Witten deformation using Lie groupoids
Omar Mohsen

































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Witten deformation using Lie groupoids
    Omar Mohsen
    
======================================

We express Witten's deformation of Morse functions using deformation to the normal cone and C^*-modules. This allows us to obtain asympotitcs of the `large eigenvalues'. This is then applied to the case of Morse-Bott functions inspired by <cit.>.

Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequialities obtained  by Connes and Fack <cit.> in a slightly more general situation.


§ INTRODUCTION
In this article, we give an application of deformation groupoids to Witten's deformation of a Morse function. A Morse function f is a real valued smooth function on a compact manifold M with nondegenerate critical points. This is a generic condition by results of Morse. In <cit.>, Morse proved the so called Morse inequalities highlighting a relation between the number of critical points of f and the Betti numbers (H^· (M)). 
Let c_i be the number of critical point of f of index i.

    (H^i(M,))≤ c_i,  ∑_i=0^k(-1)^k-i(H^i(M,))≤∑_i=0^k(-1)^k-ic_i

 


He did so by studying the level sets f^-1(]-∞,a]) and seeing how they change as a passes by a critical value. In <cit.>, Witten proposed an analytic way to prove Morse inequalities. His method consists of deforming the De Rham operator d to become d_t=e^-f/tde^f/t, and then studying the associated Laplacian Δ_t=(d_t+d^*_t)^2. Since the operator d_t is conjugate to d, it follows that (Δ_t) is isomorphic to (Δ). Hence by Hodge theory, ((Δ^i_t))=(H^i(M,)) for all t>0, where Δ^i_t denotes the deformed Laplacian acting on forms of degree i. He then proves that, as t→ 0^+, the spectrum sp(Δ_t^i) gets separated into two parts, the first part is finite and consists of an eigenvalue for each critical point of f of index i, and the second part consists of eigenvalues which converge to +∞. Morse inequalities are then corollaries of this decomposition. 
In this article, we give another point of view of Witten deformation, a global one which avoids local coordinates. By applying the deformation to the normal cone construction, one obtains  a smooth manifold whose underlying set is equal to 
    (M,(f))=M× ]0,1]⊔_a∈(f)T_aM×{0}.
 The deformed Laplacian t^2Δ_t with a slight normalisation acts on the submanifold M×{t}⊆(M,(f)). We show that this operator can be glued smoothly to the direct sum of Harmonic oscillators at every critical point acting on  ⊔_a∈(f)T_aM×{0}⊆(M,(f)). The glued operator will be called the global Witten deformation.

 The key tool that is used to be able to express the smoothness of the global Witten deformation is that of Lie groupoids. The theory of Lie groupoids in connection with pseudo-differential operators, C^*-algebras, and index theory started in the work of Connes <cit.>, and was extensively developed. See for example <cit.>. The natural projection π_:(M,(f))→ is a submersion, hence the fibers define a trivial foliation. The Lie groupoid that is used to study the global Witten deformation is the Lie groupoid associated to this foliation (see <cit.> for the general definition).

Once this operator is constructed, we show that it is a  regular operator in the sense of Baaj and Woronowicz, see <cit.>. We recall that regularity amounts to saying that continuous functional calculus can be applied to the global Witten deformation.

Furthermore by adapting the classical theorem which says that the De Rham operator on a complete manifold is locally elliptic to the context of Lie groupoids, we show that the global Witten deformation has compact resolvent in the sense of Kasparov for C^*-modules <cit.>. Note that the manifold (M,(f)) is not compact.

The compactness of the resolvent of the global Witten deformation gives immediately (see <ref>) the asymptotics of the eigenvalues of the deformed Laplacian. More precisely, we obtain 
 

Let 
    λ^p_1(t)≤λ^p_2(t)⋯
 denote the spectrum of Δ_t^p, then for every i∈, 
    lim_t→ 0^+tλ^p_i(t)=λ_i^p(0),
where λ_i^p(0) is the i'th eigenvalue of harmonic oscillator 
    Δ_0^p:=⊕_a∈(f)(d+d^*+c(d^2_a(f)))^2:⊕_a∈(f)L^2(Λ^p_ T_aM)→⊕_a∈(f) L^2(Λ^p_ T_aM),
 where L^2(Λ^p_ T_aM) is the set of all L^2 functions from T_aM to Λ^p_ T_aM, d^2_af is the associated 1-differential form on T_aM, and c is the Clifford multiplication.


Moreover, we extend the above construction and results to the case of Morse-Bott functions, inspired by Bismut's article <cit.>.



Connes and Fack <cit.>, generalised Witten's approach and Morse inequalities for foliations equipped with a holonomy invariant transverse measure. Let F be a regular foliation on a compact manifold, ν a holonomy invariant transverse measure, f a nice function (see <ref>), c_i=ν(_ℱ^i(f)) the ν-measure of longitudinally Morse critical points of f of index i, and β_i the ν-dimension of the longitudinal De Rham cohomology. One has 
    β_i≤ c_i,  ∑_i=0^k(-1)^k-iβ_i≤∑_i=0^k(-1)^k-ic_i.


The numbers β_i, c_i in the previous theorem are Von Neumann dimensions which are usually real numbers.



We extend as well our results to longitudinal Witten deformation along an arbitrary regular foliations, not necessarily equipped with a holonomy invariant transverse measure. More precisely, let F be a foliation on a compact manifold M, f:M→ a smooth function such that the longitudinal derivative d_Ff:M→ F^* is transverse to the zero section. This is a generic condition by Thom's transversality theorem. We denote by _F(f):={x∈ M:d_Ff(x)=0} the manifold of longitudinal critical points. 



Notice that the points at which _F(f) is transverse to F are the points at which f is longitudinally Morse. In general, a foliation has no compact transversal. Hence f will usually have longitudinal critical points of non Morse type. As we will see below, this adds difficulty to the study of Witten Laplacian.



As in the classical case one could define the deformed longitudinal Laplacian 
    (e^-f/td_Fe^f/t+e^f/td^*_Fe^-f/t)^2.
 which now acts on the C^*-algebra of the Lie groupoid associated to the foliation that will be denoted by 𝒢(M,F). The advantage of this is that the deformed Laplacian being longitudinally elliptic has compact resolvent in Kasparov's sense.



Using the recent observation of Debord and Skandalis <cit.>, we construct a Lie groupoid 
    (𝒢(M,F),_F(f)):=𝒢(M,F)×^*⊔𝒩^𝒢(M,F)__F(f)×{0}.




We then construct a global longitudinal Witten deformation which acts on the C^*-algebra of the Lie groupoid (𝒢(M,F),_F(f)) which on 𝒢(M,F)×{t} is equal to 
    t^2(e^-f/td_Fe^f/t+e^f/td^*_Fe^-f/t)^2
 and which is equal to a `Harmonic oscillator' on 𝒩^𝒢(M,F)__F(f)×{0} at the critical points. The Harmonic oscillator at a critical point x depends on the dimension of T_x_F(f)∩ F_x. If x is Morse (the intersection is trivial), then the Harmonic oscillator is the same as in the classical case. If not, it is slightly more complicated.
 


The methods that we used to prove in the classical case that the global Witten deformation is regular with compact resolvent can be used in the case of foliations. In <ref>, we show that the global longitudinal Witten deformation is regular and has compact resolvent as an operator on a C^*-module. In fact in <Ref>, we prove a general result that applies to any Lie groupoid from which we deduce both <ref> and <ref>.



To obtain quantitative corollaries of <ref> and recover Connes-Fack inequalities, we assume that the foliation has a holonomy invariant transverse measure and some extra genericity assumption on f, see <ref>.



In <cit.>, Connes shows that  a holonomy invariant transverse measure defines a trace on the C^*-algebra of the foliation. We show that this trace extends to a C([0,1])-valued trace on C^*((𝒢(M,F),_F(f))). This C([0,1])-trace together with  <ref> imply Connes-Fack Morse inequalities in a slightly more general situation. 



More precisely, we don't need to assume that the transverse measure is absolutely continuous with respect to the Lebesgue measure. In fact in <ref>, we show a closely related assumption that is pivotal in the construction of the C([0,1])-valued trace on C^*(𝒢(M,F),_F(f)) (see <ref>). Such condition is valid for a generic function f. We finally show that if f doesn't satisfy the mentioned above condition then the Morse inequalities for f are still true by approximating f by another function which satisfies the required conditions and whose critical points aren't far from those of f.

Finally let us remark that all our constructions and results work equally well if the Morse function is replaced by a Novikov 1-form in both the classical and the foliation case.



This paper is organised as follows :



 In section 1, we recall the notion of the deformation to the normal cone following <cit.>.



In section 2, we establish regularity and compactness of the resolvent of an abstract Witten deformation on a Lie groupoid.



In section 3, we show that the results of section 2 admits as a corollary Witten's deformation in the classical case, as well as in the case of Morse-Bott functions like in Bismut's article <cit.>.



In section 4, we apply the results of section 2 to the case of foliated manifolds.



In section 5, Quantitative corollaries of section 4 are obtained under the additional hypothesis of the existence of holonomy invariant measure. In this section we use an independent result on continuity of 1-parameter family of traces on C^*-algebras which is proven in section 6.



§ ACKNOWLEDGEMENTS

I would like to thank my Ph.D advisor G. Skandalis for the numerous suggestions and discussions. This work was supported by grants from Région Ile-de-France. I would like to also thank C. Viterbo and the Referee for their helpful remarks.


§ DEFORMATION TO THE NORMAL CONE

In this section, we recall the deformation to the normal cone construction and its relation to Lie groupoids following <cit.>. The deformation to the normal conedeformation to the normal cone of a manifold M along a submanifold V is a manifold whose underlying set is 
    (M,V):=M× ]0,1]⊔ N^M_V×{0},
 where N^M_V is the normal bundle of V inside M. The smooth structure is defined by covering (M,V) with two open sets; the first is M× ]0,1] and the second is ϕ(N^M_V)×]0,1]⊔ N^M_V×{0} where ϕ:N^M_V→ M is a tubular embedding.[To simplify the exposition, we will always assume that tubular neighbourhoods are diffeomorphisms on N^M_V.] The smooth structure on ϕ(N^M_V)×]0,1]⊔ N^M_V×{0} is given by declaring the following map a diffeomorphism 
    ϕ̃:N^M_V× [0,1]→ϕ(N^M_V)×]0,1]⊔ N^M_V×{0}
       ϕ̃(x,X,t)=(ϕ(x,tX),t)∈ M× ]0,1],   t≠ 0
       ϕ̃(x,X,0)=(x,X,0)∈ N^M_V×{0}.

This defines a smooth structure. Independence of ϕ follows by noticing that the following functions are smooth functions that generate the smooth structure:

  * the function
    (π_M,π_):(M,V)   → M×
    
    (x,t)   → (x,t),   t≠ 0
    
    (x,X,0)   → (x,0)


  * Let f∈ C^∞(M) be a smooth function which vanishes on V. Therefore df:N^M_V→ is well defined. The following function is smooth

    (f):(M,V)   →
    
    (x,t)   →f(x)/t,   t≠ 0
    
    (x,X,0)   → df_x(X)




Let  M,M' be smooth manifolds, V⊆ M, V'⊆ M' submanifolds, f:M→ M' a smooth map such that f(V)⊆ V'. Then the map defined by 

    (M,V)   →(M',V')
    
    (x,t)   → (f(x),t),   t≠ 0
    
    (x,X,0)   → (f(x),df_x(X),0)

is a smooth map that will be denoted by (f). 
 Furthermore the map  (f) is 

  * a submersion if and only if f is a submersion and f_|V:V→ V' is also a submersion.
 
  * an immersion if and only if f is an immersion and for every v∈ V, T_vV=df_v^-1(TV').
 

The map 
    N^M_V→ N^M'_V',   (x,X)   → (f(x),df_x(X))
 will be denoted by Nf.


Let M_1, M_2, M be manifolds, V_i⊆ M_i,V⊆ M submanifolds, f_i:M_i→ M smooth maps such that 

  * f_i(V_i)⊆ V for i∈{1,2}

  * the maps f_i are transverse

  * the maps f_i|V_i:V_i→ V are transverse
 Then 

  * 

  * the maps Nf_i:N^M_i_V_i→ N^M_V are transverse.

  * the natural map 
    N^M_1×_MM_2_V_1×_VV_2→ N^M_1_V_1×_N^M_VN^M_1_V_2
 is a diffeomorphism.

 
Similarly for , we have 
  * 
 
  * the maps (f_i):(M_i,V_i)→(M,V) are transverse.
 
  * the natural map 
    (M_1×_MM_2,V_1×_VV_2)→(M_1,V_1)×_(M,V)(M_2,V_2)
 is a diffeomorphism.





In the next proposition and for the rest of this article we use the notion of Lie groupoids, 𝒱ℬ-groupoids  and invariant operators on Lie groupoids.  We refer the reader to <cit.> for more details on Lie groupoids and their C^*-algebras and to <cit.> for more details on 𝒱ℬ-groupoids. There exists also a recent survey on pseudo-differential operators on Lie groupoids <cit.>. Let G⇉ G^0 be a Lie groupoid. We use the notation s and r for the source and range and for any x∈ G^0, we use G_x:=s^-1({x}). We denote the Lie algebroid of G by 𝔄G. We also denote the anchor map by ♮.

Let G be a Lie groupoid, H a Lie subgroupoid. Then


  * the space N^G_H⇉ N^G^0_H^0 is a Lie groupoid whose structure maps are Ns, Nr and whose Lie algebroid is equal to N^𝔄G_𝔄H. Furthermore, N^G_H is a 𝒱ℬ-groupoid over H.

  * the manifold (G,H)⇉(G^0,H^0) is a Lie groupoid whose structure maps are (s), (r) and Lie algebroid is equal to (𝔄G,𝔄H).



Statements 1 and 2 are direct consequences of propositions <ref> and  <ref>.

From now on, for a Lie groupoid G and a Lie subgroupoid H, we will use 𝒩^G_H to denote the space N^G_H equipped with the structure of a Lie groupoid given by <Ref>.

 Let E→ M be a vector bundle, V⊆ M a submanifold, F→ V a subbundle of the restriction of E to V. By <Ref>, the space (E,F) is a vector bundle over (M,V). Since a section of (E,F) is determined by its values on the dense set M× ]0,1]. It follows that 
    Γ((E,F))={X∈Γ(E×):X_|V×{0}∈Γ(F)},
 where Γ denotes the set of global smooth sections. 
 
 In the particular case where F is the zero bundle, it is clear that by dividing by t, we have an isomorphism from (E,V) to π_M^*E where π_M:(M,V)→ M is the projection map. It follows that to a Euclidean metric on E, one associates canonically a Euclidean metric on (E,V). 
 
 Moreover, the vector bundles (E,V)^* and (E^*,V) are canonically isomorphic by the isomorphism 
    (E^*,V)→(E,V)^*
    α→(e→1/t^2α(e))  for  t≠ 0.




  * If M is a smooth manifold, then 
    (M× M,M)=M× M× ]0,1]⊔ TM×{0}⇉ M×
 is the tangent groupoid constructed by A. Connes <cit.>. The product law is given by 
    (x,y,t)· (y,z,t)=(x,z,t),  (x,X,0)· (x,Y,0)=(x,X+Y,0).



  * Let L⊆ G^0 be a submanifold. Since N^G_L is equal to N^G^0_L⊕(ds)_|L. It follows that the groupoid 𝒩^G_L⇉ N^G^0_L is equal to 
    {(X,A,Y):X,Y∈ N^G^0_L,A∈𝔄G, Y-X=♮(A) TL},
with the structural maps 
    s(X,A,Y)=Y,  r(X,A,Y)=X,  (X,A,Y)· (Y,B,Z)=(X,A+B,Z).



 It is clear that in the definition of , the manifold [0,1] could be replaced by  or equally well by [-1,1]. In the rest of this section, we will use _(M,V)=M×^*⊔ N^M_V×{0}.

The group ^* acts on the manifold _(M,V) by the action 
    s· (x,t)=(x,ts),   s· (x,X,0)=(x,X/s,0).
 This action is free and proper on the open subset _(M,V)\ V×. The quotient is the classical blowup in differential geometry and is denoted by (M,V).

If f:M→ M' is a smooth map, V⊆ M and V'⊆ M' submanifolds such that f(V)⊆ V'. The map (f):_(M,V)→_(M',V') is ^*-invariant but it doesn't always descend to the blowup. It only descends when the map (f) is restricted to the ^*-invariant subset (f)^-1(V'×)^c. The quotient (f)^-1(V'×)^c/^* is denoted by _f(M,V). Here ^c denotes the complement set. It follows that one has a smooth map 
    (f):_f(M,V)→(M',V').


The following theorem follows from the naturality construction of (f) and is proved in <cit.>. 
Let H⊆ G be a Lie subgroupoid. The intersection 
    _r,s(G,H):=_r(G,H)∩_s(G,H)⇉(G^0,H^0)
 is a Lie groupoid with structure maps (r) and (s). Its Lie algebroid is _π(𝔄G,𝔄H), where π:𝔄G→ G^0 is the projection map.



§ A PRELIMINARY RESULT

We will deduce the properties of Witten's deformation ultimately using the following simple proposition.


Let W be a complete Riemannian manifold, #:T^*W→ TW the musical isomorphism given by the Riemannian metric, α a 1-form on W such that 

  * the form dα is bounded

  * the section of (Λ_ T^*W) given by ℒ_α^#+ℒ_α^#^* is bounded

  * α is a proper function,

then the operator d+d^*+c(α) acting on L^2(Λ_ T^*W) is a self-adjoint elliptic operator with compact resolvent, where L^2(Λ_ T^*W) is the Hilbert space of L^2 sections of Λ_ T^*W.

Here α is the function on W which sends a point x to α_x. Similarly for dα and d^*α.

 In fact we will need the Lie groupoid version of <Ref>. Before stating the extension to Lie groupoids we will need some classical results for Lie groupoids. We outline the proof of each for the reader's convenience.
 
 Let g be a Euclidean metric on the bundle 𝔄G→ G^0. For every γ∈ G, one has the isomorphism 
    T_γ G_s(γ)T_r(γ)G_r(γ)=𝔄_r(γ)G,
 where R_γ^-1 denote right multiplication by γ^-1. It follows that g defines a Riemannian metric on G_x for every x∈ G^0.
  The metric g is called complete if the induced metric on every G_x is complete.
  There exists a Euclidean metric g on 𝔄G such that for every x∈ G^0, the induced Riemannian metric on G_x is complete.


Let g be any Euclidean metric on 𝔄G, and let h:G^0→ ]0,+∞[ be a smooth function such that if x∈ G^0, then the ball in G_x of radius h(x) with center x is relatively compact. It is straightforward to verify that the euclidean metric 1/h^2g is complete. See <cit.> for more details.
 

  A consequence of the <ref> is that if G^0 is compact, then every Euclidean metric on 𝔄G is complete.
  
Extension of Chernoff's theorem  <cit.> to Lie groupoids in the case where G^0 is compact was done in <cit.> and in the case of foliation Lie groupoids in <cit.>. We present here an elementary proof for the general case.
  
Let G⇉ G^0 be a Lie groupoid, g a complete Euclidean metric on 𝔄G, E→ G^0 a Hermitian vector bundle, D a symmetric first order G-invariant differential operator on G acting on r^*E, c:G^0→ the function 
    c(x)=sup_v∈𝔄G_x^*:v=1σ(D)(x,v).
 Here σ is the principal symbol. If c is bounded above, then the closure of D is a regular self adjoint operator acting on C^*(E).


Let f∈Γ_c(r^*(E⊗ |Λ|^1/2𝔄G)). We recall that throughout this article |Λ|^α denotes the bundle of α-densities. Consider the differential equation 
    ∂_t u(γ,t)=iDu(γ,t),  u(γ,0)=f(γ),   (γ,t)∈ G×.
 By the classical theory of linear differential equations a unique C^∞ solution to this equation exists locally. By Chernoff's theorem <cit.> and our assumptions, we deduce that a solution exists globally on G_x for each x. In particular solutions to this equation exist globally on G. Furthermore the distribution kernel associated to this equation is proper for each fixed t. Let 
    V_t:Γ_c(r^*(E⊗ |Λ|^1/2𝔄G))→Γ_c(r^*(E⊗ |Λ|^1/2𝔄G)),   f→ u(·,t)

be the convolution to the left by the distribution kernel. If f,g∈Γ_c(r^*(E⊗ |Λ|^1/2𝔄G)), then 
    d/dt⟨ V_tf,V_tg⟩=⟨ iDV_tf,V_tg⟩+⟨ V_tf,iDV_tg⟩=⟨ i(D-D^*)V_tf,V_tg⟩=0.

Hence the operators V_t extend to an isometry acting on the C^*G-module C^*E. This operator is adjointable (and therefore C^*G-linear) because of the equation 
    ⟨ξ ,V_tη⟩=⟨ V_-tξ,η⟩,
 which proves as well that V_t is a unitary in ℒ(C^*E). The proposition follows then from <Ref>. 
Let S be a regular self adjoint operator acting on a C^*-module E, V_t=exp(itS), T:(T)⊆ E→ E a -linear map with a dense domain. If

  * V_t(T)=(T)

  * T⊆ S.
Then the closure of (T) is equal to (S).


By taking the closure of T, we can suppose that T is closed. Let f∈𝒮() be a Schwartz function. Since 
    f(S)=∫_-∞^∞f̂(t)V_2π tdt,
 it follows that f(S)(T)⊆(T) and Tf(S)=f(S)T. Since f(S) and Sf(S) are bounded operators and (T) is dense, it follows that {(f(S)x,Sf(S)x):x∈ E}⊆(T). 

Let 0≤ f_n≤ 1 be Schwartz functions such that f_n→ 1 uniformly on every compact. It follows that f_n(S) strongly converges to the identity. In particular if x∈(S), then f_n(S)x→ x, and Sf_n(S)x→ Sx. Hence (x,Sx)∈(T), which implies that S=T.

Under the same hypothesis as <Ref>, if furthermore D is an elliptic operator, then for every f∈ C_0(G^0) and g∈ C_0(), the operator g(D)f is compact in the sense of C^*-modules.


By a density argument it is enough to prove the proposition for f∈ C_c^∞(G), and g∈ C_c(). Let Q be a parametrix for D^2, that is D^2Q=1+R with R a G-pseudo differential operator of order ≤ -1. The support of Q can be chosen to be a subset of an  arbitrary open neighbourhood of G^0. Since D^2 is a differential operator, its Schwartz kernel is supported in a subset of G^0. In particular the support of R can be choosen as well to be a subset of an arbitrary neighbourhood of G^0. We choose the supports of R and Q so that Qf and Rf are G-invariant pseudodifferential operators with compact support. It follows from <cit.>, that Qf and Rf extend to compact operators on C^*(E). It follows from the identity 
    (1+D^2)^-1f=Qf-(1+D^2)^-1Rf+(1+D^2)^-1Qf,

that (1+D^2)^-1f is compact. Since g(x)(1+x^2) is bounded, it follows that g(D)f is compact as well.
We recall the following extension of a classical result to Lie groupoids. If X∈Γ^∞_c(𝔄G), then the operator ℒ_X:Γ^∞_c(Λ_𝔄G^*)→Γ^∞_c(Λ_𝔄G^*) is defined by Cartan's formula 
    ℒ_X=di_X+i_Xd.
 Here i_X is the interior product acting on differential forms. The operator ℒ_X is a G-operator in the sense of <cit.>. We then have The operator ℒ_X+ℒ_X^* is C^∞(G^0)-linear (i.e. a 0-order G-differential operator).

Notice that the adjoint of L_X is the adjoint in the sense of G-operators. The metric g induces a G-invariant Riemannian metric on each G_x for x∈ G^0 and X induces a G-invariant vector field on each G_x. The dual ℒ_X^* is the G-invariant dual on each G_x.


The operator ℒ is an ungraded derivation. Therefore, 
    ℒ_X(fα)=(ℒ_Xf)α+fℒ_Xα,
 where f:G^0→ is a real valued smooth function and α∈Γ_c^∞(Λ_𝔄^*G). Taking the dual one deduces that 
    fℒ_X^*(α)=(ℒ_Xf)α+ℒ_X^*(fα).
  Therefore 
    ℒ_X(fα)+ℒ_X^*(fα)=f(ℒ_X(α)+ℒ_X^*(α)).


Let G be a Lie groupoid, g a complete euclidean metric on 𝔄G, #:𝔄G^*→𝔄G the musical isomorphism given by g, α∈Γ^∞(𝔄G^*). If 

  * the form dα is bounded

  * the section of (Λ_𝔄G^*) given by ℒ_α^#+ℒ_α^#^* is bounded

  * α:G^0→ is a proper function,

then the closure of the operator d+d^*+c(α) acting on the C^*(G) Hilbert module C^*(Λ_𝔄G^*) is a regular self adjoint elliptic operator with compact resolvent in the sense of C^*-modules.

Thanks to <cit.> (see also <cit.>), <ref> implies that the Kasparov product of d+d^* seen as an element of ^0((𝔄^*G),) and c(α) seen as an element of ^0(,(𝔄^*G)) is the operator d+d^*+c(α)∈ KK^0(,), where  denotes the complex Clifford algebra.


Since 
    σ(d+d^*+c(α))(x,v)=σ(d+d^*)(x,v)=v_g_x,
 it follows that d+d^*+c(α) and d+d^* are elliptic and from <ref> that the closure of d+d^*+c(α) and d+d^* are regular self adjoint operators.



It follows from Cartan's formula that the graded commutator is equal to

    [d,i_α^#]=ℒ_α^#.


Since  
    [d,c(α)]   =[d,α∧]+[d,i_α^#]=dα∧· + ℒ_α^#.
Hence by the hypotheses of <ref>
    [d+d^*,c(α)]=[d,c(α)]+[d,c(α)]^*=dα∧· +i_(dα)^#+ℒ_α^#+ℒ^*_α^#
 is bounded, where i_(dα)^#(·) is the adjoint of dα∧·. Therefore the closure of (d+d^*)^2+c(α)^2=(d+d^*+c(α))^2-[d+d^*,c(α)] is a regular self adjoint operator.



By a classical inequality (see <cit.>), one has 
    (1+(d+d^*)^2+c(α)^2)^-1≤ (1+(d+d^*)^2)^-1
       (1+(d+d^*)^2+c(α)^2)^-1≤ (1+(c(α))^2)^-1=(1+α^2)^-1

It follows from <cit.> that there exists a,b∈ℒ(C^*Λ_𝔄G^*) such that 
    (1+(d+d^*)^2+c(α)^2)^-1/2=a(1+(d+d^*)^2)^-1/4,   (1+(d+d^*)^2+c(α)^2)^-1/2=(1+α^2)^-1/4b.
 Hence 
    (1+(d+d^*)^2+c(α)^2)^-1=a(1+(d+d^*)^2)^-1/4(1+α^2)^-1/4b.
 Since by our assumptions (1+α^2)^-1/4∈ C_0(G^0). It follows that 
    (1+(d+d^*)^2)^-1/4(1+α^2)^-1/4∈𝒦(C^*Λ_𝔄G^*)
Hence (1+(d+d^*)^2+c(α)^2)^-1 is compact as well.



Since [d+d^*,c(α)] is bounded, and 
    (1+(d+d^*+c(α))^2)^-1
       =(1-(1+(d+d^*+c(α))^2)^-1[d+d^*,c(α)])(1+(d+d^*)^2+c(α)^2)^-1,

it follows that (1+(d+d^*+c(α))^2)^-1 is compact.



 §.§ Completion

 
Let G⇉ G^0 be a Lie groupoid, g a complete Euclidean metric on 𝔄G and α∈Γ(𝔄^*G) a 1-form.

Let U be a saturated open subset of G^0, V:=U^c⊆ G^0 its complement, h:G^0→ [0,+∞[ a bounded smooth positive function such that  

  * for any x∈ V, α(x)≠ 0.

  * h^-1(0)=V 

  * dh/h_gα_g is bounded on U, where dh∈Γ(𝔄^*G) is the composition of the De Rham derivative with the anchor map.

Then 

  * if the pair g and α satisfy the hypotheses of <ref> on the Lie groupoid G, then pair g/h and α/h satisfy the hypotheses of <ref> on the Lie groupoid G_|U:=s^-1(U)=r^-1(U).

  * If for some ϵ, the set h^-1([0,ϵ]) is compact, then the converse of 1 holds.



The metric g/h is complete because h is bounded. This follows from the inclusion for any x∈ U, r>0 
    B_g/h(x,r)⊆ B_g(x,r/h_∞),

where B_g(x,r) and B_g/h(x,r) denotes the ball of radius r with center x  in G_x with respect to g and g/h.

Let us verify the equivalence 

  * First one has 
    d(α/h)_g/h=hd(α/h)_g=dα -dh/hα_g.
 Since dh/h_gα_g is bounded, it follows that d(α/h)_g/h is bounded if and only if dα_g is bounded.

  * Let X=α^#_g=(α/h)^#_g/h. The equivalence of the second condition follows from lemma <ref> and the following inequality 
    |dh(X)/h|≤dh/h_gX_g=dh/h_gα_g
Let G⇉ G^0 be  a Lie groupoid, h:G^0→ ]0,+∞[ a smooth function, g a Euclidean metric on 𝔄G and X∈Γ(𝔄G^0). If the function dh(X)/h is bounded, then the section ℒ_X+ℒ_X^*_g is bounded if and only if the section ℒ_X+ℒ_X^*_g/h is bounded.


Recall <ref>, and let x∈ G^0, h̃:=h∘ r:G_x→, α,β∈Γ(Λ^k_ T G_x^*) 
    ∫_G_x⟨ℒ_X α,β⟩_g/hd vol_g/h   =∫_G_xh̃^k⟨ℒ_X α,β⟩_g h̃^-n/2d vol_g
       =∫_G_x⟨α,ℒ_X^*(h̃^k-n/2β)⟩_g dvol_g
       =∫_G_x⟨α,h̃^n/2-kℒ_X^*(h̃^k-n/2β)⟩_g/hdvol_g/h.

It follows that 
    ℒ_X^*_g/h=h^n/2-kℒ_X^*_gh^k-n/2
 on Γ(Λ_^k𝔄G).

Notice that since the metrics g and g/h are conformal, one has 
    ℒ_X+ℒ_X^*_g/h_g=ℒ_X+ℒ_X^*_g/h_g/h∈ C^∞(G^0).
 Hence using the metric g or g/h is of no consequence on the boundness. It follows that to prove the proposition it suffices to prove that ℒ_X^*_g-ℒ_X^*_g/h_g is bounded. One has 
    ℒ_X^*_g-ℒ_X^*_g/h_g   =ℒ_X^*_g-h^n/2-kℒ_X^*_gh^k-n/2_g
       =ℒ_X-h^k-n/2ℒ_Xh^n/2-k_g
       =|h^k-n/2♮X(h^n/2-k)|=|(n/2-k-1)dh(X)/h|.

The lemma then follows.


  * The last equivalence follows from the identity 
    α/h_g/h^2=1/hα_g^2,
and the fact that α_g is non zero on V.


 suppose that V:=U^c is a smooth submanifold of G^0. This implies that for any x∈ V,  
    ♮(𝔄_xG)⊆ T_xV

If furthermore V is of codimension 1, and if h∈ C^∞(G^0) is a smooth function such that locally if G^0=^n, V=^n-1, h(x)=x_n^2 for some local coordinates, then the section of 𝔄G on U, dh/h extends to a smooth section on G^0. This follows from the fact that dh is zero on V.



§ WITTEN DEFORMATION



 §.§ Morse function

Let M be a closed manifold, f:M→ a Morse function. We denote by (f) the set of its critical points (a finite set), and by π_:(M,(f))→ [0,1], π_M:(M,(f))→ M the natural projections. By <Ref>, the following is naturally a Lie groupoid 
    G   =(M× M,(f)×(f))
       =M× M× ]0,1]⊔_a,b∈(f)T_aM× T_bM×{0}⇉(M,(f)),

whose algebroid is equal to (TM,(f)). 

Let g be a Riemannian metric on M. In <ref>, on 𝔄G=(TM,(f)) a Euclidean metric is defined which on M×{t} is equal to g/t^2 for t≠ 0 and the constant Riemannian metric g_a on T_aM×{0}. This metric is complete by the completeness of the metric g on M.



Let α be the 1-form given by <ref> 
    α=(df):(M,(f))→(T^*M,(f)).
 After identifying (T^*M,(f)) with (TM,(f))^*=𝔄G^* (see <Ref>), the form α is equal to df/t^2 on M×{t} for t≠ 0 and to d^2_af on T_aM×{0} for a∈(f).



Let us verify the condition of <Ref>. 
 
  * The form α is clearly closed.
 
  * On M×{t}, one has 
    α^#_g/t^2=df^#_g,
 where # is the musical isomorphism. Hence ℒ_α^# is independent of t. Since the Riemannian metric is multiplied by a scalar, it follows that ℒ_α^#^* doesn't depend on t as well. In other words, the section ℒ_α^#+ℒ_α^#^* on (M,(f)) is the pullback of the section ℒ_df^#_g+ℒ_df^#_g^* on M using the projection map π:(M,(f))→ M. Hence the norm of the section ℒ_α^#+ℒ_α^#^* is bounded by its boundness on M.


  * On M×{t}, one has 
    α_g/t^2=df/t^2_g/t^2=df_g/t
 and on T_aM×{0}, 
    α_g_a=d^2_af_g_a.
Hence α is a proper function on the space (M,(f)).


The operator d+d^*+c(α) acting on the C([0,1]) module C^*(Λ_(dπ_)^*) is a regular self adjoint operator with compact resolvent.


The manifold (M,(f)) gives naturally a Morita equivalence between the Lie groupoid (M× M,(f)×(f))⇉(M,(f)) and the trivial Lie groupoid [0,1]⇉ [0,1]. The corollary then follows from <Ref>.

Let d_t=e^-f/tde^f/t, Δ_t=(d_t+d_t^*)^2 be the Witten Laplacian acting on L^2(Λ_ T^*M). If 
    λ^p_1(t)≤λ^p_2(t)⋯
 denotes the spectrum of Δ_t  acting on p-forms, then the function 
    t→tλ^p_i(t)  if t≠ 0
    λ_i^p(0)  if t=0
 is continuous, where λ_i^p(0) is the i'th eigenvalue of Harmonic osccilator 
    ⊕_a∈(f)(d+d^*+c(d^2_a(f)))^2:⊕_a∈(f)L^2(T_aM,Λ_^pT_aM)→⊕_a∈(f) L^2(T_aM,Λ_^pT_aM),
 where L^2(T_aM,Λ_^p T_aM) is the set of all L^2 functions from T_aM to Λ_^p T_aM, d^2_af is the 1-differential form on T_aM, and c is the Clifford multiplication.


After normalizing the metric g/t^2, the operator (d+d^*+c(α))^2 on M×{t} is equal to t^2Δ_t^2. The corollary then follows from <Ref>.Let E be a C[0,1] module, L∈𝒦(E) a compact operator, μ_i(t) denote the singular value of the operator |L_t|. The function t→μ_i(t) is continuous.

 
By <cit.>, one has if T_1,T_2 are compact operators, then for every i∈, 
    |μ_i(T_1)-μ_i(T_2)|≤T_1-T_2.
It follows that if H is a Hilbert space and E=H⊗ C([0,1]) is a constant C([0,1])-module the result follows. For general modules the results follows from Kasparov stabilisation theorem <cit.>.


 <Ref> is false if L is only supposed to be in ℒ(E), and L_t is compact for every t. For example if E=C_0(]0,1]) and L the identity.



The calculation of the spectrum of the harmonic oscillator in <Ref> is a classical calculation. In particular we have If Q is a quadratic form on a real euclidean vector space V with signature (p,q) with p+q=n and 
    ξ_1≤…≤ξ_q<0< ξ_q+1≤…≤ξ_n,
 denote the eigenvalues of Q, then the spectrum of 
    (d+d^*+c(Q))^2:L^2(V,Λ_^kV^*)→ L^2(V,Λ_^kV^*)
 is the weighted set[the union is with multiplicity] 
    ∐_J⊆{1,…,n}:|J|=k
    α∈^n{∑_j∈ JΔ{1,…,q}|ξ_j|+∑_j=1^nα_j|ξ_j|}.

Here c(Q) means the Clifford multiplication by the 1-form associated to Q.






If C_i denotes the number of critical points of f, then for every k, 
    ∑_i=0^k(-1)^k-iC_i≥∑_i=0^k(-1)^k-i H^i(M,)



Multiplication by e^f is an isomorphism between the complex (Ω^*(M),d+df) and (Ω^*(M),d). The corollary follows from Hodge theory and <Ref>.



  * It is clear that the above proof verbatim works for Novikov Morse 1-forms.

  * We could have equally well used the groupoid (M× M,(f)). The main difference would be that C^*((M× M,(f))) is not Morita equivalent to C([0,1]) which makes the arguments slightly more complicated. One would need to use traces to obtain <ref> as done in <ref>.




 §.§ Morse-Bott functions

Let f:M→ be a Morse-Bott function. In this section it is simpler to work with _[-1,1] than _[0,1] to avoid difficulties with the boundary, see <ref>. To simplify the notation we will use  to denote _[-1,1]. As in <ref>, the groupoid 
    G   =(M× M,(f)×(f)) ⇉(M,(f))
       =(M,(f))×_[-1,1](M,(f))⇉(M,(f))
 will be used to describe a global Witten deformation. 

The Lie algebroid of G is (TM,T(f)). This vector bundle is isomorphic but not canonically to the vector bundle π^*TM. In particular a Riemannian metric on TM doesn't automatically give a Riemannian metric on (TM,T(f)). In the other hand we can use a natural compactification of the space G^0 and <ref>.

By <ref>, let  
    K=_r,s(M× M× [-1,1],(f)×(f)×{0})⇉(M× [-1,1],(f)×{0}).


One can immediately see that 
    K^0=M× [-1,1]\{0}⊔ M\(f)×{0}⊔ℙ(N^M_(f))⊔ N^M_(f)×{0}
 and 
    K   =M× M×  [-1,1]\{0}⊔ M\(f)× M\(f)×{0}
       ⊔ (N^M_(f)\{0}× N^M_(f)\{0})/^*⊔ N^M_(f)× N^M_(f)×{0}.
 It follows that (M,(f)) can be seen as a saturated open subset of K^0. Furthermore G=K_|(M,(f)).

The set 
    V=K^0\(M,(f))=M\(f)×{0}⊔ℙ(N^M_(f))=(M (f))
 is a submanifold of K^0 of codimension 1. Let h∈ C^∞(K^0) be as in <ref>, a smooth non-negative function such that h^-1(0)=V and locally h is the square of a distance function to V. The choice of h is unique up to multiplication by a positive smooth function on K^0.  Let (g_i)_i=1^k be a finite family of  smooth functions on M such that 

  * g_i((f))=0 and ∩_i g_i^-1(0)=(f).

  * for every x∈(f) the linear map T_xM/T_x(f)→^k given by X→ (dg_1(X),…,dg_i(X)) is injective.
The following function is an example of a function h:K^0→ 
    h=h(x,t)= t^2/∑_i g_i(x)^2+t^2  if  (x,t)∈ M×^*
    h(x,X,0)=1/∑ dg_i(X)^2 +1  if  (x,X,0)∈ N^M_(f)
    0   if not



On the manifold (M,(f)), one has the differential form 
    α=(df):(M,(f))→(T^*M,T^⊥(f))=(TM,T(f))^*.


The form hα extends to a section of 𝔄K^* furthermore it is nowhere zero on V.


This clearly doesn't depend on the choice of h. So it is enough to prove it for h given in <ref>.

Let U_i be an open cover of M, such that f is constant on U_i∩(f) for every i. It follows that (U_i×,U_i∩(f)×{0}) and (U_i,U_i∩(f)) is an open cover of (M×,(f)×{0}) and (M,(f)) respectively. Hence to prove the lemma it is enough to suppose that f is constant on (f). Let c denote f((f)). The following function ϕ:(M,(f))→ is well defined and smooth by results in <ref>
    ϕ=ϕ(x,t)=f(x)-c/t^2  if  (x,t)∈ M×^*
    ϕ(x,X,0)=1/2d^2f_x(X)  if  (x,X)∈ N^M_(f).

Furthermore it is straightforward to see that dϕ=α∈Γ(𝔄G^0), where d as usual denotes the composition of the De Rham derivative with the anchor map.

One has 
    hα=hdϕ=d(hϕ)-hϕdh/h.
 By <ref>, dh/h extends to a smooth section on K^0. 

The function hϕ extends to a smooth function on K^0. On M×^*, one has 
    hϕ(x,t)=f(x)-c/∑_ig_i^2(x)+t^2.
 This function clearly extends smoothly to (M×,(f)×{0}). This proves the first part concerning the extension of hα to K^0.

It is immediate to see that the extension of hα to K^0 is equal to df/∑ g_i^2 on M\(f)×{0} which is clearly nonzero for every point in M\(f)×{0}.

Finally we restrict our attention to the form hα on 
    (N^M_(f)\{0}× N^M_(f)\{0})/^*⊔ N^M_(f)× N^M_(f)   ⇉ℙ(N^M_(f)⊕)
       = ℙ(N^M_(f))⊔ N^M_(f).


By Morse-Bott lemma, we can suppose that M=(f)× L, with L a vector space and f=Q a quadratic form on L. The Lie algebroid of the groupoid  
    ((f)×(f))×((L\{0}× L\{0})/^*⊔ L× L)
    ⇉(f)×ℙ(L⊕)= (f)×(ℙ(L)⊔ L)
 is equal to T(f)×⊔_l∈ℙ(L⊕)(l,L). It is immediate to see that the value of the extension of hα at (X,T)∈ T(f)×⊔_ l∈ℙ(L)(l,TL) is given by 
    hα(X,T)=1/∑ dg_i(a)^2Q(a,T(a)),
 where a is any nonzero element in l. Hence, it is clear that hα is nowhere zero on ℙ(N^M_(f)). 

Let g be a euclidean metric on 𝔄K. The restriction to (M,(f)) of the metric g/h together with the form α satisfies the properties of <ref>. In particular d+d^*+c(α) is regular with compact resolvent on C^*(G)-C^*-modules


The metric g is complete by <ref>. It follows from <ref> that dh_g/h extends to a smooth function on K^0, hence it is bounded by compactness of K^0. And by <ref>, hα_g is also bounded. Since K^0 is compact, the metric g with the form hα trivially satisfy the conditions of <ref>. It follows from <ref> that g/h and α satisfy the conditions of <ref>. The result then follows.








 





The groupoid G is Morita equivalent to [-1,1]. Hence C^*(G) can be replaced by C([-1,1]) in the previous proposition.

 The groupoid (M× M,(f)) can also be used to obtain a global Witten deformation for Morse-Bott function. For that groupoid the algebroid (TM,(f)) is canonically isomorphic to π^*(TM), hence there are no difficulties with the choice of the metric. On the other hand the fiber of its C^*-algebra at 0 is canonically Morita equivalent to C_0(T^*(f)). This makes it more difficult to obtain precise statements about eigenvalues.
 



 §.§ The second approach

The Lie algebroid of G_2 is (TM,(f)). It is isomorphic to π^*TM by <ref>. It follows that a complete Riemannian metric g on TM induces a complete euclidean metric on 𝔄G_2. The 1-form 
    α=(df):(M,(f))→(TM,(f))

is well defined.

The following is a corollary of <ref> following the same reasoning as the proof of <ref> The operator d+d^*+c(α) acting on C^*G_2-C^*-module C^*λ_𝔄^*G_2.

It follows from <ref>.2 that the fiber 𝒩^M× M_(f) at t=0 of the groupoid G_2 is equal to 
    {(x,X,A,Y):x∈(f), A∈ T_xM,     X,Y∈T_xM/T_x(f),
       Y-X=A T_x(f) }.

Up to the choice of a supplemntary subbundle to T(f) inside TM, the fiber at 0 is equal to the union over x∈(f) of the pair groupoid (𝒩^M_(f))_x×(𝒩^M_(f))_x and the abelian group T_x(f).

Hence the C^*-algebra of the fiber of G_2 at 0 is (not canonically) Morita equivalent to C_0(T^*(f)).




For t∈ [0,1], we have a trace τ_t on the C^*-algebra C^*G_2.

For t≠ 0, it is given by the formula for f∈ C^∞(G_2), 
    τ_t(f)=1/t^(M)∫_x∈ Mf(x,x,t)dg.


For t=0, it is given by the following formula for f∈ C^∞_c(G_2) 
    τ_0(f)=∫_x∈(f)∫_T_xM/T_x(f)f(x,X,0,X,0)dg_x dg
 where the integral is against the volume coming from the Riemannian metric g. 

  * The previous two formulas are much more natural if written using densities because then no choice of g would appear.

  * The trace τ_0 on the C^*-algebra C^*𝒩^M× M_(f) corresponds under the Morita equivalence to the natural trace on C_0(T^*(f)) coming from the Liouville measure on the cotangent bundle.


Let f∈ C^∞_c(G_2). The function t→τ_t(f) is continuous.


First the traces τ_t only depend on the restriction of f the space of objects (M,(f)). By writing (M,(f)) as the union of local charts and using a partition of unity, it suffices to prove this proposition for functions f∈ C^∞_c(U) for a family of charts covering (M,(f)). The proof is then a computation depending on the type of chart.

  * If the chart is U=M×^*, in this case the continuity is evident.

  * If C is a connected component of (f), then Morse-Bott lemma implies that there exists a local chart V in M around C such that V⊆^n×^m and 
    f(x,y)=y_1^2+… +y^2_i-…-y_n^2,
 where i is the index of f at C. The submanifold (f) is then identified with ^n×{0}∩ V. A local chart (M,(f)) is then U={(x,t):t≠ 0 ,x/t∈ V}⊔^n×^m×{0}. A function f is smooth with compact support on U is equivalent to the existence of a smooth function g with compact support on ^n×^m×  such that 
    f(x,y,t)=g(x,y/t,t),  f(x,X,0)=g(x,X,0).

It then follows 





§ WITTEN DEFORMATION ON FOLIATIONS

Let F⊆ TM be a subbundle (not necessarily integrable) of the tangent bundle of a closed manifold M, f:M→ a smooth function. We are interested in the set 
    _F(f):={x∈ M:df_x(F_x)=0}.




Let x_0∈_F(f), X∈Γ^∞(TM), Y∈Γ^∞(F). One defines 
    d^2_x_0f(X,Y):=(XYf)(x_0).
  The number d^2_x_0f(X,Y) only depends on X(x_0) and Y(x_0). In other words d^2_x_0f:T_x_0M× F_x_0→ is a well defined bilinear form. 

This is clear for X. Let Y'∈Γ^∞(F) be another section such that Y'(x_0)=Y(x_0). It follows that Y'-Y could be written as the sum of elements of the form gZ, where g:M→ is a function that vanishes at x_0, and Z∈Γ^∞(F). Hence 
    X(Y+gZ)f(x_0)=XYf(x_0)+X(g)(x_0)Zf(x_0)+g(x_0)XZf(x_0)=XYf(x_0),

where the second term vanishes because Zf(x_0)=0 and the third because g(x_0)=0.

Let Z=F^⊥⊆ T^*M. The section df:M→ T^*M is transverse to Z if and only if for every x∈_F(f), the bilinear map d^2_xf is of maximal rank, that is the induced linear map d^2_xf:T_xM→ F_x^* is surjective. Furthermore if this is the case, then _F(f) is a smooth manifold whose tangent bundle is T_F(f)=(d^2_xf).


Let h:T^*M→ F^* be the map which restricts a linear map on TM to F. It is clear that h∘ df is equal to the longitudinal derivative of f. Furthermore the inverse image by h of the zero section is equal to Z. It follows that df:M→ T^*M is transverse to Z if and only if the longitudinal derivative d_Ff:M→ F^* is transverse to the zero section.

Let c(t) be a path in M, with c(0) a point such that d_Ff_c(0)=0. The derivative of the function d_Ff_c(t):→ F^* at zero is then a linear map on T_c(0)F^*=T_c(0)M⊕ F^*_c(0). The value of this linear map at a vector in F_c(0) can be computed using a section of X of F. It follows that the derivative of d_Ff_c(t) at time 0 applied to X(c(0)) is equal to d/dt_|t=0Xf(c(t)). This is precisely the definition of d^2_c(0)f(c'(0),X(0)). The proposition is then clear.

By Thom's multijet transversality theorem (see <cit.>), the transversality condition of <Ref> is satisfied generically. We now fix such a function f, and suppose that F is integrable.


 The manifold _F(f) is of complementary dimension to F. It is transverse to F at a point x∈_F(f) if and only if the critical point x of the function f|l_x is non degenerate, where l_x is the leaf containing x. In particular, if the foliation doesn't admit a closed transversal, then there exist no smooth function which is Morse on each leaf.


Let 
    𝒢(M,F)={(γ(0),[γ],γ(1))|γ:[0,1]→ M,γ'(t)∈ F ∀ t∈ [0,1]}
 be the holonomy groupoid of F, where [γ] denotes the holonomy class of γ.


Let 
    G=(𝒢(M,F),_F(f))⇉(M,_F(f))
 be the deformation of foliation groupoid 𝒢(M,F) along the submanifold of its units _F(f). In this section and the next one all deformation to the normal cone are fibered on [0,1]. By <Ref>, this is a Lie groupoid whose Lie algebroid is equal to (F,_F(f)). Recall that by <Ref>, 
    𝒩^𝒢(M,F)__F(f)={(X,A,Y):X,Y∈𝒩^M__F(f),A∈ F,  Y-X=A T_F(f) }.




Let g be a Euclidean metric on F. The Lie algebroid (F,_F(f)) admits then a Euclidean metric by <Ref>. On M×{t}, it is equal to g/t^2, and on 𝒩^M__F(f) it is equal to g_|_F(f). This metric is complete because for t≠ 0, the metric g is complete on each leaf, and for t=0, it is complete by the description of 𝒩^𝒢(M,F)__F(f) given above.




 Let 
    α=(d_Ff):(M,_F(f))→(F^*,_F(f)),
 where d_F is the longitudinal derivative. The map α is regarded as a section of 
    𝔄G^*=(F,_F(f))^*=(F^*,_F(f)).
 See <Ref> for the last equality. On M×{t}, α=d_Ff/t^2, and on 𝒩^M__F(f)×{0} it is equal to d^2f.





Let us show that the hypotheses of <Ref> hold.


  * It is clear that the form α is closed.

  * On M×{t}, one has 
    α^#_g/t^2=d_Ff^#_g,
 where # is the musical isomorphism. Hence ℒ_α^# is independent of t. Since the Riemannian metric is multiplied by a scalar, it follows that ℒ_α^#^* doesn't depend on t as well. In other words, the section ℒ_α^#+ℒ_α^#^* on (M,_F(f)) is the pullback of the section ℒ_d_Ff^#_g+ℒ_d_Ff^#_g^* on M using the projection map π:(M,_F(f))→ M. Hence the norm of the section ℒ_α^#+ℒ_α^#^* is bounded by its boundness on M.

  * On M×{t}, one has 
    α_g/t^2=d_Ff/t^2_g/t^2=d_Ff_g/t
 and on N^M__F(f)×{0}, 
    α_g=d^2f_g_a.
 Transversality of d_Ff to the zero section is then equivalent to properness of α as a function on (M,_F(f)).

The closure of the operator d+d^*+c(α) acting on C^*(Λ_𝔄G^*) is a regular self adjoint operator with a compact resolvent.

By <Ref>, the operator d+d^*+c(α) defines an element in ^0(,C^*G). By regarding the evaluation  at 0 and at 1 of the previous element, one deduces:
The Euler characteristic e(F) of ℱ as an element in (,C^*(𝒢(M,F))) can be represented by the element d+d^*+c(α) in ^0(,C^*𝒩^𝒢(M,F)__F(f)). More precisely, 
    e(F)=Ind^𝒢(M,F)__F(f)([d+d^*+c(α)]),
where 
    Ind^𝒢(M,F)__F(f):K_0(C^*(N^𝒢(M,F)__F(f)))→ K_0(C^*𝒢(M,F))
 is defined in <cit.>. It is also defined in <cit.> in a slightly different formalism where it is denoted by i_!.

It is clear that one can replace f with a 1-form α∈Γ(F^*) such that d_Fα=0 and α:M→ F^* is transverse to the zero section.




 §.§ Quantitative result

In this section, we will show that when a transverse measure is used to obtain a quantitative result from <ref>, one obtains Connes-Fack Morse inequalities. More precisely, we suppose that M is a compact smooth manifold, F a foliation of dimension p and codimension q equipped with a holonomy invariant transverse measure v. 
We refer to <cit.> for more details on transverse measure from the point of view of Von Neumann algebras. As in <ref>, a Euclidean metric g on F is chosen. The measure ν together with g give rise to a measure on M that will be denoted μ=g×ν.

 A smooth function f:M→ is called nice if 
 

  * all longitudinal critical points of f are either Morse or of Birth-death type. See <cit.> for the definition of Birth-death singularity.

  * the set of leaves on which f isn't Morse is of ν-measure zero.


It is shown in <cit.> and <cit.> that any smooth function can be C^0 approximated by nice functions. Furthermore if (F)≤(F) then the approximation can be supposed C^1. In what follows we suppose that f is a nice function.

 Morse lemma for a birth-death critical point x_0 states the following; there exists a foliated chart where x_0=0∈ U⊆^p×^q such that 
    f(x,y)=f(0,y)+1/3x_1^3-y_1x_1+ x_2^2…+x^2_p-i-x^2_p-i+1-… -x_p^2,
 and _F(f)={(x,y):y_1=x_1^2,x_2,…,x_p=0}. Here i is called the index of x_0.  We will suppose that g is locally Euclidean in such foliated charts. This is a nonrestrictive condition since everything below is about densities and not Euclidean metrics. The eigenvalues of the longitudinal Hessian of f at a point (x,y)∈_F(f) are equal to 2 with multiplicity p-1-i, -2 with multiplicity i, and 2x_1 with multiplicity 1.



Such local coordinates will be called nice local charts. For convenience we will suppose that U contains [-1,1]^n.



 We denote by G=(𝒢(M,F),_F(f)). By <cit.>, for t≠ 0, ν defines a semi-finite lower semi-continuous trace on C^*(G) by the formula, 
    τ_t(ϕ)=1/t^(p)∫_Mϕ(·,t) dμ,
 where ϕ∈ C_c^∞(G), and the integral is on the subspace of the space of objects M×{t}⊆(M,_F(f)). Notice that the factor 1/t^(p) comes from the fact that we work in this article with functions instead of densities. 
 


 For t=0, let _F(f)⊆_F(f) be the open dense subset of points where _F(f) is transversal to the foliation. On _F(f), the vector bundle F_|_F(f) is transverse to T_F(f). We identify the vector bundle N^M__F(f) with F_|_F(f). The trace τ_0 on C_c^∞(𝒩^𝒢(M,F)__F(f)) is defined by first restricting a function ϕ∈ C_c^∞(𝒩^𝒢(M,F)__F(f)) to the open subset 𝒩^M__F(f) of the space of objects 𝒩^M__F(f), then using the formula 
    τ_0(ϕ)=∫_x∈_F(f)∫_F_xϕ dg_xdν.
 Equivalently the restriction of C^*(𝒩^𝒢(M,F)__F(f)) to _F(f) is equal to a fiber bundle over _F(f) with the fiber over each point x∈_F(f) equal to 𝒦(L^2(F_x)). The trace is then the canonical trace on each fiber followed with the trace against the transverse measure on the base.
 
 This trace extends to the C^*-algebra. On the other hand this trace is not necessarily finite C^∞_c(G). In fact in <ref>, we prove that τ_0 is finite if and only if for every birth-death singularity x_0, and nice local chart around x_0, the following integral is finite 
    ∫_{y∈ [-1,1]^q:y_1>0}1/√(y_1)<+∞.

  
Let us remark that in a nice local chart the singular values of the longitudinal Hessian of f at a point (x,y)∈_F(f) are equal to 2 with multiplicity p-1 and 2√(y_1) with multiplicity 1. It follows that the integrability of the function 1/√(y_1) around every birth-death singularity is equivalent to the finiteness of the integral 
   
    ∫__F(f)1/E_p(d^2_Ff)dν<∞,
 where E_p(d^2_Ff) is the pth singular value of the longitudinal Hessian (the restriction of d^2f defined in <ref> to F× F). 



  * The trace τ_0 is finite if and only if  ∫__F(f)1/E_p(d^2_Ff)dν<∞.

  * Suppose that ∫__F(f)1/E_p(d^2_Ff)dν<∞, and let ϕ be a smooth function with compact support on (𝒢(M,F),_F(f)). The function t→τ_t(ϕ) is continuous .



By a partition of unity argument, it suffices to verify the proposition around a local neighbourhood of every critical point x∈_F(f). The case of Morse singularity is straightforward. We will only do the case of a birth-death singularity. Let x_0∈_F(f) be a birth-death singularity. In a nice chart around x_0, we have 
    f(x,y)=f(0,y)+1/3x_1^3-y_1x_1+ x_2^2…+x^2_p-i-x^2_p-i+1-… -x_p^2
 and _F(f)={(x,y):y_1=x_1^2,x_2,…,x_p=0}.  We can assume that ϕ∈ C^∞_c((^p×^q, _F(f))) is a smooth function. Being smooth with compact support is equivalent to the existence of a smooth function with compact support ϕ̃∈ C^∞_c(^p×^q×) such that for t≠ 0, we have 
    ϕ(x,y,t)=ϕ̃(x_1,x_2/t,…,x_p/t,y_1-x_1^2/t,y_2,…,y_q,t).
  

Notice that ν({0}×^q-1)=0 by the conditions on f.

In the following equation we will use the notation x',y' for the vectors (x_2,…,x_p), (y_1,…,y_q). For t≠ 0, one has

    τ_t(ϕ)   =∫_{y∈^q:y_1>0}∫_^p1/t^pϕ(x,y,t)dxdν(y)
       =∫_{y∈^q:y_1>0}∫_^p1/t^pϕ̃(x_1,x'/t,y_1-x_1^2/t,y',t)dxdν(y)
       =∫_{y∈^q:y_1>0}∫_^p1/tϕ̃(x_1,x',y_1-x_1^2/t,y',t)dxdν(y)

In the last identity we divide the integral into the sum of the integral over the region where x_1≥ 0 and the integral over the complement and then used the change of variables z=y_1-x_1^2/t, to obtain 
    τ_t(ϕ)   =∫_{y∈^q:y_1>0}∫_^p-1∫_-∞^y_1/t1/2√(y_1-tz)ϕ̃(√(y_1-tz),x',z,y',t)dzdx'dν(y)
       +∫_{y∈^q:y_1>0}∫_^p-1∫_-∞^y_1/t1/2√(y_1-tz)ϕ̃(-√(y_1-tz),x',z,y',t)dzdx'dν(y)


Similarly one checks as well that 
    τ_0(ϕ)   =∫_{y∈^q:y_1>0}∫_^p1/2√(y_1)ϕ̃(√(y_1),x',z,y',0)dx'dzdν(y)
       +∫_{y∈^q:y_1>0}∫_^p1/2√(y_1)ϕ̃(-√(y_1),x',z,y',0)dx'dzdν(y).

The first part of the proposition is then clear and the second follows from the dominated convergence theorem.

For any nice function f:M→ and ϵ>0, there exists a nice function h:M→ such that 

  * ∫__F(h)1/E_p(d^2_Fh)dν<+∞.

  * |ν(^i_F(h))-ν(^i_F(f))|≤ϵ, where ^i_F(f), ^i_F(h) is the set of Morse critical points of index i of f and h respectively.



Let f be a nice function. We will modify the function f in local charts around birth-death singularities to make the condition  ∫__F(f)1/E_p(d^2_Ff)dν<∞ hold.

In a nice local chart of a critical point of birth-death singularity type, one has 
    ∫_-1^1∫_-x^11/√(y+x)dν(y)dx   =∫_-1^1∫_-y^11/√(y+x)dxdν(y)
       =∫_-1^12√(y+1)dν(y)<+∞.

Equally well ∑_x∈ [-1,1]ν({x}× [-1,1]^q-1)≤ν([-1,1]^q)<+∞. It follows that for almost all ϵ∈ [-1,1] with respect to the Lebesgue measure, the function 1/√(y-ϵ) is ν-integrable and ν({ϵ}× [-1,1]^q-1)=0. 
If one takes such ϵ, and ϕ:^n→^n any diffeomorphism which is equal to the identity outside [-1,1]^n and which is equal on [-1/2,1/2]^n to the diffeomorphism 
    (y,x)→ (y_1-ϵ,y_2,…,x),
 then the function f∘ϕ is a nice function such that ∫__F(f∘ϕ)1/E_p(d^2_F(f∘ϕ))dν is finite around the critical point. Doing the above procedure for a finite cover of birth-death singularities finishes the proof.

Let f be a nice function, c_i=ν(_ℱ^i(f)) the ν-measure of critical points of index i, and β_i the ν-measure of (Δ^i). One has 
    ∑_i=0^k(-1)^k-ic_i≥∑_i=0^k(-1)^k-iβ_i



By  <ref>, it is enough to prove the corollary under the extra hypothesis that ∫__F(f)1/E_p(d^2f_x)dμ(x) is finite. Let ϕ∈ C^0([0,+∞[) be a decreasing continuous function with compact support which is equal to 1 on the interval [0,1]. Algebraic Morse inequalities imply that 
    ∑_i=0^k(-1)^k-iτ_t(ϕ(Δ^i_t,f))≥∑_i=0^k(-1)^k-iτ_t((Δ^i_t,f))
 for any t≠ 0, where Δ^i_t,f is the Witten deformation constructed in <ref> at time t acting on forms of degree i. Since the multiplication e^f/t gives an isomorphism from the deformed De Rham complex at time t with the De Rham complex. It follows that τ_t((Δ^i_t,f))=β_i.

 It follows from <ref>, that <ref> can be applied to the traces τ_t. Hence <ref> together with <ref> imply that t→τ_t(ϕ(Δ^i_t,f)) is a continuous function. It follows that 
    ∑_i=0^k(-1)^k-iτ_0(ϕ(Δ^i_0,f))≥∑_i=0^k(-1)^k-iβ_i.
 Applying the previous inequality to ϕ(·/ϵ), and taking the limit when ϵ goes to zero one obtains 
    ∑_i=0^k(-1)^k-iτ_t((Δ^i_0,f)))≥∑_i=0^k(-1)^k-iβ_i.
 By definition one has τ_t((Δ^i_0,f))=c_i.
 
 
 
  * Notice that the above argument works equally well with a longitudinally Novikov 1-form, a 1-closed which is longitudinally closed and which locally is the longitudinal derivative of a nice function. The only difference between the Novikov 1-form case and the above is that one would obtain the following inequalities Let α be a longitudinally nice Novikov 1-form, c_i=ν(_ℱ^i(α)) the ν-measure of critical points of index i, and β_i(t) the ν-measure of (Δ_t^i) where Δ_t=(d+d^*+tc(α))^2. 

One has 
    ∑_i=0^k(-1)^k-ic_i≥lim sup_t→ 0∑_i=0^k(-1)^k-iβ_i(t)


 
In the classical case of a compact manifold the spectrum is discrete and β_i(t) is integar valued which implies that β_i(t) is constant for t small enough. The limit is then called the Novikov betti number of α. In the foliation case, I don't see any reason for β_i(t) to converge as t→ 0.
 
 



§ CONTINUOUS FAMILY OF TRACES

Recall that a lower-semi continuous trace on a C^*-algebra A is a lower semi-continuous function τ:A^+→ [0,+∞] such that 

  * for every a,b∈ A^+, λ,μ∈^+, τ(λ a+μ b)=λτ(a)+μτ(b).

  * for every unitary u∈ A, τ(uau^*)=τ(a).


It is a classical theorem (see for example <cit.>) that for a trace τ, the span of elements {x∈ A^+: τ(a)<+∞} is an ideal on which τ admits a unique linear extension. In  <cit.>, the following inequality is also proved 
    |τ(xy)|≤τ(|x|)y∀ x,y∈ A.


For each t∈ [0,1], let τ_t:A^+→ [0,+∞] be a lower semi-continuous trace, 𝒜⊆ A  a dense *-subalgebra such that for every x∈𝒜, t→τ_t(x) is a finite continuous function. Then






  * if a∈ A such that sup_t∈ [0,1]τ_t(|a|)<+∞, then for every b∈ A, t→τ_t(ab) is a continuous function.

  * If a∈ A^+, f∈ C^0([0,+∞[) a continuous function which vanishes on a neighbourhood of 0, then t→τ_t(f(a)) is a finite continuous function.

It follows from 1, that if c∈ A, then the function t→τ_t(f(a)c) is continuous.


  * If a∈ A^+, then t→τ_t(a) is lower semi-continuous.
 





 


 


  * By writing b=b_1+b_2+ib_3-ib_4 with b_i positive, we can suppose that b is positive. Let a_n, b_n be sequence of elements of 𝒜 that converge to a and b respectively. Furthermore we can suppose that b_n is positive for all n. Since 𝒜 is an algebra, t→τ_t(a_nb_m) is a finite continuous for every n,m. It follows from the inequality 
    |τ_t(a_nb_m)-τ_t(ab_m)|≤τ_t(b_m)a_n-a
 and the fact that t→τ_t(b_m) is a finite continuous function that as n→∞, t→τ_t(a_nb_m) converges uniformly to t→τ_t(ab_m). Hence t→τ_t(ab_m) is a finite continuous function for every m. It follows from the inequality 
    |τ_t(ab_m)-τ_t(ab)|≤τ_t(|a|)b_m-b≤sup_t∈ [0,1]τ_t(|a|)b_m-b
 that t→τ_t(ab_m) converges uniformly to t→τ_t(a b). Hence t→τ_t(ab) is a finite continuous function.

  * We will use notation from <cit.>. In particular, for a trace τ, a∈ A^+ we will denote by 
    E_s^τ(a):=inf{α∈ ]0,+∞[,τ(P_]α,+∞[(a)≤ s}
 the s-number of a with respect to the trace τ, where P is the spectral projection.

It is proved in <cit.>, that 
    |E_s^τ(a)-E_s^τ(b)|≤a-b, τ(f(a))=∫_0^∞ f(E_s^τ(a))ds
 for any continuous function f∈ C^0([0,+∞[), a,b∈ A^+. 





Let α>0, ϕ_α(x)=max(x-α,0)∈ C^0([0,+∞[). If b∈𝒜^+ such that a-b≤α. Since ϕ_α is increasing and E_s^τ_t(a)≤ E_s^τ_t(b)+α, one has  
    τ_t(ϕ_α(a))   =∫_0^+∞ϕ_α(E_s^τ_t(a))ds
       ≤∫_0^+∞ϕ_α(E_s^τ_t(b)+α)ds
       =∫_0^+∞E_s^τ_t(b)ds=τ_t(b)

It follows that sup_t∈ [0,1]τ_t(ϕ_α(a))<+∞.

Let f∈ C^0([0,+∞[) be a continuous function which vanishes on [0,2α[. One can write f=ϕ_αg with g a continuous function such that g(0)=0. It follows from 1 and the above that t→τ_t(f(a)) is continuous.







  * This follows from 2 and the monotone convergence theorem (see  <cit.>) by writing the function f(x)=x as the supremum of functions satisfying the conditions of 3.


Part 2 of <ref> equivalently says that for every x in the Pedersen ideal [Smallest dense two-sided ideal of A.], t→τ_t(x) is a finite continuous function.

 
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