Ann. I. H. Poincaré – AN 30 (2013) 961–968
www.elsevier.com/locate/anihpc
Addendum to “Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric”
[Ann. I. H. Poincaré – AN 27 (3) (2010) 857–876]
Erasmo Caponio
a,∗,1, Miguel Ángel Javaloyes
b,2, Antonio Masiello
a,1aDipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy
bDepartamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain Received 27 September 2012; received in revised form 12 March 2013; accepted 12 March 2013
Available online 12 September 2013
Abstract
We give the details of the proof of equality (29) in Caponio et al. (2010)[3].
©2013 Elsevier Masson SAS. All rights reserved.
Résumé
On donne les détails de la preuve de l’équation (29) dans Caponio et al. (2010)[3].
©2013 Elsevier Masson SAS. All rights reserved.
MSC:58E05; 53C60; 53C22
Keywords:Morse theory; Critical groups; Finsler metrics
1. Introduction
In [3, Eq. (29)], we claim that the relative homology groups H∗(E˜|cX∩ ˜O∗,E˜|cX∩ ˜O∗\ {0}) and H∗(E˜c∩ O˜∗,E˜c ∩ ˜O∗ \ {0}) are isomorphic, where, we recall, X =C01([0,1], U ), U is a neighbourhood of 0∈ Rn,
DOI of original article:http://dx.doi.org/10.1016/j.anihpc.2010.01.001.
* Corresponding author.
E-mail addresses:caponio@poliba.it(E. Caponio),majava@ugr.es,ma.javaloyes@gmail.com(M.Á. Javaloyes),masiello@poliba.it (A. Masiello).
URL:http://goo.gl/6sVkT(E. Caponio).
1 Supported by M.I.U.R. research project PRIN09 “Metodi Variazionali e Topologici nello Studio di Fenomeni Nonlineari” and INDAM–
GNAMPA research project “Analisi Geometrica sulle Varietà di Lorentz ed Applicazioni alla Relatività Generale”.
2 Supported by MINECO (Ministerio de Economía y Competitividad) project MTM2012-34037, Regional J. Andalucía Grant P09-FQM-4496 and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007–2010).
0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.anihpc.2013.03.005
E:˜ H01([0,1], U )→R,E(x)˜ =1
0 G(s, x,˜ x)˙ ds, 0∈H01([0,1], U )is a non-degenerate critical point ofE,˜ c= ˜E(0), E˜c= {x∈H01([0,1], U )| ˜E(x)c}andO˜∗is a neighbourhood of 0 inH01([0,1], U ).
For this we refer to the following result by Palais[12, Theorem 16]:
Theorem 1. (See Palais[12].) LetV1 andV2 be two locally convex topological vector spaces,f be a continuous linear map fromV1onto a dense linear subspace ofV2and letObe an open subset ofV2andO˜ =f−1(O). IfV1 andV2are metrizable thenf˜=f| ˜O:O˜→Ois a homotopy equivalence.
As a consequence, ifEis a Banach space which is dense and continuously immersed in a Hilbert space H and (A, B)is a pair of open subsets ofHwithB⊂A, then the relative homology groupsH∗(A, B)andH∗(A,˜ B), where˜ A˜=A∩EandB˜=B∩E, are isomorphic.
In this addendum we would like to make clear how the above result can be applied to get H∗E˜|cX∩ ˜O∗,E˜c|X∩ ˜O∗\ {0}∼=H∗E˜c∩ ˜O∗,E˜c∩ ˜O∗\ {0}
.
Although it is not difficult to find some open subsets which are homotopically equivalent, with respect to theH1 topology, to the ones involved in the computations of the critical groups (cf., for example,[5, Ch. III, Corollary 1.2]), it is not trivial to ensure, after applying Palais’s result, that the intersections of these subsets withXcontinue to be homotopically equivalent in theC1topology.
Actually, the equality between the critical groups of a Dirichlet functional with respect to theH1andC1topology is not a novelty (cf.[5,6,10]). Anyway, there are some issues for the functional E˜ that we would like to point out.
First,E˜ is notC2with respect to theH1topology (this is a very general phenomenon for smooth, at most quadratic in the velocities Lagrangians, cf.[1, Prop. 3.2]); moreover, asG˜ is not everywhere twice differentiable,E˜ is not also twice Gateaux differentiable at anynon-G-regular˜ curve (see Definition2). Secondly, although its flow is well defined onX, the gradient ofE˜ is not of the type identity plus a compact operator, thus we cannot immediately state that it possesses theretractible propertyin[4, §III], which ensures that the deformation retracts involved in the computation of the critical groups are also continuous in X, where the Palais–Smale condition does not hold. To overcome this problem, we extend a result in [1], constructing a smooth vector field, which is a pseudo-gradient inU \ ¯B(0, r), whereU is a neighbourhood of 0 inH01([0,1], U )andB(0, r)¯ is the closure of a ball, and whose flow satisfies the retractible property.
The proof we give in the next section (without Lemma4, which becomes superfluous) also holds for anysmooth Lagrangian on[0,1] ×T M, whereM is a finite dimensional manifold, which is fiberwise strongly convex and has at most quadratic growth in each fibre. We can also consider, with minor modifications, more general boundary conditions as the curves joining two given submanifolds inM. The Lagrangian action functional will be then defined on the Hilbert manifold of the H1 curves between the two submanifolds. As we have already mentioned above, such functional is in general not C2. Assuming that at least one of the submanifolds is compact and that all the critical points are non-degenerate, we can obtain, as in[3, Theorem 9], the Morse relations for the solutions of the corresponding Lagrangian system. In this case, the number of the conjugate instants along a geodesic, counted with their multiplicity, is replaced by the number of the “focal instants” with respect to one of the two submanifold (counted with multiplicities) along a solution plus the index of a bilinear symmetric form related to the other submanifold[7].
We recall that a Morse complex for the action functional of such kind of Lagrangian, whose homology is isomorphic to the singular homology of the path space between the two submanifolds, has been obtained in[1].
2. Proof of the isomorphism between the critical groups inH1andC1 We recall that the LagrangianG:˜ [0,1] ×U×Rn→ [0,+∞)is given by
G(t, q, y)˜ =F2
ϕ(t, q),dϕ(t, q) (1, y)
,
whereF is a Finsler metric on then-dimensional smooth manifoldMandϕ:[0,1] ×U→Mis defined asϕ(t, q)= expγ0(t)Pt(q); here, exp is the exponential map with respect to any auxiliary Riemannian metrichonM,γ0is the geodesic of(M, F )in which we want to compute the critical groups,Pt:U→Tγ0(t)Mis given byPt(q1, . . . , qn)= n
i=1qiEi(t), where{Ei}i∈{1,...,n} aren-orthonormal smooth vector fields alongγ0 andU is the Euclidean ball of radiusρ/2, whereρis the minimum of the injectivity radii (with respect to the metrich) at the pointsγ (t),t∈ [0,1].
The setZwhereG˜ is not twice differentiable is defined by the equation dϕ(t, q)[1, y] =0 and then it corresponds to the subset of[0,1] ×U×Rnwhere the LagrangianG(t, q, y)˜ =F2(ϕ(t, q),dϕ(t, q)[(1, y)])vanishes. We recall also that for each(t, q)∈ [0,1] ×Uthere is only oney∈Rnsuch that dϕ(t, q)[(1, y)] =0. Indeed, dϕ(t, q)[(1, y)] =
∂tϕ(t, q)+∂qϕ(t, q)[y]and, as∂qϕ(t, q)is one-to-one,y∈Rnis the only vector such that
∂qϕ(t, q)[y] = −∂tϕ(t, q).
We recall also that theϕ defines a smooth injective mapϕ∗:H01([0,1], U )→Ωp0,q0(M),ϕ∗(x)(t)=ϕ(t, x(t )), such thatE˜=E◦ϕ∗, whereEis the energy functional ofF, i.e.E(γ )=121
0 F2(γ ,γ )˙ dt andΩp0,q0 is the Hilbert manifold of theH1curves onMbetweenp0andq0. Observe that the curve of constant value 0 is mapped byϕ∗to the geodesicγ0(hence 0 is a critical point ofE).˜
From the fact thatF2is fiberwise positively homogeneous of degree 2 andϕis a smooth map, it follows that there exists a constantc1, depending only onU, such that
G˜qq(s, q, y)c1
1+ |y|2
, G˜qy(s, q, y)c1 1+ |y|
, G˜yy(s, q, y)c1, (1) for every(s, q, y)∈ [0,1] ×U×Rn\Z, where| · |and · are, respectively, the Euclidean norm and the norm of bilinear forms onRn.
Moreover, sinceF2is fiberwise strongly convex, there exists a positive constantc2such that
G˜yy(s, q, y)[w, w]c2|w|2, (2)
for each(s, q, y)∈ [0,1] ×U×Rn\Zandw∈Rn.
Definition 2.A curvex∈H01([0,1], U )is said to beG-regular˜ if the set of pointst∈ [0,1]where(t, x(t),x(t ))˙ ∈Z is negligible.
Letα:Rn→Rbe a smooth function such thatα|U =1, α|UC =0, whereUis an open subset ofRn such that 0∈UandU¯⊂U. Consider the LagrangianL:R×Rn×Rn→R,L(t, q, y)=α(q)G(t, q, y)˜ +(1−α(q))|y|2. Clearly, by the definition ofα, 0 is also a critical point of the action functionalAL(x)=121
0L(s, x,x)˙ ds. Notice also that, likeE,˜ AL:H01([0,1],Rn)→Ris aC1functional with locally Lipschitz differential.
LetBbe a closed ball inH01([0,1],Rn), centred in 0 and containing curves that have support inU.
AsL = ˜GonR×U×Rn, we have thatAL|B= ˜E|B. SinceE˜ satisfies the Palais–Smale condition (see[2]), we also have thatAL satisfies the Palais–Smale condition inB.
Moreover, from (1) it follows thatE˜ is twice Gateaux differentiable at anyG-regular curve˜ x∈H01([0,1], U )and then the same property is satisfied byAL.
Observe that, as the endpoints of the geodesicγ0 are not conjugate, then we can assume thatB is an isolating neighbourhood of the critical point 0. Moreover, the non-conjugacy assumption implies also that 0 is a non-degenerate critical point ofE, that is, the kernel of the operator˜ A, which represents the second Gateaux differential at 0 of both E˜andAL, with respect to the scalar product·,·inH01([0,1],Rn), is empty.
The following proposition has been obtained in[1, Lemma 4.1 and formula (4.8)]for the action functional of aC2, time-dependent, fiberwise strongly convex, at most quadratic in the velocities, Lagrangian onT M.
Proposition 3.There exist a neighbourhoodUof 0inH01([0,1],Rn)(that we can assume it is contained inB)and a positive constantμ0, such that the linear vector fieldx∈U→Ax, satisfies the inequality
dAL(x)[Ax]μ0∇AL(x)2
0, (3)
for eachx∈U.
Here · 0is theH01norm. In our setting, the LagrangianL is not twice differentiable onZ⊂T Mand this leads to some differences between the proof of[1, Lemma 4.1]and ours, which we outline in Lemmata4,7and8.
Lemma 4.Letx be a smooth curve(non-necessarilyG-regular)˜ inH01([0,1], U ). Then the curvest∈ [0,1] →sx(t ) can be non-G-regular for˜ sin a subset of[0,1]which is at most countable.
Proof. We recall thatw∈H01([0,1], U ),w=w(t), is notG-regular if˜ (t, w(t),w(t))˙ ∈Z for eacht in a subset of positive Lebesgue measure in[0,1]. Now, forx:[0,1] →U, smooth andx(0)=x(1)=0, let us consider the map f:[0,1] × [0,1] →M defined asf (s, t )=ϕ(t, sx(t)). Observe that for eacht¯∈ [0,1],s→f (s,t )¯ is the affinely parametrized geodesicσt¯of the Riemannian metrichdefined byσt¯(s)=ϕ(¯t , sx(t ))¯ =expγ (¯t )(sx(¯t ))(fort¯=0 and
¯
t=1 the geodesics are constant) while, for eachs¯∈ [0,1],t→f (s, t )¯ is the curveγs¯ corresponding tosx¯ by the map ϕ∗ (for s¯=0 and s¯=1, we get respectively γ0, the geodesic of (M, F ), and the curve γ1=ϕ∗(x)). Thus f =f (s, t )defines a geodesic congruence and, then,s→Jt(s)=∂tf (s, t )= ˙γs(t)defines a Jacobi field alongσt
for eacht ∈(0,1)wherex(t)=0. Observe that at the instantst¯wherex(¯t )=0 (if they exist),σ¯t is constant and equal to γ0(¯t ). Since there is only oney ∈Rn such that(¯t ,0, y)∈Z and suchy cannot be equal to 0 (otherwise 0=dϕ(¯t ,0)[1,0] =∂tϕ(¯t ,0)+∂qϕ(¯t ,0)[0] =∂tϕ(¯t ,0)= ˙γ0(¯t )=0), there can be at most ones∈(0,1]such that (¯t , sx(¯t ), sx(¯˙ t ))=(¯t ,0, sx(¯˙ t ))∈Z. Now let us assume that fors, s∈(0,1],s=s, the curvessx andsx are not G-regular. From what we have recalled above, this is equivalent to the fact that the curves˜ γs andγs have velocity vector fields vanishing on, respectively,Zs⊂ [0,1]andZs⊂ [0,1]with|Zs|,|Zs|>0. We claim thatZs∩Zs= ∅.
Indeed, if there existst¯∈Zs∩Zs, thenx(¯t )must be different from 0 and this implies that the Jacobi fieldJt¯is well defined and equal to 0 at the instants s ands. Thus the points σt¯(s)andσt¯(s)are conjugate alongσt¯, but this is impossible (see, e.g.,[8, Prop. 2.2, p. 267]) because such geodesic has length less than the injectivity radius atγ0(¯t ).
Therefore the setZ ofs∈ [0,1]such that|Zs|>0 is at most countable. Indeed, by contradiction, assume thatZ is uncountable and consider the setAh= {s∈(0,1]: |Zs|>h1}. Since h∈NAh=Z, there must exist at least one k∈Nsuch thatAk is uncountable. Thus, for infinitely manys∈ [0,1], we would have disjoint subsetsZs ⊂ [0,1]
having measure greater than 1h, which is impossible. 2
Remark 5.From Lemma4, it also follows that any smooth non-G-regular curve˜ x∈H01([0,1], U )is the limit, in the H1topology, of some sequence(xk)⊂H01([0,1], U )of smoothG-regular curves. Indeed, it is enough to consider a˜ sequence(sn)⊂ [0,1]such thatsn→1 andsnxisG-regular.˜
Remark 6.From (2), the second Gateaux differential ofE˜ at aG-regular curve˜ xis represented by a linear bounded self-adjoint operator onH01([0,1],Rn)of the typeAx=Bx+KxwhereBxis a strictly positive definite operator and Kxis compact. Moreover from (1), if a sequence ofG-regular curves˜ {xn}converges to aG-regular curve˜ xin theH1 topology thenKxnconverges toKxin the norm topology of the bounded operators andBxnconverges strongly toBx, i.e.Bxn[ξ] →Bx[ξ]for eachξ∈H01([0,1],Rn)(cf. Claims 1 and 2 of the proof of Lemma 4.1 in[1]). We recall that from[3, Lemma 2],A≡A0is given byI+K(that is,B0is the identity operator).
The following two results are analogous to, respectively, Eq. (4.5) and Claim 3 in[1].
Lemma 7.Let(xn)⊂H01([0,1], U )be a sequence of smoothG-regular curves such that˜ xn→0in theH1topology.
Then
dE(x˜ n)[Axn] = 1 0
Bsx1/2n+Ksxn2
xn, xn ds+o
xn20
, asn→ ∞.
Proof. Eqs. (1)–(2) imply thatG(t, q, y)˜ satisfies assumptions(L1)and(L2)on page 605 of[1], for each(t, q, y)∈ [0,1] ×U×Rn\Z. Hence the lemma follows arguing as in[1, Lemma 4.1], taking into account that
dE(x)˜ [Ax] =
∇ ˜E(x), x+K(x)
= 1
0
d ds
∇ ˜E(sx), x+K(x) ds
= 1 0
(Bsx+Ksx)x, x+K(x)
ds. (4)
In fact, dsd∇ ˜E(sx)=(Bsx+Ksx)[x]at the pointsswhere the curvet∈ [0,1] →sx(t )isG-regular. From Lemma˜ 4, the set of pointss∈ [0,1]wheresxis notG-regular is at most countable.˜ 2
The next lemma follows as in Claim 3 of[1, Lemma 4.1], recalling Remark6and the fact that 0 is a non-degenerate critical point ofE.˜
Lemma 8.There exist a numberμ >0 and a neighbourhoodUof0inH01([0,1], U )such that, for each smooth andG-regular curve˜ x∈U, the spectrum of the self-adjoint operatorBx1/2+Kxis disjoint from[−μ, μ].
Proof of Proposition 3. Since AL|B= ˜E|B, it is enough to prove the proposition for the functional E. From˜ Lemmata7and8, we get that there exists a positive constantμ1, such that
dE(x)˜ [Ax]μ1x20, (5)
for each smoothG-regular curve˜ x∈U. From Remark5and the continuity of dE˜ andAwith respect to theH1 topology, inequality (5) can be extended to any smooth curve in U and then, since smooth curves are dense in H01([0,1], U ), to anyx∈U. As∇ ˜Eis a locally Lipschitz field and∇ ˜E(0)=0, we get
dE(x)˜ [Ax]μ0∇ ˜E(x)2
0,
for some positive constantμ0and for allxin some neighbourhoodUof 0. 2
Now letη0:H01([0,1],Rn)→ [0,1]be a smooth bump function such that suppη0⊂U andη0(x)=1, for all x∈U, whereU is an open neighbourhood of 0 inH01([0,1],Rn)withU ⊂U. Let us consider the vector field on H01([0,1],Rn)defined as
Y (x)= −η0(x)Ax−
1−η0(x)
∇AL(x).
We point out that we cannot state thatY is a pseudo-gradient vector field because we are not able to prove that Ax0μ2dAL(x)
0, (6)
for some constantμ2> μ0and allxin some neighbourhood of 0.3Anyway (3) implies thatY satisfies the inequality dAL(x)
Y (x)
−μ∇AL(x)2
0, (7)
for eachx∈H01([0,1],Rn), whereμ=min{μ0,1}. As we will show in Lemma9, inequality (7) (together with the remark in footnote3) is enough to get a deformation result as in[11, Lemma 8.3]. For allx∈H01([0,1],Rn), let (ω−(x), ω+(x))be the maximal interval of definition of the solution of
ψ˙ =Y (ψ ),
ψ (0)=x. (8)
Observe that this problem is well defined becauseY is a locally Lipschitz vector field inH01([0,1],Rn), sinceAand
∇AL are. Furthermore, (7) implies thatAL is decreasing along the flow ofY and as,Y|U = −A= −I−K, such flow is given by
ψ (x, t )=e−tx− t
0
e−t+sK
ψ (x, s)
ds (9)
forx∈U, whereasψ (x, t )∈U. The following lemma is an adaptation of Lemma 8.1 in[11]to the flow of the vector fieldY.
3 Actually using thatAL satisfies the Palais–Smale condition and 0 is an isolated critical point ofAL, we can prove thatYsatisfies (6) in any open subsetU\ ¯B(0, r), whereB(0, r)is an open ball strictly contained inU, for a constantμ2depending onU\ ¯B(0, r).
Lemma 9.LetV be a closed neighbourhood of0contained inU. Then there existε >0and an open neighbourhood O⊂V of0inH01([0,1],Rn)such that ifx∈O, then the solutionψ (x,·)of(8)either stays inV for allt∈ [0,+∞) or it stays inV at least untilAL(ψ (x, t))becomes less thanc−ε(wherec=AL(0)= ˜E(0)).
Proof. Observe that, sinceY|V = −A,ψ (x,·)is defined for all times until it lies inV. LetB(0, ρ)be the ball of radiusρcentred at 0 such thatB(0, ρ)¯ ⊂V and let
C =
x∈H01
[0,1],Rn : ρ
2 x0ρ
. SinceC⊂B, it is free of critical points and then
δ= inf
x∈C∇AL(x)
0>0, (10)
becauseAL satisfies the Palais–Smale condition onC. Moreover Y (x)
0= Ax0ρA0ρA0
δ ∇AL(x)
0, (11)
for eachx∈C. Letν:=ρAδ0 andO=B(0, ρ/2)∩ ˚
ALc+μδρ4ν . Ifx∈Ois such thatψ (x,t )¯ does not belong toV for somet >¯ 0, then there exist 0< t1< t2< ω+(x)such thatψ (x, t )∈C, for allt∈(t1, t2)andψ (x, t1)0=ρ/2, ψ (x, t2)0=ρ. It follows that
AL
ψ (x, t2)
=AL
ψ (x, t1) +
t2
t1
dAL
ψ (x, t ) Y
ψ (x, t ) dt
AL(x)−μ
t2
t1
∇ALψ (x, t )2
0dt
c+μδρ
4ν −μδ
t2
t1
∇ALψ (x, t )
0dt
c+μδρ
4ν −μδ ν
t2
t1
Yψ (x, t )
0dt
c+μδρ
4ν −μδ
ν ψ (x, t2)
0−ψ (x, t1)
0
=c+μδρ 4ν −μδρ
2ν =c−μδρ
4ν . (12)
In the first inequality above, we have used the fact thatAL is decreasing in the flow of (8) and inequality (7); in the second one, the fact thatx∈O⊆ALc+μδρ4ν and (10); in the third one, inequality (11); in the last one, the following chain of inequalities:
t2
t1
Yψ (x, t )
0dt=
t2
t1
ψ(x, t)˙
0dt
t2
t1
ψ(x, t)˙ dt 0
ψ (x, t2)
0−ψ (x, t1)
0.
Thus the conclusion follows withε=μδρ4ν . 2
Let V be the subset of H01([0,1],Rn) given asV = x∈Oψ (x,[0, ω+(x))), where O is the neighbourhood of 0 associated to V by Lemma 9. Since O is open, from standard results in ODE theory (cf. for example [9, Corollary 4.2.10]),V is also an open subset ofH01([0,1],Rn). From Lemma9,AL−1((c−ε, c+ε))∩V \ {0}
is contained inV ⊂U and it is free of critical points.
Lemma 10. For every x ∈ AL−1([c, c+ε)) ∩V, either there exists a unique T (x) ∈ [0, ω+(x)) such that AL(ψ (x, T (x)))=corω+(x)= +∞andψ (x, t )→0, inH01([0,1],Rn), ast→ +∞.
Proof. If AL(ψ (x, t)) > c, for allt ∈ [0, ω+(x)), then from Lemma9,ω+(x)= +∞ andψ (x, t )∈V, for each t∈ [0,+∞). From inequality (12),
+∞
0
∇ALψ (x, t )2
0dt 1 μ
AL(x)−c
<+∞,
hence lim inft→+∞∇AL(ψ (x, t))20=0 and the Palais–Smale condition implies the existence of a sequence{tn} converging to+∞such thatψ (x, tn)→0. Hence the conclusion follows from Lemma9. 2
By Lemmata9and10, as in[11, Lemma 8.3], we get thatALc ∩V is a strong deformation retract of ˚
ALc+ε/2∩V. Analogously, ALc−ε∩V is a strong deformation retract of both ALc ∩V \ {0}and ˚
ALc−ε/2∩V. Using that, for A⊂B⊂C, ifB is a strong deformation retract ofC, thenH∗(B, A)∼=H∗(C, A)and ifAis a strong deformation retract ofB, thenH∗(C, A)∼=H∗(C, B)(for the last property, see for example[13, PropertyH6–β]), we obtain
H∗
ALc ∩V ,ALc ∩V\ {0}∼=H∗ ˚
ALc+ε/2∩V , ˚
ALc−ε/2∩V
. (13)
LetO=ϕ∗(O)andγ0=ϕ∗(0), then C∗(E, γ0)=H∗
Ec∩O, Ec∩O\ {γ0}∼=H∗
(E◦ϕ∗)c∩O, (E◦ϕ∗)c∩O\ {0}
=H∗E˜c∩O,E˜c∩O\ {0}
=H∗
ALc ∩O,ALc ∩O\ {0}
∼=H∗
ALc ∩V ,ALc ∩V \ {0}
, (14)
last equivalence, by the excision property of the singular relative homology groups. By Palais’s theorem above we get H∗ ˚
ALc+ε/2∩V , ˚
ALc−ε/2∩V∼=H∗ ˚ ALc+ε/2
C01([0,1],Rn)∩V , ˚ ALc−ε/2
C10([0,1],Rn)∩V .
The above equivalence, together with (13) and (14), implies that C∗(E, γ0)∼=H∗ ˚
ALc+ε/2
C01([0,1],Rn)∩V , ˚ ALc−ε/2
C01([0,1],Rn)∩V .
It remains to prove that these last relative homology groups are isomorphic to the critical groups in X = C01([0,1], U ). To this end, let us consider the Cauchy problem (8), withx ∈C1([0,1],Rn)∩AL−1((c−ε/2, c+ ε/2))∩V. SinceAL−1((c−ε/2, c+ε/2))∩V ⊂V ⊂U, it holds (9) and the orbitψ (x,·), defined byx, is also in C01([0,1],Rn).
As a consequence, the strong deformation retracts that we have considered above are well defined in C01([0,1],Rn)× [0,1]and by the continuity of the flow (9) with respect to the C1 topology, we immediately de- duce that they are also continuous at each point different from(0,1). Clearly, the continuity at the point(0,1)with respect to the product topology ofC01([0,1],Rn), with theC1topology, andR, with the standard one, comes into play only for the deformation mapη: ˚
ALc+ε/2∩V × [0,1] → ˚
ALc+ε/2∩V of ˚
ALc+ε/2∩V inALc ∩V, which is given by η(x, t )=
ρ(x,t−t1) ift∈ [0,1), lims→+∞ρ(x, s) ift=1, where ρ: ˚
ALc+ε/2∩V × [0,+∞)→ ˚
ALc+ε/2∩V is the map defined as follows: if AL(x) > c and there exists T (x) >0 such thatAL(ψ (x, T (x)))=c, then
ρ(x, t )=
ψ (x, t ) ift∈ [0, T (x)], ψ (x, T (x)) ift∈(T (x),+∞),
ifψ (x, t )→0 ast→ +∞, thenρ(x, t )=ψ (x, t )and ifAL(x)c, thenρ(x, t )=x, for allt∈ [0,+∞). Since the flowψ1of the linear vector fieldx→ −Ax= −I x−Kx is given by (9) andKis bounded fromH01([0,1],Rn)to C01([0,1],Rn), we have
t
0
e−t+sK
ψ1(x, s) ds
C1
e−t t
0
esK
ψ1(x, s)
C1dsCe−t t
0
esψ1(x, s)
0ds.
Thus, ifψ (x, t )→0 inH1, ast→ +∞, then, from Lemmata9and10,ψ (x, t )=ψ1(x, t). Hence, for everyε >0, there exists t >¯ 0 such that for allt >t,¯ ψ (x, t )0< εand then the last function in the above inequalities can be estimated, fort >t¯, as
e−t t
0
esψ (x, s)
0ds=e−t
¯
t
0
esψ (x, s)
0ds+e−t t
¯ t
esψ (x, s)
0ds
e−t
¯
t
0
esψ (x, s)
0ds+ε
1−e−tet¯ .
Thusψ (x, t )→0 also with respect to theC1topology, giving the continuity of the mapηat the point(0,1)also with respect to the product of such a topology and the Euclidean one on the interval[0,1].
In conclusion we have that the following groups are isomorphic H∗ ˚
ALc+ε/2
C01([0,1],Rn)∩V , ˚ ALc−ε/2
C01([0,1],Rn)∩V
∼=H∗ ALc
C10([0,1],Rn)∩V ,ALc
C01([0,1],Rn)∩V \ {0} . By excision, these last relative homology groups are isomorphic to
H∗ ALc
C10([0,1],Rn)∩O,ALc
C01([0,1],Rn)∩O\ {0}
and then, since the curves inOhave their support inU, to H∗E˜c
C01([0,1],U )∩O,E˜c
C01([0,1],U )∩O\ {0} .
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