363Vol.36,No.3 19935ACTA MATHEMATICA SINICA May,1993 Morse*1(310027)M n bM.M Morse Morse M M MorseM MMorse Morse h-Riemann(ideal boundary)([1] ).[2]L.Siebenmann n≥6n-Siebenmann[2]L.S.Husch T.M.Price[3],G.O’Brien[4],T. W.Tucker[5]T.Chapman L.Siebenmann[6],Morse§1.h−M(C∞)n-bM.N n-i:M→N x∈M x i(x)M N i:M→N N−M⊂bN,M N M V=bN−bM M(missing boundary).1.1M n-bM.MM h-N0N1M V0=bN0−bM V1=bN1−bM.V i N i C i,i=0,1.C0=α(V0×[0,1]),*1990513129036α:V0×(−ε,1]→N0x∈V0α(x,1)=x;C1=β(V1×[0,1]),β:V1×(−ε,1]→N1x∈V1β(x,1)=x;V0=α(V0×1) V1=β(V1×1).N1−(N0−α(V0×(0,1]))N1C1⊂N1−(N0−α(V0×(0,1])).α(V0×[12,1))⊂β(V1×[0,1)).W=N1−(N0−α(V0×[0,1])),bW=α(V0×0) β(V1×1).(W,α(V0×0),β(V1×1))h-H:W×I→WH(x,t)=α(y,t),t∈[0,12],x=α(y,s)∈α(V0×[0,t]);=x,t∈[0,12],x/∈α(V0×[0,t]);=β(z,r+2(t−12)(1−r)),t∈[12,1],x=α(y,s)∈α(V0×[0,12])α(y,12)=β(z,r)∈β(V1×[0,1]);x=β(z,r)∈β(V1×[0,1])−α(V0×[0,12]),H H0=id r=H1:W→β(V1×1)β(V1×1)β(V1×1)Wα(V0×0)W F:W×I→WF(x,t)=β(y,1−2t),t∈[0,12],x=β(y,s)∈β(V1×[1−2t,1]);=x,t∈[0,12],x/∈β(V1×[1−2t,1]);=α(z,r−2(t−12)r),t∈[12,1],x=β(y,s)∈β(V1×[0,1])β(y,0)=α(z,r)∈α(V0×[0,12]);x=α(z,r)∈α(V0×[0,12])−β(V1×[0,1]).(invertible cobordism)([7]2)h-1.2n=3,4M MS.Smale h-[8].1.3M V0n≥5,M W h(π1V0)([9])3Morse291 [7,p.250].§2.MorseM,M Morse M MbM MorseM M M Morseµ(M)M MorseMorse MorseM.W.Hirsch[10]Morse MorseMorseM bM.M M=M∪{∗},∗M M{∗}MGδUrysohn([11]),f:M→[0,1]f−1(0)=bM f−1(1)=∗.ε:M→[0,1)ε(x)=12min{f(x),1−f(x)}.f Mε-{∗}→{1},g:M→[0,1].g M M g−1(0)=bM g−1(1)=∗.[12]Theorem3.1.2g M12ε-Mh:M→[0,1]h−1(0)=bM h−1(1)=∗.h2.1M bM.f:M→[0,1],(a)f(b)f M(c)f(d)f−1(0)=bM f−1(1)=∗.2.1M Morse M Morseµ(M)M Morse2.2M 2.1M=φbM=φ,µ(M)≥1.M=φbM=φM Morse2.3M k nµ(M)≥k.D1,...,D k n–S n k n–Σ0=S n Σj=Σj−1−{Int(D j)},j=1,...,k,...→H q(Σj)→H q(Σj−1)→H q(Σj−1,Σj)∂→H q−1(Σj)→...29236H q(Σj−1,Σj)≈H q(D j,bD j),j>10→H n(Σj−1,Σj)→H n−1(Σj)→H n−1(Σj−1)→0,j>1βn−1(Σj)=βn−1(Σj−1)+1,βn−1(Σ1)=0βn−1(Σ2)=1.βn−1(Σk)=k−1.MΣk,βn−1(M)=k−1.Morse M Morseβλ(M)≤Cλ,λ=0,1,...,n.µ(M)≥k.2.4M n-{p i}p.µ(M)=∞.2.5K S n Cantor M=S n−K.M n-µ(M)=∞.2.4 2.52.6M(fake cell),µ(M)>1.f M MorseM§3.Mµ(M)MMMorse3.1M MM N.f:N→[0,1]Morse f−1(0)=bMf−1(1)=bN−bM. 2.1.N fM M g:M→[0,1]g|M=f|M g(∗)=1.g M Morse MM M M Morse f.a f0<a<1.M a=f−1[0,a]L a=f−1(a).M a Mn L a M(n−1)bM a=bM L a.3.2a0<a<1f f−1[a,1)L a×[a,1).3Morse293 a<b<1b,[13]Theorem3.4f−1[a,b]L a×[a,b].b→1f−1[a,1)∼=L a×[a,1).3.3a f−1[a,1]∗L a C(∗,L a).3.1. 3.2,f−1[a,b)∼=L a×[a,b)∼=L a×[a,1)∼=f−1[a,1).M b=f−1[0,b]M=f−1[0,1)M3.1.2.4 2.5µ(M)<+∞,M M2.4M 2.5[1]Ker´e kj´a rt´o,B.,Vorlesungen uber Topologie I,Springer-Verlag,Berlin,1923.[2]Siebenmann,L.,The Obstruction to Finding a Boundary for an Open Manifold of Dimension Greater thanFive,Ph.D.Thesis,Princeton University,1965.[3]Husch,L.S.and Price,T.M.,Finding a boundary for a3-manifold,Ann.of Math.,91(1970),223—235.[4]O’Brien,G.,The missing boundary problem for smooth manifolds of dimension greater than or equal to six,Topology and its Applications,16(1983),303—324.[5]Tucker,T.W.,Non-compact3-manifolds and the missing-boundary problem,Topology,13(1974),267—273.[6]Chapman,T. A.and Siebenmann,L.,Finding a boundary for a Hilbert cube manifold,Acta Math.,137(1976),171—208.[7]Stallings,L.,On infinite processes leading to differentiability in the completment of a point,Differential andCombinatorial Topology,Princeton University Press,Princeton,1965.[8]Smale,S.,On the structure of manifolds,Amer.J.Math.,84(1962),387—399.[9]Cohen,M.M.,A Course in Simple Homotopy Theory,Springer-Verlag,New York,1970.[10]Hirsch,M.W.,On imbedding differentiable manifolds in euclidean spaces,Ann.of Math.,73(1961),566—571.[11]Dugundji,J.,Topology,Allyn and Bacon,Inc.,Boston,1966.[12]Hirsch,M.W.,Differential Topology,Springer-Verlag,New York,1976.[13]Milnor,J.,Lectures on the h-Cobordism Theorem,Princeton University Press,Princeton,1965.。
