice)37 b(that)h(the)g(connected)f(sum)g (of)h(t)n(w)n(o)e(Engel)h(manifolds)g Fi(M)2258 4852 y Fc(1)2295 4840 y Fi(;)14 b(M)2413 4852 y Fc(2)2487 4840 y Fk(do)r(es)37 b(not)g(admit)h(an)f(Engel)f(structure)0 4940 y(since)h(the)h(Euler)f(c)n(haracteristic)e(of)j Fi(M)1305 4952 y Fc(1)1341 4940 y Fk(#)p Fi(M)1491 4952 y Fc(2)1566 4940 y Fk(is)f Fj(\000)p Fk(2.)g(This)h(can)f(b)r(e)g (corrected)g(using)g(an)g(additional)g(summand)1858 5131 y(5)p eop %%Page: 6 6 6 5 bop 0 257 a Fi(S)56 227 y Fc(2)111 257 y Fj(\002)19 b Fi(S)251 227 y 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