Representation Theorem for Free Continuous Lattices

则称 d 是 X 上的度量（距离）函数，称 X 为度量（距离）空间．有时为了明确，记为 ( X , d ) ．
度量空间的子集合 E ，仍以 d 为 E 上度量构成的度量空间称为 ( X , d ) 的子空间．
例 1 对于 n 维空间Φ n 中的点 x = (x1, , xn ) 和 y = ( y1, , yn ) ，定义
利用 Zorn 引理可以证明： 任一线性空间必存在极大线性无关集合，这一集合即是 X 的 Hamel 基．换句话说，任一线性空间必存在 Hamel 基．
凸集和子空间是线性空间中时常用到的子集． X 的子集 E 称为是凸的，若 ∀x, y ∈ E ，
0 ≤ r ≤ 1 ， rx + (1 − r) y ∈ E ．对于任一集合 E ⊂ X ，记
容易验证 X 是线性空间． 今后对于有限维空间，无穷序列空间和函数空间将分别采用以上规定的线性运算．许多
在经典分析、代数、复变、实变、微分方程中遇到的空间都是线性空间。 注意：定义 1 与线性代数中关于线性空间的叙述是一致的，但是其内涵要比线性代数中
广泛得多。因为在线性代数中限定所考虑的对象为 n 数组。这一点很重要，例如在线性代数 中有一个结论：任何 n +1 个向量必线性相关。对于现在的空间，这一结论却不必成立。
实际上在Φ
n
上还可以定义其他度量，例如
d1 ( x,
y)
=
max
1≤i≤n
xi
−
yi
，此时 (Φ n , d1) 仍是度
量空间．但须注意应把 (Φ n , d1) 与 (Φ n , d ) 视为不同的度量空间．此外注意今后当说到Φ n 是
度量空间时，总意味着它带有欧氏度量.

a r X i v :0808.1209v 1 [m a t h .G T ] 8 A u g 2008ON THE PONTRYAGIN-STEENROD-WU THEOREMDu ˇs an Repov ˇs ,Mikhail Skopenkov and Fulvia SpaggiariAbstract.This paper is on the homotopy classification of maps of (n +1)-dimensional manifolds into then -dimensional sphere.For a continuous map f :M n +1→S n define the degree deg f ∈H 1(M n +1;Z )to be the class dual to f ∗[S n ],where [S n ]∈H n (S n ;Z )is the fundamental class.We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem:Theorem.Let M be a connected orientable closed smooth (n +1)-manifold,n ≥3.Then the map deg :πn (M )→H 1(M ;Z )is1-to-1(i.e.,bijective),if w 2(M )·ρ2H 2(M ;Z )=0;2-to-1(i.e.,each element α∈H 1(M ;Z )has exactly 2preimages)—otherwise.The proof is based on the Pontryagin-Thom construction and a geometric definition of the Stiefel–Whitney classes w i (M ).1.IntroductionThroughout this paper let M be a connected orientable closed smooth manifold of dimension m =n +k .Denote by L k (M )the set of k -dimensional framed links in M up to framed cobordism.By the Pontryagin-Thom construction,the set L k (M )is in 1-to-1correspondence with the set πn (M )=[M ;S n ]of continuous maps M →S n up to homotopy.The main purpose of this paper is to describe L 1(M )=πn (M )for k =1and in the ”stable range”n ≥3.The description of πn (M )was reduced in [Pon39][Ste47](see also [FoFu86;§30.3])to a calculation with Steenrod squares,which was done by Wu (cf.[FoFu86;§30.2.D]).In this paper we present a short proof of this Pontryagin-Steenrod-Wu classification theorem.There are reasons to believe that this is Pontryagin’s original proof,which he never published,because he went straight ahead to the general case –when M is an arbitrary polyhedron (cf.Theorem 1.2below and the remark after its statement).This classification is based on the notions of natural orientation on a framed link and degree of a framed link,defined as follows.Take a point x on a framed link L and let f 1,...,f n be the frame at this point.The basis e 1,...,e k of T x (L )is said to be positive ,if the basis e 1,...,e k ,f 1,...,f n of T x (M )is positive.The degree deg L of L is the homology class (with integral coefficients)of positively oriented L .So we have a mapdeg:L k (M )→H k (M ;Z ).The Hopf-Whitney theorem (1932-35)asserts that this map is bijective for k =0and surjective for k =1.2 D.REPOVˇS,M.SKOPENKOV AND F.SPAGGIARITheorem1.1.(a)Let M be a connected orientable closed smooth(n+1)-manifold,n≥3.The degree map deg:L1(M)→H1(M;Z)is1-to-1(i.e.,bijective),if w2(M)·ρ2H2(M;Z)=0;2-to-1(i.e.,each elementα∈H1(M;Z)has exactly2preimages)—otherwise.(b)Let M be a connected orientable closed smooth(n+2)-manifold,n≥3.Then an elementαlies in the image of deg:L2(M)→H2(M;Z)if and only if w2(M)·ρ2α=0.Here·is the multiplication H k(M;Z/2Z)×H k(M;Z/2Z)→Z/2Z andρ2:Z→Z/2Z is reduction modulo2.However,in the proof of Theorem1.1it is convenient to replace the cohomological Stiefel-Whitney classes by their homological duals.These classes are denoted by the same letters w i and¯w i, and their geometric definition(equivalent to other definitions)is recalled below.Then·in the above (and in all the below)formulae is to be understood as the intersection product H i(M)×H j(M)→H i+j−m(M).Notice that the conditionρ2β·w2(M)=0in case(a)of Theorem1.1cannot be replaced by w2(M)=0(e.g.,for M=R P4).Theorem1.2(Pontryagin).(a)Let M3be a connected orientable closed smooth3-manifold.Then for eachα∈H1(M3;Z)there is a1-1correspondence between the sets deg−1αand Z2d(α),where d(α) is the divisibility of the projection ofαto the free part of H1(M3;Z).(b)Let M be an orientable closed smooth4-manifold.Then an elementαlies in the image of deg:L2(M)→H2(M;Z)if and only ifα·α=0.Theorem1.2.b can be proved analogously to our proof of Theorem1.1.b below.Our methods can be used to prove Theorem1.2.a which was stated without proof in[Pon39].In fact,Theorem1.2.a was not included in[Pon39](published in English),but only in the abstract(published in Russian), without any indication of its proof.A short geometric proof of this result is published,for example,in [CRS07].2.Geometric definition of homology Stiefel-Whitney classesTake a general position system of s smooth tangent vectorfields on M.LetΣ⊂M be the set of points at which these vectorfields are not linearly independent.By transversality[DNF79;§10.3],Σis a pseudomanifold in M.The Stiefel-Whitney class w m+1−s(L)∈H s−1(M;Z/2Z)is the class of the pseudomanifoldΣ(this is thefirst obstruction to existence of a linear independent system of s tangent vectorfields on M).This definition can be easily generalized to the case when tangent vectorfields in T M are replaced by vectorfields in an arbitrary vector bundle with the base M.If L⊂M is a submanifold,then such classes for the normal bundle of L in M and for the restriction of T M to L are denoted by¯w2(L)and w2(M)|L,respectively.We will also use relative versions of these classes.For example,suppose that L⊂M is an l-submanifold with boundary and a system f of m−l−1linearly independent normal vectorfields is given on∂L.Then we can extend f to an arbitrary general position system of normal vectorfields on L.Define¯w2(L,f)∈H l−2(L;Z/2Z)to be the class of the(l−2)-pseudomanifold,on which these extended vectorfields are not linearly independent(this is thefirst obstruction to extension of f to a linear independent system on L).We will omit f from the notation,if no confusion could arise.3.Proof of Theorem1.1.bTake anyα∈H2(M;Z).Realizeαby an orientable2-submanifold L⊂M.Clearly,α∈Im deg if and only if L can be framed(for some choice of L).We may assume that L is connected.Indeed,if some disconnected L can be framed,then the submanifold,which is the connected sum of all connected components of L,can also be framed and realizes the same homological class.In this paragraph we show that L can be framed if and only if¯w2(L)=0.By the definition of ¯w2(L)this condition is necessary.In order to prove the sufficiency assume that¯w2(L)=0.Since n≥3 and dim L=2,it follows that there is an orthonormal system of vectorfields f1,...,f n−1which are normal to L.Since L2and M n+2are orientable,it follows that the normal bundle to L is orientable. Fix an orientation of this bundle.Taking a unit vectorfield f n ortogonal to f1,...,f n−1and such thatON THE PONTRYAGIN-STEENROD-WU THEOREM3 the basis f1,...,f n is positive(with respect to the specified orientation of the bundle),we obtain the required framing.Now the theorem follows from the equalities¯w2(L)=w2(M)|L=w2(M)·[L]=w2(M)·ρ2α.Here thefirst equality follows by the Wu formula of Stiefel-Whitney classes of the sum of two bundles: w2(M)|L=w2(L)+w1(L)·¯w1(L)+¯w2(L),in which w2(L)=w1(L)=0because L is an orientable 2-manifold(thefirst equality can also be proved directly).The second equality follows by the above geometric definition because L is connected(we identify H0(L;Z/2Z)∼=Z/2Z∼=H0(M;Z/2Z)).4.Proof of Theorem1.1.aTake an elementα∈H1(M;Z).Let L1,L2⊂M be a pair of framed1-submanifolds such that deg L1=deg L2.Denote by[L1],[L2]∈L1(M)their classes.Since L1and L2are homologous, it follows by general position that there is an embedded2-dimensional cobordism L⊂M×I(not framed)between them:∂L=L1⊔L2.Clearly,[L1]=[L2]if and only if for some L the framing of∂L extends to L.Assume further that L is connected.Let us show that the framing of∂L extends to that of L if and only if¯w2(L)=0.By definition of the relative Stiefel-Whitney classes this condition is necessary.Let us prove the sufficiency.Assume that¯w2(L)=0.Since n≥3and dim L=2,it follows that the orthonormal system of thefirst n−1 vectorfields of the framing of∂L extends to L.Since L and M×I are orientable,and L1,L2are naturally orientable,it follows that there is an orientation of the normal bundle of L in M×I restricted to the given orientations on L1and L2.So we can add one more unit vectorfield to the constructed ortonormal system on L to obtain a positive basis at each point of L(with respect to the specified orientation of L).So the required extension of the framing of∂L to L has been constructed.Completion of the proof of Theorem1.1.a in the case when w2(M)·ρ2H2(M,Z)=0.If ¯w2(L)=0then there is nothing to prove.Assume now that¯w2(L)=1.Here¯w2(L)∈H0(L;Z/2Z)∼= Z/2Z,because L is connected.Further we identify all groups H0(X;Z/2Z)isomorphic to Z/2Z with Z/2Z.Let us construct a new cobordism L′between L1and L2such that¯w2(L′)=0.Take an element β∈H2(M;Z)such that w2(M)·ρ2β=1.Let K be a connected orientable general position2-submanifold realizing the classβ.We may assume that K⊂M×12is a submanifold realizing the class w2(M),thefirst equality follows from geometricdefinition above.Put L′=L♯K(L∩K=∅by general position).By Claim4.1below¯w2(L′)=¯w2(L)+¯w2(K)=0,and this case of the theorem is proved.Claim4.1.Suppose that K2,L2⊂M n+2is a pair of disjoint connected orientable submanifolds and a frame of K and L is given on∂K and∂L,respectively.Then¯w2(K♯L)=¯w2(K)+¯w2(L),where the groups H0(X;Z/2Z)are identified with Z/2Z for X=K♯L,K and L.Proof of Claim4.1.Take a pair of small2-disks k⊂K and l⊂L.Let kl∼=S1×I be a narrow tube such that∂kl=∂k⊔∂l and kl is tangent to both disks k and l.Fix a trivial frame of k and l(and, consequently,of∂k and∂l).By the above geometric definition it follows easily that¯w2(K♯L)=¯w2(K−k)+¯w2(kl)+¯w2(L−l). On the other hand,one can check analogously that¯w2(K)=¯w2(K−k)+¯w2(k)and¯w2(L)=¯w2(L−l)+¯w2(l).Since¯w2(kl)=¯w2(k)=¯w2(l)=0,it follows that¯w2(K♯L)=¯w2(K)+¯w2(L). Completion of the proof of Theorem1.1.a in the case when w2(M)·ρ2H2(M,Z)=0.It suffices to show that forfixed[L1]the map[L2]→w2(L)is well-defined and is a bijection deg−1α→Z/2Z.Let us prove that the map is well-defined.Let L′1and L′2be a pair of framed submanifolds of M framed cobordant to L1and L2respectively.Let L′be a(not framed)cobordism between them.It suffices to prove that w2(L)=w2(L′)in case when L1and L′1,L2and L′2,L and L′are in general position.4 D.REPOVˇS,M.SKOPENKOV AND F.SPAGGIARIAssume that L1,L′1⊂M×1,L2,L′2⊂M×0and L,L′⊂M×[0,1].Change the sign of thefirst vectorfield belonging to the framings of L′1and L′2.Denote the obtained framed submanifolds by−L′1 and−L′2,respectively.Denote by¯w2(−L′)the relative Stiefel-Whitney class of L with the reversed framing of∂L′.Then ¯w2(−L′)=−¯w2(L′).Further,both L1⊔(−L′1)and L2⊔(−L′2)are framed cobordant to zero,i.e.to an empty submanifold.Let L+⊂M×[1,+∞)and L−⊂M×(−∞,0]be the corresponding framed cobordisms.Then¯w2(L+)=¯w2(L−)=0.By general position L∩L′=∅.Denote by K=L∪L+∪L′∪L−By the above geometric definition it follows easily that¯w2(K)=¯w2(L)+¯w2(L+)+¯w2(−L′)+¯w2(L−)=¯w2(L)−¯w2(L′).It suffices to show that¯w2(K)=0.Letβbe the cohomological class of image of K under the projection M×R→M.Analogously to the proof of the previous case of the theorem we see that ¯w2(K)=¯w2(M)·ρ2β=0,hence w2(L)=w2(L′)and our map deg−1α→Z/2Z is well-defined.Now let us prove that our map is injective.It suffices to show that if L′2is a framed1-submanifold and L′is a connected2-dimensional embedded cobordism(not framed)between L1and L′2such that ¯w2(L)=¯w2(L′),then[L2]=[L′2].Indeed,we may assume that L1⊂M×0,L2⊂M×1,L′2⊂M×(−1),L⊂M×[0,1]and L′⊂M×[−1,0].Then L∪L′is a cobordism between L2and L′2.By the above geometric definition it follows that¯w2(L∪L′)=¯w2(L)+¯w2(L′)=0,hence L∪L′can be framed.So our map deg−1α→Z/2Z is injective.Let us prove that our map is surjective.It suffices to show that some[L2]is mapped to1.Since M is orientable,it follows there exists a framing f1of L1.Fix a homeomorphism L1∼=S1.Denote by f1(x)the choice of the framing at the point x∈S1.Take a mapϕ:S1→SO(n)realizing a nonzero element ofπ1(SO(n))∼=Z/2Z(which is true because n≥3).Define a new framing f2of L1 by the formula f2(x)=ϕ(x)f1(x).The obtained framed submanifold is the required submanifold L2.Indeed,take L=L1×I.Then ¯w2(L)=1.Indeed,assume the converse.Then the frames of L1and L2can be extended to the frame of L1×I.This frame gives the homotopy betweenϕand the constant map in SO(n),which contradicts the choice ofϕ.This contradiction completes the proof of Theorem1.1.a.AcknowledgmentsWe thank A.B.Skopenkov and the referee for comments and suggestions.References[CRS07]M.Cencelj,D.Repovˇs and M.Skopenkov,Classification of framed links in3–manifolds,Proc.Indian Acad.Sci.(Math.Sci.)117:3(2007),301–306,arXiv:math-gt/0705.4166.[DNF79] B.A.Dubrovin,S.P.Novikov and A.T.Fomenko,Modern geometry:Methods and applications,Nauka, Moscow,1979.(in Russian)[FoFu89]A.T.Fomenko and D.B.Fuchs,A course in homotopy theory,Nauka,Moscow,1989.(in Russian)[Pon39]L.S.Pontryagin,Homologies in compact Lie groups,Mat.Sbornik6(1939),no.3,389–422.[Pon76]L.S.Pontryagin,Smooth manifolds and their application to homotopy theory,Nauka,Moscow,1976.(in Russian)[Ste47]N.E.Steenrod,Products of cocycles and extensions of mappings,Ann.of Math.(2)48(1947),290–320.Institute for Mathematics,Physics and Mechanics,University of Ljubljana,P.O.Box2964,1001 Ljubljana,Slovenia.e-mail:dusan.repovs@uni-lj.siDepartment of Differential Geometry,Faculty of Mechanics and Mathematics,Moscow State Uni-versity,Moscow,Russia119992.e-mail:skopenkov@rambler.ruDipartimento di Matematica,Universita‘di Modena e Reggio Emilia,via Campi213/B,41100Modena, Italy.e-mail:spaggiari.fulvia@unimo.ita r X i v :0808.1209v 1 [m a t h .G T ] 8 A u g 2008(n +1)nf :M n +1→S ndeg f ∈H 1(M ;Z )f ∗[S n ][S n ]∈H n (S n ;Z )M(n +1)n ≥3deg :πn (M )→H 1(M ;Z )w 2(M )·ρ2H 2(M ;Z )=0α∈H 1(M ;Z )w i (M )Mn +kL k (M )kML k (M )πn (M )=[M ;S n]M →S nL k (M )=πn (M )k =1n ≥3L k (M )MMxLf 1,...,f ne 1,...,e kT x (L )e 1,...,e k ,f 1,...,f n T x (M )deg LLLdeg :L k (M )→H k (M ;Z ).k =0k =1M(n +1)n ≥3deg :L 1(M )→H 1(M ;Z )w 2(M )·ρ2H 2(M ;Z )=0α∈H 1(M ;Z )M(n +2)n ≥3αdeg :L 2(M )→H 2(M ;Z )ρ2α·w 2(M )=0·H k (M ;Z 2)×H k (M ;Z 2)→Z 2w i¯w i·H i (M )×H j (M )→H i +j −m (M )ρ2β·w2(M)=0w2(M)=0M=R P4M3α∈H1(M3;Z)deg−1αZ2d(α)d(α)αH1(M3;Z)Mαdeg:L2(M)→H2(M;Z)α·α=0s MΣ⊂MΣM w n+2−s(L)∈H s−1(M;Z2)Σs MMM L⊂ML M T M L¯w i(L)w i(M)|LL⊂M×I lL1L2f∂L=L1∪L2m−l−1L¯w2(L,f)∈H l−2(L;Z2)(l−2)∂L L fα∈H2(M;Z)αLα∈Im degL LL LαL LL¯w2(L)=0¯w2(L)=0Lf1,...,f n−1n≥3dim L=2L2M n+2L Mf1,...,f n−1f nL¯w2(L)=w2(M)|L=w2(M)·[L]=w2(M)·ρ2α.Z2H0(X;Z2) Z2w2(M)|L=w2(L)+w1(L)·¯w1(L)+¯w2(L)w2(L)=w1(L)=0LLLα∈H1(M;Z)L i deg L i=αL1,L2⊂M deg L1= deg L2=α[L1][L2]L1(M)L1L2L⊂M×I,∂L=L1⊔L2.[L1]=[L2]L∂LL M L∂L L¯w2(L)=0¯w2(L)=0n−1L n≥3dim L=2 L M×I L1L2L L1L2n−1LLw2(M)·ρ2H2(M,Z)=0L1L2L¯w2(L)=0¯w2(L)=1¯w2(L)∈H0(L;Z2)∼=Z2LL′L1L2¯w2(L′)=0β∈H2(M;Z)w2(M)·ρ2β=1K⊂M×1¯w2(−L′)L′∂L′¯w2(−L′)=−¯w2(L′)L1⊔(−L′1)L2⊔(−L′2)L+⊂M×[1,+∞)L−⊂M×(−∞,0]¯w2(L+)=¯w2(L−)=0L∩L′=∅K=L∪L+∪L′∪L−¯w2(K)=¯w2(L)+¯w2(L+)+¯w2(−L′)+¯w2(L−)=¯w2(L)−¯w2(L′).w2(K)=0K M M×R→MβK¯w2(K)=¯w2(M)·ρ2β=0[L2]→w2(L)L′2L′L1L′2¯w2(L)=¯w2(L′) [L2]=[L′2]L1⊂M×0L2⊂M×1L′2⊂M×(−1)L⊂M×[0,1] L′⊂M×[−1,0]L∪L′L2L′2¯w2(L∪L′)=¯w2(L)+¯w2(L′)=0[L2]→w2(L)[L2] 1L1∼=S1Mf1L1f1(x)x∈S1ϕ:S1→SO(n)π1(SO(n))∼=Z2n≥3f2L1f2(x)=ϕ(x)f1(x)L2L1f2L=L1×I¯w2(L)=1f1f2L1×IϕS1→SO(n)ϕ。

! !
Abstract
x1 Introduction
Let be a C 1 di eomorphism of a compact Riemannian manifold M . If we can restrict to a (locally maximal) hyperbolic set then we call : ! a hyperbolic map. A familiar example is a (transitive) Anosov di eomorphism, in this case = M . More generally, an Axiom A di eomorphism restricted to a basic set in the non-wandering set is hyperbolic. We give a complete de nition in x2. Let f : ! R be Holder. It is often important to determine when f is a coboundary, i.e. f = u ? u for some Holder u : ! R. This was rst studied by Livsic L1, L2] and, in more general circumstances, has been the subject of several recent papers NP, NT, PP2, P, Sc, W1, W2]. Let m be the Gibbs measure corresponding to a Holder continuous function (see x2). In L2], Livsic showed that, for any Holder f : ! R, a measurable solution u : ! R to the coboundary equation f = u ? u a.e. has a Holder continuous version, i.e. u is necessarily equal a.e. to a Holder function u0 for which f = u0 ? u0 everywhere. It is natural to try to generalise this result by replacing R with an arbitrary connected Lie group G. More speci cally, we ask for which connected Lie groups and Holder f does the following property hold: Property (R) (Regularity). If f : ! G is Holder and u is a measurable solution to the coboundary equation

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Example. Consider the following 0-1 knapsack problem: max 8x1 + 11x2 + 6x3 + 4x4 s.t. 5x1 + 7x2 + 4x3 + 3x4 ≤ 14 x ∈ {0, 1}4 .
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Solving Integer Knapsack by B&B
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Best First Depth-First Hybrid Strategies Best Estimate
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One way to minimize overall solution time is to try to minimize the size of the search tree. We can achieve this by choosing the subproblem with the best bound (lowest lower bound if we are minimizing). A candidate node is said to be critical if its lower bound is also a lower bound of the original IP. Every critical node will be processed no matter what the search order Best first is guaranteed to examine only critical nodes, thereby minimizing the size of the search tree. Drawbacks of Best First