Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds
Masaki Taniguchi
math.GT







Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds
    Masaki Taniguchi
    December 30, 2023
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We construct an obstruction to the existence of embeddings of a homology 3-sphere into a homology S^3× S^1 under some cohomological condition. The obstruction are defined as an element in the filtered version of the instanton Floer cohomology due to <cit.>. We make use of the -fold covering space of homology S^3× S^1 and the instantons on it.





§ INTRODUCTION

 There are two typical studies of gauge theory for 4-manifolds with periodic end by C.H.Taubes <cit.> and  J.Lin <cit.>. They gave a sufficient condition to exist a natural compactification of the instanton and the Seiberg-Witten moduli spaces for such non-compact 4-manifolds. The condition of Taubes is the non-existence of non-trivial SU(2) flat connection of some segment of the end. The condition of Lin is the existence of a positive scalar curvature metric on the segment of the end. As an application of the existence of the compactification, Taubes showed the existence of uncountable family of exotic ^4 and Lin constructed an obstruction to the existence of a positive scalar curvature metric.
 
 
 
 In this paper, we also give a similar sufficient condition for the instanton moduli spaces. The condition is an uniform bound on the L^2-norm of curvature of instantons. When we bound the L^2-norm of curvature, we use an  invariant which is a generalization of the Chern-Simons functional. Under this condition, we prove a compactness theorem (Theorem <ref>). 
 
 
 As the main theorem of this paper, we construct an obstruction of the existence of embeddings of a homology 3-sphere into a homology S^3× S^1 with some cohomological condition (Theorem <ref>). 
 
To formulate the obstruction, we need a variant of the instanton Floer cohomology. The variant is the filtered instanton Floer cohomology HF^i_r whose filtration was essentially considered by R.Fintushel-R.Stern in <cit.>. The obstruction is an element of the filtered instanton cohomology. We denote the element by [θ^r]. The element [θ^r]∈ HF^1_r is a filtered version of [θ] which was already defined by S.K.Donaldson <cit.> and K.Frøyshov <cit.>. The class [θ] is defined by counting the gradient lines of the Chern-Simons functional which converge to the trivial flat connection. In order to show [θ^r] is actually an obstruction of embeddings, we count the number of the end of the 1-dimensional instanton moduli space for 4-manifold which has both of cylindrical and periodic end. For the couting, we use the compactness theorem (Theorem <ref>).


This paper is organized as follows. In Section <ref>, we give a precise formulation of our main theorem (Theorem <ref>). In Section <ref>, we prepare several notations and constructions which are used in the rest of this paper.
In particular, we introduce the filtered instanton Floer homology HF^r_i and the obstruction class [θ^r]. We also review Fredholm, moduli theory for 4-manifolds with periodic end. In Section <ref>, we generalize the Chern-Simons functional and introduce the invariants Q^i_X. In Section 5, we prove the compactness theorem(Theorem <ref>). We use Q^i_X to control the L^2-norm of curvature. In Section 6, we deal with technical arguments about the transversality and the orientation for the instanton moduli spaces for 4-manifolds with periodic end. In Section 7, we prove Theorem <ref>.
 
 
 Acknowledgements:
The author is grateful to Mikio Furuta for his suggestions. The author would like to thank Hokuto Konno for useful conversations.


















§ MAIN THEOREM

Let X be a homology S^3× S^1, i.e. , X is a closed 4-manifold equipped with an isomorphism ϕ:H_*(X,) H_*(S^3× S^1,) in this paper. Then X has an orientation induced by the standard orientation of S^3× S^1 and ϕ.  Let Y be an oriented homology S^3. We construct an obstruction of embeddings f of Y into X satisfying  f_*[Y]=1∈ H_3(X,) as an element in the filtered instanton Floer cohomology. We use information of the compactness of the instanton moduli spaces for periodic-end 4-manifold in a crucial step of our construction.  In order to formulate our main theorem, we need to prepare several notations.


For any manifold Z, we denote by P_Z the product SU(2) bundle. The product connection on P_Z is written by θ.

    (Z):= {SU(2)-connections on P_Z},


    ^flat(Z):={SU(2)-flat connections on P_Z}⊂(Y),


    (Z):= (Z) /_0(Z,SU(2)),


    R(Z):=^flat /_0(Z,SU(2))⊂(Z),

and

    R(Z):= ^flat(Z)/(Z,SU(2)),

where _0(Z,SU(2)) is a set of smooth functions with mapping degree 0. When Z is equal to Y, the Chern-Simons functional cs_Y:(Y) is defined by 

    cs_Y(a):=1/8π^2∫_Y Tr(a∧ da +2/3a∧ a∧ a).

It is known that cs_Y decends to a map (Y), which we denote by the same notation cs_Y.

We denote the number of elements in R(Y) by l_Y. If R(Y) is not a finite set, we set l_Y=∞.

We will use the following assumption on Y in our main theorem(Theorem <ref>).

All SU(2) flat connections on Y are non-degenerate, i.e. the first cohomology group of the next twisted de Rham complex:

    0 ^0(Y)⊗d_a^1(Y)⊗d_a^2(Y)⊗d_a^3⊗ 0

vanishes for [a] ∈ R(Y).


All flat connections on the Brieskorn homology 3-sphere Σ(p,q,r) are non-degenerate.(<cit.>)

Under Assumption <ref>, l_Y is finite (<cit.>).


In this paper without the use of Assumption <ref>, we will introduce the following invariants:


  * HF^i_r(Y) for Y and r ∈ (∖ cs_Y(R(Y)))∪{∞} in Definition <ref>  satisfying HF^i_∞(Y)=HF^i(Y),

  * [θ^r] ∈ HF^1_r(Y) for Y and r ∈ (∖ cs_Y(R(Y)))∪{∞} in Definition <ref> satisfying [θ^∞]=[θ] ∈ HF^1(Y), and

  * Q^i_X ∈_≥ 0∪{∞} for i ∈ and X in Definition <ref> (When X is homotopy equivalent to S^3× S^1, Q^i_X=∞ for all i ∈).





Our main theorem is:
Under Assumption <ref>, if there exists an embedding f of Y into X with f_*[Y]=1∈ H_3(X,)  then [θ^r] vanishes for any r∈ [0,min{Q^2l_Y+3_X, 1}] ∩ (∖ cs_Y(R(Y))∪{∞}) 

In particular, if there exists an element 

    r∈ [0,min{Q^2l_Y+3_X,1}] ∩ (∖ cs_Y(R(Y))∪{∞})

satisfying 0≠ [θ^r], Theorem <ref> implies that there is no embedding from Y to X with f_*[Y]=1 ∈ H_3(X,).

Let X be a closed 4-manifold which is homotopy equivalent to S^3× S^1.
There is no embedding f of Σ(2,3,6k-1) into X satisfying f_*[Σ(2,3,6k-1)] =1∈ H_3(X,) for a positive integer k satisfying 1≤ k ≤ 12.
 
The proof of Example <ref> is given in the end of Subsection <ref>.



§ PRELIMINARIES

In this section, we review the (filtered) instanton Floer theory and moduli theory on the periodic end 4-manifolds.

 §.§ Holonomy perturbation(1)

In this subsection we review classes of perturbations which were considered in <cit.>, <cit.> to define the instanton Floer homology.


Let Y and P_Y be as in Section <ref>. We fix a Riemannian metric g_Y on Y. We define the set of embeddings from solid tori to Y by

    _d:= { (f_i)_1≤ i ≤ d :S^1× D^2 Y  | f_i: orientation preserving embedding }.

Fix a two form d𝒮 on D^2 supported in the interior of D^2 with ∫_D^2d𝒮=1. We denote by C^l(SU(2)^d,)_ the adjoint invariant C^l-class functions from SU(2)^d to  and define 

    ∏(Y):= ⋃_d ∈_d× C^l(SU(2)^d,)_.


We use the following notation,

    ^*(Y):={[a]∈(Y) | a is an irreducible connection} ,

where (Y) is defined in Section <ref>.
For π = (f,h)∈∏(Y), the perturbed Chern-Simons functional cs_Y,π:^*(Y)  is defined by

    cs_Y,π(a)= cs_Y(a)+ ∫_x ∈ D^2 h((a)_f_1(-,x),… ,(a)_f_d(-,x))d𝒮,

where (a)_f_i(-,x) is the holonomy around the loop t ↦ f_i(t,x) for each i ∈{1,…,d}. If we identify  with its dual by the Killing form, the derivative of h_i=pr_i^*h is a Lie algebra valued 1-form over SU(2) for h ∈ C^l(SU(2)^d,)_. Using the value of holonomy for the loops {f_i(x,t)| t∈ S^1}, we obtain a section _f_i(t,x)(a) of the bundle P over f_i. Sending the section _f_i(t,x)(a) by the bundle map induced by h_i': P P, we obtain a section h_i'(_f_i(t,x)(a)) of P over f_i. 
We now describe the gradient-line equation of cs_Y,π with respect to L^2 metric :

    ∂/∂ t a_t=_a cs_Y,π = *_g_Y(F(a_t)+∑_1≤ i ≤ d h'_i((a_t)_f_i(t,x))(f_i)_*_2^*d𝒮),

where _2 is the second projection _2:S^1× D^2  D^2 and *_g_Y is the Hodge star operator. We denote _2^*d𝒮 by η.
We set

    R(Y)_π:= {a ∈(Y)  |F(a)+∑_1≤ i ≤ d h'_i((a)_f_i(t,x))(f_i)_*η=0 },

 and 
 
    R^*(Y)_π:= R(Y)_π∩^*(Y).

The solutions of (<ref>) correspond to connections A over Y× which satisfy an equation:

    F^+(A)+ π(A)^+=0,

where 


  * The two form π(A) is given by 

    ∑_1≤ i ≤ d h'_i((A)_f̃_i(t,x,s))(f̃_i)_* (_1^* η).


  * The map _1 is a projection map from (S^1× D^2) × to S^1× D^2.

  * The notation + is the self-dual component with respect to the product metric on Y×.

  * The map f̃_i: S^1× D^2× Y× is f_i× id. 

We also use several classes of the perturbations.
A class of perturbation ∏(Y)^flat is defined by a subset of ∏(Y) with the conditions:


  * cs_Y coincides  with cs_Y,π on a small neighborhood of critical points of cs_Y 

  * R(Y)=R(Y)_π,
 
for all element in ∏(Y)^flat.

If the cohomology groups defined by the complex (12) in <cit.> satisfies H^i_π,a=0 for all [a] ∈R(Y)_π∖{ [θ]} for π, we call π non-degenerate perturbation. 
If π satisfies the following conditions, we call π regular perturbation. 


  * The linearization of (<ref>)

    d^+_A+dπ^+_A : ^1(Y×)⊗_L^2_q^+(Y×)⊗_L^2_q

 is surjective for [A] ∈ M(a,b)_π and  all irreducible critical point a,b of cs_Y,π.
 
  * The linearization of (<ref>)
 
    d^+_A+dπ^+_A : ^1(Y×)⊗_L^2_q,δ^+(Y×)⊗_L^2_q,δ

is surjective for [A] ∈ M(a,θ)_π and  all irreducible critical point a of cs_Y,π.
 
 Here the spaces  M(a,b)_π and M(a,θ)_π are given in (<ref>) in Subsection <ref>, L^2_q is the Sobolev norm and L^2_q,δ is the weighted Sobolev norm which is same one as in Subsection 3.3.1 in <cit.>.








 §.§ Filtered instanton Floer (co)homology

In this subsection, we give the definition of the filtration of the instanton Floer (co)homology by using the technique in <cit.>. First, we give the definition of usual instanton Floer homology.



Let Y be a homology S^3 and fix a Riemannian metric g_Y on Y. Fix a non-degenerate regular perturbation π∈∏(Y).
Roughly speaking, the instanton Floer homology is inifinite dimensional Morse homology with respect to 

    cs_Y,π :^*(Y) .


Floer defined : R^*(Y)_π, called the Floer index. 
The (co)chains of the instanton Floer homology are defined by

    CF_i:= { [a] ∈R^*(Y)_π | (a)=i }(CF^i:= (CF_i,)).

The boundary maps ∂ :CF_i  CF_i-1(δ:CF^i CF^i+1) are given by

    ∂ ([a]) := ∑_b ∈R^*(Y)_π with (b)=i-1# (M(a,b)/) [b] (δ:=∂^*),
 
where M(a,b) is the space of trajectories of cs_Y,π from a to b. We now write the explicit definition of M(a,b). Fix a positive integer q≥3. Let A_a,b be an SU(2) connection on Y × satisfying A_a,b|_Y× (-∞,1]=p^*a and A_a,b|_Y× [1,∞)=p^*b where p is projection Y× Y.


    M(a,b)_π:={A_a,b+c   | c ∈Ω^1(Y×)⊗_L^2_q with (<ref>)}/ (a,b),

where (a,b) is given by 

    (a,b):={ g ∈(P_Y×)⊂(ℂ^2)    _L^2_q+1,loc | ∇_A_a,b(g) ∈ L^2_q }.

The action of (a,b) on {A_a,b+c   | c ∈Ω^1(Y×)⊗_L^2_q with (<ref>)} given by the pull-backs of connections. The space  has an  action on M(a,b) by the translation. Floer show that M(a,b)/ has structure of a compact oriented 0-manifold whose orientation is induced by the orientation of some determinant line bundles and ∂^2=0 holds.
The instanton Floer (co)homology HF_*(Y)(HF^*(Y)) is defined by 

    HF_*(Y):= ∂  / ∂ (HF^*(Y):= δ  / δ).



Second, we introduce the filtration in the instanton Floer homology. This filtration is essentially considered by Fintushel-Stern in <cit.>. We follow Fintushel-Stern and use the class of perturbations which they call ϵ-perturbation defined in Section 3 of <cit.>. They constructed -graded Floer homology whose chains are generated by the critical points of cs_Y,π with cs_Y(a)∈ (m,m+1). We now consider Floer homology whose chains generated by the critical points of cs_Y,π with cs_Y(a)∈ (-∞,r). 


Let R(Y) be as in Section <ref> and Λ_Y be ∖ cs_Y|_R(Y). For r ∈Λ_Y, we define the filtered instanton homology HF^r_*(Y) (HF^*_r(Y)) by using ϵ-perturbation.  For r ∈Λ_Y, we set ϵ= inf_a ∈R(Y) |cs_Y(a)-r| and choose such a ϵ-perturbation π.  
The chains of the filtered instanton Floer (co)homology are defined by

    CF^r_i:= { [a] ∈R^*(Y)_π | (a)=i, cs_Y,π(a)<r } (CF^i_r:= (CF_r^i,)).

The boundary maps ∂^r :CF_i^r  CF_i-1^r(resp. δ^r:CF^i_r CF^i+1_r) are given by the restriction of ∂ to CF_i^r(resp. δ^r:=(∂^r)^*). This maps are well-defined and (∂^r)^2=0 holds as in Section 4 of <cit.>.
The filtered instanton Floer (co)homology HF^r_*(Y)(resp. HF_r^*(Y)) is defined by 

    HF^r_*(Y):= ∂^r  / ∂^r  (resp. HF^*_r(Y):= δ^r  / δ^r).


We can also show HF^r_i(Y) and HF^i_r(Y) are independent of the choices of the perturbation and the metric by  similar discussion in <cit.>.
For r ∈Λ_Y, we now introduce obstruction classes in HF^1_r(Y). These invariants are  generalizations of  [θ] ∈ HF^1(Y) considered in Subsection 7.1 of <cit.> and Subsection 2.1 of <cit.>. 
For r ∈Λ_Y, we set homomorphism 

    θ^r :CF^r_1

 by

    θ^r(a):= # (M(a,θ)_π/).

As in <cit.> and <cit.>, we use the weighted norm on M(a,θ)_π to use Fredholm theory. From the same discussion for the proof of (δ^r)^2=0, we can show δ^r (θ^r)=0. Therefore it defines the class [θ^r] ∈ HF^1_r(Y). We call the class [θ^r] obstruction class.

The class [θ^r] does not depend on the small perturbation and the metric. The proof is similar to the proof for original one [θ]. Now we give the proof of Example <ref>.



Because X is homotopy equivalent to S^3× S^1, Q^i_X=∞ for i ∈. If the element [θ^1] ∈ HF^1_r(-Σ(2,3,6k-1)) does not vanish for r=1, we can apply Theorem <ref>. Frøyshov showed 0 ≠ [θ] ∈ HF^1(-Σ(2,3,6k-1)) by using the property of h-invariant of Proposition 4 in <cit.>. Then we get nonzero homomorphism θ:CF^r_1(Y) for r=∞. ( If r=∞, HF^1_r is the usual instanton Floer cohomology by the definition.) 
By using calculation about the value of the Chern-Simons functional of Section 7 in <cit.>, 
 we can see θ^1:CF_1^r is nonzero for r=1 and CF_i^r is zero for r=1 and i∈ 2. This implies 
 
    0 ≠ [θ^1] ∈ HF^1_r(-Σ(2,3,6k-1))  for  r=1.

 





 §.§ Fredholm theory and moduli theory on 4-manifolds with periodic end

 In <cit.>, Taubes constructed the Fredholm theory of some class of elliptic operators on 4-manifolds with periodic ends. He also extend moduli theory of SU(2) gauge theory on such non-compact 4-manifolds.
In this subsection, we review Fredholm theory of a certain elliptic operator on 4-manifolds with the periodic ends  as in <cit.> and define the Fredholm index of the class of operators, which gives the formal dimension of a suitable instanton moduli space on such non-compact 4-manifolds. First we formulate the 4-manifolds with periodic ends.

Let Y be an oriented homology S^3 as in Section <ref>. 
Let W_0 be an oriented homology cobordism from Y to -Y. We get a compact oriented 4-manifold X by pasting W_0 with itself along Y and -Y. We give several notations in our argument.


  * The manifold W_i is a copy of W_0 for i ∈

  * We denote ∂(W_i) by Y^i_+∪ Y^i_- where Y^i_+(resp. Y^i_-) is equal to Y(resp. -Y) as oriented manifolds.

  * For  (m,n)∈ (∪{ -∞} ) × (∪{∞}) with m<n, we set

    W[m,n]:=∐_m≤ i ≤ n W_i  / {Y^j_- ∼ Y^j+1_+ j ∈{m,⋯ ,n}}.




We denote by W the following non-compact 4-manifold

    W:= Y× (-∞,0] ∪ W[0,∞]/{∂ (Y× (-∞,0]) ∼ Y^0_+}.
 
For a fixed Riemannian metric g_Y on Y, we choose a Riemannian metric g_W on W which satisfies


  * g_W|_Y× (-∞,-1]=g_Y× g^stan_.

  * The restriction g_W|_W[0,∞] is a periodic metric.

There is a natural orientation on W[0,∞] and W induced by the orientations of W_0.  The infinite cyclic covering space of X can be written by 

    X≅ W[-∞,∞].

Let T be the deck transformation of X which maps each W_i to W_i+1. By restriction, T has an action on W[0,∞]. We use the following smooth functions τ and τ' on W

    τ, τ': W

satisfying 


  * τ (T|_W[0,∞](x))= τ(x)+1, τ'(T|_W[0,∞](x))=τ'(x)+1 for x ∈ W[0,∞].

  * τ|_Y× (-∞,-2]=0, τ'(y,t)= -t for (y,t) ∈ Y× (-∞,-2].

 By the restriction of τ, we have a function on W[0,∞] which we denote by same notation τ.


In this subsection, we review the setting of the configuration space of fields on W and define the Fredholm index of a kind of operator on W. We fix π∈∏(Y) in Subsection <ref> and assume that π is a non-degenerate perturbation. Let P_W be the product SU(2) bundle. 
For each element [a] ∈R(Y), we fix an SU(2) connection A_a on P_W which satisfying A_a|_Y× (-∞,-1] =_1^*a and A_a|_W[0,∞]=θ. If a is an irreducible (resp. reducible) connection, we define the space of connections on P_W by

    ^W(a)_δ := { A_a+c  | c ∈Ω^1(W)⊗_L^2_q,δ},

 
    ( resp. ^W(a)_(δ,δ) := { A_a+c  |  c ∈Ω^1(W)⊗_L^2_q,(δ,δ)} )

where Ω^1(W)⊗_L^2_q,δ(resp. Ω^1(W)⊗_L^2_q,(δ,δ)) is the completion of Ω^1(W)⊗ with L^2_q,δ-norm(resp. L^2_q,(δ,δ)-norm), q is a natural number greater than 3, and δ is a positive real number. For f ∈Ω^i(W)⊗ with compact support, we define L^2_q,δ-norm(resp. L^2_q,(δ,δ)-norm) by 

    ||f||^2_L^2_q,δ:= ∑_0≤ j ≤ q∫_W e^δτ|∇_θ^j f |^2 dvol ,


    (resp.  ||f||^2_L^2_q,(δ,δ):= ∑_0≤ j ≤ q∫_W e^δτ'|∇_θ^j f |^2 dvol  )

where ∇_θ is covariant derivertive with respect to the product connection θ. We use a periodic metric ||-|| on the bundle. Its completion is denoted by Ω^i(W)⊗_L^2_q,δ.
We define the gauge group 

    ^W(a)_δ:= { g ∈(P_W)_L^2_q+1,loc| ∇_A_a(g) ∈ L^2_q,δ}


    (  resp. ^W(a)_(δ,δ):= { g ∈(P_W)_L^2_q+1,loc| ∇_A_a(g) ∈ L^2_q,(δ,δ)}),

which has the action on ^W(a)_δ induced by the pull-backs of connections.
The space ^W(a)_δ(resp. ^W(a)_(δ,δ)) has structure of Banach Lie group and the action of ^W(a)_δ on ^W(a)_δ(resp. ^W(a)_(δ,δ) on ^W(a)_(δ,δ)) is smooth.
The configuration space for W is defined by 

    ^W(a)_δ := ^W(a)_δ /^W(a)_δ (resp.  B^W(a)_(δ,δ) := ^W(a)_(δ,δ) /^W(a)_(δ,δ)).

Let s be a smooth function from W to [0,1] with 

    s|_Y× (-∞ -2]=1, s|_Y× [-1,0] ∪_Y W[0,∞]=0.

We define the instanton moduli space for W by 

    M^W(a)_π,δ:={[A] ∈^W(a)_δ |  ℱ_π(A)=0 }
 
where ℱ_π is the perturbed ASD-map 

    ℱ_π(A):=F^+(A)+sπ(A).


For each A ∈^W(a)_δ, we have the bounded linear operator:

    d^*_L^2_δ_A+dℱ_A: ^1(W)⊗_L^2_q,δ (^0(W)  ⊕^+(W))⊗_L^2_q-1,δ


    ( resp. d^*_L^2_(δ,δ)_A+dℱ_A: ^1(W)⊗_L^2_q,(δ,δ) (^0(W)  ⊕^+(W))⊗_L^2_q-1,(δ,δ)).


Taubes gave a criterion for the operator d^*_L^2_δ_A+dℱ_A=d^*_L^2_δ_A+ d^+_A+sdπ^+_A in (<ref>)(resp. (<ref>)) to be Fredholm in Theorem 3.1 of <cit.>.


There exists a descrete set D in  with no accumulation points such that (<ref>)(resp. (<ref>)) is Fredholm for each δ in ∖ D.

The discrete set D is defined by 

    D:={δ∈ |    the cohomology groups  H^i_z are acyclic for all z with  |z|=e^δ/2}.

The cohomology groups H^i_z are given by the complex:

    0^0(X) ⊗d_θ,z^1(X)⊗d^+_θ,z^+(X)⊗ 0,

where
 
    d_θ,z:^0(X) ⊗^1(X)⊗
and
 
    d^+_θ.z :^1(X)⊗^+(X)⊗

are given by 
 
    d_θ,z(f)=z^τ d_p^*θ (z^-τ (p^*f)),  d^+_θ,z(f)=z^τ d^+_p^*θ (z^-τ(p^*f)),

 where p is the covering map X X.
 (We fix a branch of ln z to define z^τ = e^τ ln z.) In above definition,  d_θ,z(f) and d^+_θ,z(f) are sections of p^*P_X, however these are invariant under the deck transformation, we regard d_θ,z(f) and d^+_θ,z(f)
as sections on P_X. 
 
 
 The operators d_θ,z and d^+_θ.z in (<ref>) depend on the metric on X and τ however the cohomology groups H_i^z are independent of the choice of them.
We now introduce the formal dimension of the instanton moduli spaces:
Suppose a is an irreducible critical point of cs_Y,π.
From Theorem <ref>, there exists δ_0>0  such that (<ref>) is Fredholm for any δ∈ (0,δ_0) and A ∈^W(a)_δ. We define the formal dimension _W(a) of the instanton moduli spaces for W by the Fredholm index of (<ref>). For the case of a=θ, we also set _W(a) as the Fredholm index of (<ref>).


The formal dimension _W(a) is calculated by using the following proposition.

Suppose a is an irreducible critical point of cs_Y,π.
The formal dimension _W(a) is equal to the Floer index (a) of a. If a is equal to θ, _W(a)= (a)=-3.


First we take a compact oriented 4-manifold Z with ∂ Z=-Y. It is easy to show there is an isomorphism H^*(W[0,∞]) ≅ H^*(S^3). We define Z^+:= Z∪_Y W[0,∞] and fix a periodic Riemannian metric g_Z^+ satisfying g_Z^+|_W[0,∞]=g_W. In Proposition 5.1 of <cit.>, Taubes computed the Fredholm index of d^+_θ+ d^*_L^2_δ_θ as a operator on Z^+ in the situation that H_*(W[0,∞],)≅ H_*(S^3,)(The proof is given by using the admissibility of each segment W_0, however Taubes just use the condition H_*(W[0,∞],)≅ H_*(S^3,).):

    (d^+_θ+ d^*_θ) =-3(1-b_1(Z)+b^+(Z))

for a small δ. Fix an SU(2)-connection A_a,θ on W with A|_Y× (-∞,-1]=a, A|_W[0,∞]=θ and an SU(2)-connection B_a on Z∪_Y Y× [0,∞) 
  By the similar discussion about gluing of the operators on cylindrical end in Proposition 3.9 of <cit.>, we have 

    (d_B_a^*+d_B_a^+)+  (d_A_a,θ^*+d_A_a,θ^+)=  (d_θ^*+d_θ^+).

Donaldson show that (d_B_a^*+d_B_a^+) is equal to -(a)-3(1-b_1(Z)+b^+(Z)) in Proposition 3.17 of <cit.>.
The second statement is similar to the first one.






§ CHERN-SIMONS FUNCTIONAL FOR HOMOLOGY S^3× S^1

For a pair (X,ϕ) consisting of an oriented 4-manifold and non-zero element 0≠ϕ∈ H^1(X,), we  generalize the Chern-Simons functional to a functional cs_(X,ϕ) on the flat connections on X. We define the invariants Q^i_X ∈_≥0∪{∞} for i∈ by using the value of cs_(X,ϕ). In our construction, cs_(X,ϕ) cannot be extended to a functional for arbitrary SU(2) connections on X.

Let X be an oriented closed 4-manifolds equipped with 0 ≠ϕ∈ H^1(X,) and p:X^ϕ X be the -hold covering of X corresponding to ϕ∈ H^1(X,)≅ [X,B]. Recall that the bundle P_X and the set R(X) as in Section <ref>. Let f be a smooth map representing the class ϕ∈ H^1(X,)≅ [X,S^1], and f̃ is a lift of f.

[Chern-Simons functional for a homology S^3× S^1]
We define the Chern-Simons functional for X as the following map

    cs_(X,ϕ):R(X)×R(X) ,


    cs_(X,ϕ)([a],[b]):= 1/8π^2∫_X^ϕ Tr(F(A_a,b)∧ F(A_a,b)),

where a,b are flat connections on P_X and A_a,b is an SU(2)-connection on P_X^ϕ:=X^ϕ× SU(2) which satisfies A_a,b|_f̃^-1(-∞,-r]=p^*a and A_a,b|_f̃^-1([r,∞))=p^*b for some r>0.



We have an alternative description of cs_(X,ϕ)([a],[b]) when a closed oriented 3-manifold Y is given as a sub-manifold of X satisfying i_*[Y]=PD(ϕ) ∈ H_3(X,), where i is the inclusion Y X. Such Y is given as an inverse image of a regular value of f. We can take Y to be connected, and we assume this. We denote by W_0 the cobordism from Y to itself obtained from cutting X open along Y. Since Y is connected and ϕ≠ 0, W_0 is also connected. Then we choose the idenitification of X^ϕ and …∪_Y W_0 ∪_Y W_1 ∪_Y …. We have the following formula.


    cs_(X,ϕ)([a],[b])=cs_Y([i^*a])-cs_Y([i^*b])




Let A_a,b the SU(2)-connection on P_X^ϕ in Definition <ref>.
Take a natural number N large enough to satisfy 

    f̃^-1([-r,r]) ⊂ W[-N,N],

for which we have

    ∫_X^ϕ Tr(F(A_a,b)∧ F(A_a,b))= ∫_W[-N,N]Tr(F(A_a,b)∧ F(A_a,b)).


    =cs_Y(i^*_+A_a,b)-cs_Y(i^*_-A_a,b).

Here  i_+(resp. i_-) is inclusion from Y(resp. -Y) to X, and we use the Stokes theorem.



When X is equal to Y× S^1, this map cs_(X,ϕ):R(X)×R(X) essentially coincides with the restricton of Chern-Simons functional cs_Y on Y by the following sense.
For [a]∈R(Y× S^1), the restriction [i^*a] ∈R(Y) satisfies 

    cs_Y([i^*a])=cs_(Y× S^1,PD([Y]))([a],[θ]),

where i is a inclusion Y=Y× 1  Y× S^1 and PD is the Poincaré duality.

This is a corollary of Lemma <ref>. We have the following well-definedness.


cs_(X,ϕ) does not depend on the choices of f, representatives a and b, and A_a,b.

This is also a consequence of Lemma <ref> 
Let X be a closed oriented 4-manifold equipped with ϕ∈ H^1(X,). 
The invariant Q̃_(X,ϕ) is defined by

    .9
    ∞                                              if R^*(X)= ∅,
    
    inf{|cs_(X,ϕ)([a],[θ])+m| | m∈ , [a] ∈R^*(X) } if R^*(X) ≠∅,

where R^*(X) is the subset of the classes of the irreducible connections in  R(X).










We now give a definition of Q^i_X ∈_≥ 0∪{∞}.

Suppose that X is a homology S^3× S^1 and i is a positive integer. Let X be -fold covering space over X corresponding to the 1∈ H^1(X,)≅_PD H_3(X,).
We set X^i:= X/ i. Since the quotient map p^i:XX^i is a -fold covering, this determine a class ϕ^i ∈ H^1(X^i,).
We define Q^l_X∈_≥ 0∪{∞} by 

    Q^l_X:=min_0≤ i ≤ lQ̃_(X^i,ϕ^i).



We show the following lemma which is used in the proof of Key lemma(Lemma <ref>).

Suppose that γ is a flat connection on W[m,n] satisfying the following conditions.


  * γ|_Y^m_+≅γ|_Y^n_-.

  * There exists u ∈ satisfying |cs_(W[m,n],PD[Y^m_+])(r(γ),θ)+ u|< Q^n-m+1_X, where r(γ) is a flat connection on W[0,k] given by pasting γ with itself along Y^0_+ ∪ Y^k_-.

Then γ is gauge equivalent to θ.

 Suppose γ is not  gauge equivalent to θ.
The calculation 

    H_1(W[m,n])≅ 0

 and holonomy correspondence 

    R(W[m,n]) ≅ (π_1(W[m,n]),SU(2))/ conjugate

imply that there is no reducible SU(2) connection on W[m,n] except θ. Therefore γ is an irreducible connection on W[m,n].
Because W[m,n] X is the (n-m+1)-fold covering space of X,

    Q^n-m+1_X ≤|cs_(W[m,n],PD[Y^m_+])(r(γ),θ)+ u|

holds by the definition of Q^i_X. This is a contradiction.





§ COMPACTNESS

The compactness of the instanton moduli spaces for non-compact 4-manifolds is treated in <cit.>, <cit.>,<cit.> for cylindrical end case and in <cit.> for periodic end case. In <cit.> and <cit.>, they consider the instanton moduli spaces with the connections asymptotically convergent to the trivial connection on the end.  

 We also follow their strategy by using Q^2l_Y+3_X defined in the previous section. More explicitly, in this section we explain a compactness result for the instanton moduli spaces for a non-compact manifold W[0,∞] with periodic end.




 §.§ Key lemma

Let W_0, W[0,∞] be the oriented Riemannian 4-manifolds introduced in the beginning of Subsection <ref>. By pasting W_0 with itself along its boundary Y and -Y, we obtain a homology S^3× S^1 which we denote by X. We consider the product SU(2)-bundle P_W[0,∞] on W[0,∞].
For q≥3 and δ>0, we define the instanton moduli space M^W[0,∞]_δ by 

    M^W[0,∞]_δ := {θ +c ∈^1(W[0,∞])⊗_L^2_q,δ | F^+(θ+c)=0 }/ ,

where  is the gauge group

    :={g ∈(P_W[0,∞]) ⊂(ℂ^2)_L^2_q+1,loc | dg ∈ L^2_q,δ},

and the action of  is given by the pull-backs of connections. For f∈Ω^i(W[0,∞])⊗ with compact support, we define L^2_q,δ norm by the following formula

    ||f||^2_L^2_q,δ:= ∑_0≤ j ≤ q∫_W[0,∞] e^δτ|∇_θ^j f|^2  dvol ,

where ∇_θ is the covariant derivertive with respect to the product connection. We use the periodic metric |-| which is induced from the Riemannian metric g_W. Its completion is denoted by Ω^i(W[0,∞])⊗_L^2_q,δ.
 
 
 Our goal of this section is to show the next theorem under the above setting.
Under Assumption <ref> the following statement holds.
There exist δ'>0 satisfying the following property. Suppose that δ is a non-negative number less than δ' and {A_n} is a sequence in M^W[0,∞]_δ satisfying 

    sup_n ∈||F(A_n)||^2_L^2(W[0,∞]) < min{8π^2,Q^2l_Y+3_X}.


Then for some subsequence {A_n_j}, a positive integer N_0 and some gauge transformations {g_j} on W[N_0,∞], the sequences {g_j^*A_n_j} converges to some A_∞ in L^2_q,δ(W[N_0,∞]).

The proof of Theorem <ref> is given in the end of Subsection <ref>.














We use the following estimate.

For a positive number c_1>0, there exists a positive number c_2>0 satisfying the following statement.

For any SU(2)-connection a on Y^0_+ and any flat connection γ on W[0,k] satisfying the following conditions


  * sup_x ∈ Y^+_0∑_0≤ j ≤ 1|∇^(j)_γ(a-(l^0_+)^*γ)(x)|<c_1.

  * γ|_Y^0_+ ≅γ|_Y^k_-.

, the inequality

    | cs_Y([a])-cs_(W[0,k],PD[Y^0_+])(r(γ),θ)| ≤ c_2 sup_x ∈ Y^+_0∑_0≤ j ≤ 1|∇^(j)_(l^0_+)^*γ(a-(l^0_+)^*γ)(x)|^2

holds, where r(γ) is a flat connection on W[0,k] given by pasting γ with itself along Y^0_+ ∪ Y^k_- and l^0_+:Y^0_+ W_0 is the inclusion.



Lemma <ref> imply

    |  cs_Y([a])-cs_(W[0,k],PD[Y^0_+])(γ,θ)|
    =|cs_Y(a)-cs_Y((l^0_+)^*γ)|.

Since (l^0_+)^*γ is a flat connection on Y^0_+, we have

    =1/8π^2| ∫_Y^0_+Tr( (a-(l^0_+)^*γ) ∧ d_(l^0_+)^*γ(a-(l^0_+)^*γ) + 2/3(a-(l^0_+)^*γ) ^3|


    ≤1/8π^2vol_Y (sup_x ∈ Y^0_+|∇_(l^0_+)^*γ(a-(l^0_+)^*γ) (x)||(a-(l^0_+)^*γ)|+ 2/3sup_x ∈ Y^0_+|(a-(l^0_+)^*γ)|^3)


    ≤ c_2 sup_x ∈ Y^0_+∑_0≤ j ≤ 1|∇^(j)_(l^0_+)^*γ(a-(l^0_+)^*γ)(x)|^2.





Next lemma gives us a key estimate. We use Q^2l_Y+3_X to obtain an estimate of the difference between an ASD-connection and the trivial flat connection on the end W[0,∞].
Suppose that Y satisfies Assumption <ref>. There exists a positive number c_3 satisfying the following statement.


For A ∈ M^W[0,∞]_δ satisfying 1/8π^2||F(A)||^2_L^2(W[0,∞]< min{1, Q^2l_Y+3_X}, there exists a positive number η_0 which depends only on the difference min{1, Q^2l_Y+3_X}-1/8π^2||F(A)||^2_L^2 such that  the following condition holds.


Note that if K is sufficiently large, the inequality ||F(A)||^2_L^2(W_k)< η_0 is satisfied for every k>K. When K satisfies this property, there exist gauge transformations g_k over W[k,k+2] such that 

    sup_x∈ W[k,k+2]∑_0≤ j≤ q+1|∇^j_θ(g_k^*A|_W[k,k+2]-θ)(x)|^2


    ≤ c_3||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])

holds for k>K+l_Y+3.


For k>K+l_Y+3, we apply Lemma 10.4 in <cit.> to A|_W[k-l_Y-1,k+l_Y+2]. 
Then we obtain gauge transformations g_k and flat connections γ_k over W[k-l_Y-1,k+l_Y+2] satisfying

    sup_x ∈ W[k-l_Y-1,k+l_Y+2]∑_0≤ j≤ q+1|∇^j_γ(g_k^*A|_W[k-l_Y-1,k+l_Y+2]-γ_k)(x)|^2


    ≤ c_3||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])≤ (2l+5)c_3η

for a small η.
By using the pull-backs of γ_k from W[k-l_Y-1,k-1](resp. W[k+2,k+l_Y+2]) to Y^i_+, we get the flat connections over Y. Then we get l_Y+1 flat connections by using this method. Under the assumption that l_Y=# R(Y), same flat connections appear by the pigeonhole principle. We choose two numbers  k(1)^±<k(2)^± which satisfy (l^k(1)^±_+)^*γ_k ≅(l^k(2)^±_+)^*γ_k as connections on Y, where k(1)^+ and k(2)^+ are elements in {k-l_Y-1,⋯ , k-1} (resp. k(1)^-,k(2)^- ∈{k+2,k+l_Y+2}). The map l^k_±:Y^k_± W_k is the inclusion.



Suppose ||F(A)||^2_L^2(W_k)<η holds for k>K+l_Y+3. For sufficiently small η, the flat connection γ_k is isomorphic to  θ.

The properties of k(1)^± and k(2)^±, (<ref>) and Lemma <ref> imply

    | cs_Y((l^k(1)^±_+)^* g_k^*A) -cs_((W[k(1)^±,k(2)^±],PD[Y^k(1)^±_+])(r(γ_k),θ)|≤ (2l+5)c_3η c_2.

We also have

    cs_Y((l^k(1)^±_+)^* g_k^*A)= cs_Y(l^k(1)^±_+)^*A)+deg(g_k|_Y^k(1)^±_+)

 
    = ||F(A)||^2_L^2(W[k(1)^±,∞])+deg(g_k|_Y^k(1)^±_+).

 
We choose η_0 satisfying the following condition: 

    (2l+5)c_3η_0 c_2 <  Q^2l_Y+3_X - 1/8π^2∫_W[k(1)^±,∞]|F(A)|^2,

where the right hand side is positive by the assumption of A.
We obtain 

    |cs_(W[k(1)^±,k(2)^±], PD[Y^k(1)^±])(r(γ_k),θ)+deg(g_k|_Y^k(1)^±_+)|< Q^2l_Y+3_X

 from (<ref>), (<ref>) and (<ref>). Then Lemma <ref> imply γ|_W[k(1)^±,k(2)^±]≅θ.


Similarly the inequality

    sup_x ∈ W[k(2)^-,k(1)^+]∑_0≤ j≤ q|∇^j_γ_k(g_k^*A|_W-γ_k|_W[k(2)^-,k(1)^+])(x)|≤ (2l_Y+5)c_3η_0

holds over W[k(2)^-,k(1)^+].
From above discussion, l^k(2)^-_+^*γ_k and i^k(1)^+_+^*γ_k are gauge equivalent.
By Lemma <ref>, we also get 

    | cs_Y((l^k(2)^-_+)^*g_k^*A)-cs_(W[k(2)^-,k(1)^+], PD[Y^k(2)^-_+])(r(γ_k),θ)|≤ (2l_Y+5)c_3η_0c_2.

By choice of η_0, we have 

    |cs_(W[k(2)^-,k(1)^+], PD[Y^k(2)^-_+])(r(γ_k),θ)+ deg(g_k|_Y^k(2)^-_+)|< Q^2l_Y+3_X

and get γ_k|_W[k(2)^-,k(1)^+]≅θ by using Lemma <ref> as above.





 §.§ Chain convergence

We introduce the following notion which is crucial for our proof of the compactness theorem(Theorem <ref>).
 [Chain decomposition of a sequence of the instantons]
For a fixed number η>0 and a sequence {A_n}⊂ M^W[0,∞]_δ satisfying 

    sup_n∈||F(A_n)||^2_L^2(W[0,∞])<∞,

when a finitely many sequences {s^j_n(η)}_1≤ j ≤ m of non-negative numbers satisfies

    ||F(A_n)||^2_L^2(W_s)>η  s=s^j_n  for some j,

and

    s^1_n < … < s^m_n ,

we call {s^j_n}_1≤ j ≤ m chain decomposition of {A_n}⊂ M^W[0,∞]_δ for η. 

For any sequence {A_n} satisfying sup_n∈||F(A_n)||^2_L^2(W[0,∞])<∞ and any η>0, we can show the existence of a chain decomposition for η>0 if we take a subsequence of {A_n}.

First we give the next technical lemma.

Let A be a L^2_q ASD-connection on W[k,∞] satisfying 

    1/8π^2 ||F(A)||^2_L^2(W[k,∞])<1.

Then there exists a positive number c_4 which depends only on the difference 1-1/8π^2 ||F(A)||^2_L^2(W[k,∞])  such that the following statement holds.


Suppose there exists a gauge transformation g on Y^k_+ satisfying 

    ∑_0≤ j≤ 1sup_x ∈ Y^k_+| ∇^j_θ (g^*(l^k_+)^*A-θ)(x) |^2 ≤ c_4.

Then g is homotopic to the identity gauge transformation.



By the property of cs_Y, we have

    |deg(g)|= |cs_Y(g^*(l^k_+)^*A)-cs_Y((l^k_+)^*A)|


    ≤ |cs_Y(g^*(l^k_+)^*A)|+||F(A)||^2_L^2(W[k,∞])


    ≤∑_0≤ j≤ 1sup_x ∈ Y^k_+| ∇^j_θ (g^*(l^k_+)^*A-θ)(x)|^2+||F(A)||^2_L^2(W[k,∞]).

We define c_4< 1/2(1-||F(A)||^2_L^2(W[k,∞])), then 

    |deg(g)| < 1

holds. This implies the conclusion.





The next proposition is a consequence of Lemma <ref> and <ref>.

Let η be a positive number and {s^j_n}_1≤ j ≤ m be a chain decomposition for η of a sequences {A_n} in M^W[0,∞]_δ with 

    1/8π^2sup_n ∈||F(A_n)||^2_L^2(W[0,∞]) <min{1,Q^2l_Y+3_X}.

Then there exists a subsequence {A_n_i} of {A_n} such that sup_i∈  |s^m_n_i|<∞ holds.   




Suppose there exists η_0>0 which does not satisfy the condition of Proposition <ref>. There exist a chain decomposition {s^j_n(η_0) }_0≤ j≤ m of {A_n} which satisfies s^m_n∞ as n ∞. We take a sufficiently small η>0, we will specify η later. Choose a subsequence of {A_n} which allows a chain decomposition for η. For simplify, we denote the subsequence by same notation {A_n}. We denote the chain decomposition of {A_n} for η by  {t^j_n(η)}_1≤ j ≤ m'.
There are two cases for {t^j_n(η)}_1≤ j ≤ m':


  * There exists j”∈{0,⋯, m'-1} satisfying t^j”_n - t^j”+1_n -∞.

  * There is no j”∈{0,⋯, m'-1} satisfying t^j”_n - t^j”+1_n -∞.

We define a sequence by 

    u_n(η):=⌊t^j”_n(η)+t^j”+1_n(η)/2⌋∈

for the first case and by

    u_n(η):=0

for the second case, where ⌊ - ⌋ is the floor function.
Applying Lemma <ref> to A_n|_W[u_n(η),u_n(η)+2], we get the gauge transformation g_n on W[u_n(η)-l_Y-2,u_n(η)+l_Y+3] satisfying 

    sup_x ∈ W[u_n(η),u_n(η)+2]∑_1≤ j ≤ q+1|∇^j_θ(g_n^*A_n|_W[u_n,u_n(η)+2] - θ)|^2≤ c_3(2l_Y+5)η,

for small η and large n. 
 Because A_n is the ASD-connection for each n, we have

    1/8π^2∫_W[u_n,∞]Tr(F(A_n)∧ F(A_n))=1/8π^2∫_W[u_n,∞]|F(A_n)|^2>η_0

for large n.
On the other hand, by the Stokes theorem

    1/8π^2∫_W[u_n,∞]Tr(F(A_n)∧ F(A_n))=cs_Y(l^u_n_+^*A_n)

holds. 
By Lemma <ref>, we have 

    cs_Y(l^u_n_+^*A_n)=cs_Y(l^u_n_+^*g_n^* A_n)

for small η. Therefore (<ref>) and Lemma <ref> gives 

    |cs_Y(l^u_n_+^*A_n)|≤ c'c_3(2l_Y+5)η.

We choose η satisfying

    c'c_3(2l_Y+5)η<1/8π^2η_0.

For such η, we have 
    |cs_Y(l^u_n_+^*A_n)|<1/8π^2η_0.

On the other hand, η_0 satisfies 
    1/8π^2η_0 <1/8π^2∫_W[u_n,∞]|F(A_n)|^2=|cs_Y(l^u_n_+^*A_n)|
 which is a contradiction.





 §.§ Exponential decay

In the instanton Floer theory, there is an estimate called exponential decay about the L^2-norm of curvature of the instanton over cylindrical end.  We give a generalization of the exponential decay estimate over W[0,∞]. In the end of this subsection, we also give a proof of Theorem <ref>.


There exists a constant c_5 satisfying the following statement.
For A ∈ M^W[0,∞]_δ satisfying 1/8π^2||F(A)||^2<min{1,Q^2l_Y+3_X}, there exists η_1>0  which depends only on the difference  min{ Q^2l_Y+3_X,1} -1/8π^2||F(A)||^2 such that the following condition holds.


Let K>0 be a positive number satisfying ||F(A)||^2_L^2(W_k)< η_1 for any k>K, the inequality 

    ||F(A)||^2_L^2(W[k,k+m])


    ≤ c_3( ||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])+||F(A)||^2_L^2(W[k+m-l_Y-2,k+m+l_Y+3])

holds for k>K+l_Y+3.


Let η be the positive number in Lemma <ref> which depends only the difference Q^2l_Y+3_X-1/8π^2||F(A)||^2. Then for k>K+l_Y+3, we have the following inequalities

    sup_x ∈ W_k∑_0≤ j≤ q|∇^j_θ(g_k^*A-θ)(x)|^2≤ c_3||F(A)||^2_L^2(W[k-l_Y-2,k+l_Y+3])


    ≤ c_3(2l_Y+5)η_1

and

    sup_x ∈ W_k+m∑_0≤ j≤ q|∇^j_θ(g_k+m^*A-θ)(x)|^2≤  c_3||F(A)||^2_L^2(W[k+m-l_Y-2,k+m+l_Y+3]


    ≤ c_3(2l_Y+5)η_1.

These inequality (<ref>), (<ref>) and Lemma <ref> imply that for sufficiently small η_1, the gauge transfromation g_k|_Y^+_k(resp. g_k+m|_Y^-_k+m) is homotopic to the constant gauge transformation.
Hence, there exists a gauge transformation ĝ on W[k,k+m] satisfying ĝ|_W_k=g_k and ĝ|_W_k+m=g_k+m, moreover, since A is the ASD connection, we have

    ||F(A)||^2=||F(ĝ^*A)||^2_L^2(W[k,k+m])=8π^2 (cs_Y((l^k_+)^*g_k^*A))-cs_Y((l^k+m_-)^*g_k+m^*A)).

 Applying the inequalities (<ref>) and (<ref>) again, we get the conclusion.



There exists δ'>0 satisfying the following statement.


Suppose A is an element in M^W[0,∞]_δ satisfying the assumption of Lemma <ref>.
Then there exists c_5(K)>0 satisfying the following inequality. 

    ||F(A)||^2_W[k-l_Y-2,k+l_Y+3]≤ c_5(K)e^-kδ'.
 
for k>K+l_Y+3.



This is a consequence of Lemma <ref> and Lemma 5.2 in <cit.> by applying q_i = ||F(A)||^2_W[i-l_Y-2,i+l_Y+3].



By using a similar argument in Lemma 4.2 and Lemma 7.1 of <cit.> we have:

For a positive number c_7, there exists a constant c_8 satisfying the following statement holds.


Suppose we have an L^2_q connection A, the gauge transformations g_k on W[k-1,k+1] satisfying

    ∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g_k^*A|_W[k-1,k+1]-θ)|^2


    ≤ c_7||F(A)||^2_L^2(W[k-l-3,k+l+2]).

for any non-negative integer k.


Then there exists the positive integer n_0 and a gauge transformation g on W[n_0,∞] satisfying the following condition:

    ∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g^*A|_W[k-1,k+1]-θ)|^2


    ≤ c_8||F(A)||^2_L^2(W[k-l-3,k+l+2])

for k>n_0.

We use this lemma to prove the next proposition.

There exists δ'>0 satisfying the following condition.
Let K>0 be a positive number and {A_n} be a sequence in M^W[0,∞]_δ satisfying the following properties:


  * 0< min{Q^2l_Y+3_X,1}- sup_n∈1/8π^2||F(A_n)||^2.

  * There exists a chain decomposition {s^n_j} of {A_n} for η_2 satisfying 

    sup_n∈|s^m_n(η_*)|<∞

  for 

    η_2:= inf_n∈{η | constants which depend on A_n in Lemma <ref> and <ref>}.





Then there exist a positive integer N_0, gauge transformations {g_j} on W[N_0,∞] and subsequence {A_n_j} of {A_n} such that {g_j^*A_n_j} converge to some A_∞ in L^2_q,δ(W[N_0,∞]) for any 0≤δ<δ'.




If we apply the Lemma <ref> to A_n, there exists gauge transformations g^n_k on W[k-1,k+1] satisfying the following condition: for k>l_Y+K+3,

    ∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g^n_k^*A_n|_W[k-1,k+1]-θ)|^2


    ≤ c_3||F(A_n)||^2_L^2(W[k-l-3,k+l+2]).


On the other hand, we have 

    ||F(A_n)||^2_L^2(L^2(W[k-l-3,k+l+2])≤ c_6(K)e^-δ' k

by using the exponential decay estimate(Proposition <ref>).
Using (<ref>), we can show that n_0 uniformly with respect to n in Lemma <ref>.
So there exist a large natural number N_0 and a gauge transformation on W[N_0,∞] for each n satisfying

    ∫_W[k-1,k+1]∑_0≤ j≤ q+1|∇^j_θ(g_n^*A_n|_W[k-1,k+1]-θ)|^2


    ≤ c'_8||F(A_n)||^2_L^2(W[k-l-3,k+l+2])≤ c_6(K)c'_8e^-δ'k,

where the last inequality follows from (<ref>).




We set g_n^*A_n=θ +a_n. Then we have

    ||a_n||_L^2_q+1,δ(W[N_0,∞])=∑_0≤ j≤ q+1∫_W[N_0,∞]  e^δτ|∇^j_θ(a_n)|^2


    ≤∑_0≤ j≤ q+1∑_N_0 ≤ i≤∞ e^iδ∫_W_i |∇^j_θ(a_n)|^2.

Putting this estimate and (<ref>) together, we have 

    ||a_n||^2_L^2_q+1,δ(W[k,∞] )≤ c_9e^(δ-δ')k

for k>N_0.
 We take a subsequence of {a_n} which converges on any compact set in L^2_q(W[k,∞]) by using the Relich Lemma. We denote the limit in L^2_q,loc by a_∞.
Then the exponential decay (<ref>) and a standard argument implies that {a_n} converges a_∞ on W[N_0,∞] in L^2_q,δ-norm. 

We now give the proof of Theorem <ref>.



We choose η_2 in Proposition <ref>.  After taking subsequence of {A_n}, we consider the chain decomposition {s^j_n}_1≤ j ≤ m for η_2 of {A_n}. 
From Proposition <ref>, {s^j_n} has upper bound by some K>0 after taking a subsequence of {A_n} again. So we can apply Proposition <ref>, we get the conclusion.




§ PERTURBATION AND ORIENTATION

To prove the vanishing [θ^r]=0 in Theorem <ref>, we use the moduli spaces M^W(a)_π,δ and need the transversality for the equation F^+(A)+sπ(A)=0. We also need the orientability of M^W(a)_π,δ.




 §.§ Holonomy perturbation(2)

In <cit.>, Donaldson introduced the holonomy perturbation with compact support for irreducible ASD-connections. Combining the technique in <cit.> and the compactness theorem (Theorem <ref>), we get sufficient perturbations to achieve required transvesality.  
 
Let  π be an element in ∏(Y) and a be a critical point of cs_Y,π.
We use the following notations: 


  * Γ (W):={  l:S^1 × D^3  W  | l: orientation preserving embedding}.

  * Λ^d(W):= {(l_i, μ^+_i)_1≤ i ≤ d∈Γ(W)^d× (^+(W)⊗))^d  | suppμ^+_i ⊂ l_i }.

  * Λ(W):= ⋃_d∈Λ^d(W).


Let χ:SU(2)𝔰𝔲(2) be

    χ(u):=u-1/2tr(u)id

and fix μ^+_i ∈Ω^+(W)⊗𝔰𝔲(2) supported on l_i(S^1× D^3) for i∈{1,⋯ ,d}.
For ϵ∈^d, we set

    σ_Ψ(A,ϵ):= ∑_1≤ i ≤ dϵ_i  χ(_x ∈ l_i(S^1× D^3) (A))μ^+_i,

where _x∈ l_i(S^1× D^3) is a holonomy around the loop t ↦ l_i(t,y_x) satisfying x=l_i (t_x,y_x) for some t_x and ϵ=(ϵ_i)_1≤ i ≤ d.
For Ψ=(l_i,μ_i)_1≤ i ≤ d∈Λ, Donaldson defined the holonomy perturbation of the ASD-equation:

    ℱ_π,Ψ(A,ϵ):=F^+(A)+sπ(A)+σ_Ψ(A,ϵ)=0.

The map σ_Ψ(-,ϵ) is smoothly extended to the map ^W(a)_δΩ^+(W)⊗𝔰𝔲(2)_L^2_q-1,δ and the map ^W(a)_(δ,δ)Ω^+(W)⊗𝔰𝔲(2)_L^2_q-1,(δ,δ). For Ψ and ϵ∈^d, the perturbed instanton moduli space are defined by 

    M^W(a)_π,Ψ,ϵ,δ:={ c ∈^W(a)_δ |ℱ_π,Ψ(c,ϵ)=0 }

in the case of a∈R^*(Y)_π and

    M^W(a)_π,Ψ,ϵ,(δ,δ):={ c ∈^W(a)_(δ,δ) |ℱ_π,Ψ(c,ϵ)=0 }

in the case of Stab(a)=SU(2).
For a fixed ϵ∈^d, if the operator 

    d(ℱ_π,Ψ)_(A,0):T_A^W(a)_δ×^d Ω^+(W)⊗_L^2_q-1,δ.

 is surjective for all [A] ∈ M^W(a)_δ,π,Ψ,ϵ, we call (Ψ,ϵ) regular perturbation for a∈R^*(Y)_π. 
 

Let FM^W(a)_δ,π,Ψ be the family version of the perturbed instanton moduli spaces defined by

    FM^W(a)_δ,π,Ψ:={ (c,ϵ) ∈^W(a)_δ×^d  |ℱ_π,Ψ(c,ϵ)=0 }.




Suppose that Y satisfies Assumption <ref>.
There exists δ'>0 such that for a fixed δ∈ (0,δ'), the following statement holds.
Suppose π is a holonomy perturbation which is non-degenerate and regular. Let a be an irreducible critical point of cs_Y,π with cs_Y(a)<min{Q^2l_Y+3_X,1}.
We assume the next three hypotheses for (π,a).


  * For [A] ∈ M^W(b)_π,δ,

    1/8π^2sup_n∈ ||F(A)+sπ(A)||^2_L^2(W) <min{1 ,Q^2l_Y+3_X},

 where b is an element of  R(Y)_π with  cs_Y,π (b) ≤ cs_Y,π(a). 

  * The linear operator

    d^+_θ+  sdπ^+_θ:T_θ^W(θ)_(δ,δ)×^d Ω^+(W)⊗_L^2_q-1,(δ,δ)

is surjective.

  * M(c)_π,δ is empty set for c∈R_π(Y) satisfying cs_Y,π(c)<0.


Then there exist a small number η>0 and a perturbation Ψ such that the map

    dℱ_π,Ψ :T^W(a)_δ×^d Ω^+(W)⊗_L^2_q-1,δ.

is surjective for all point in ℱ_π,Ψ^-1(0)∩ (^W(a)_δ× B^d(η)).



First we show that the surjectivity of dℱ_π,Ψ at the point in  ℱ_π,Ψ^-1(0)∩ (^W(a)_δ×{ 0} ).
Second, we show that there exists a positive number η>0 such that dℱ_π,Ψ is surjectivie at the point in  ℱ_π,Ψ^-1(0)∩ (^W(a)_δ× B^d(η)).
We name the critical point of cs_Y,π by

    0=cs_Y,π( θ=a_0 ) ≤ cs_Y,π(a_1)≤ cs_Y,π (a_2 ) ⋯≤ cs_Y,π(a_w=a).

The proof is induction on w and there are four steps.  
  For an irreducible element A ∈(a_w)_δ with 0≠ (d(ℱ_π,Ψ)_(A,0)), there exists Ψ(A)={l^A_i,μ^A_i }_1≤ i ≤ d(A) such that d(ℱ_π,Ψ)_(A,0)| ^d(A) generates the space (d^+_A+sdπ^+ _A).

The proof is essentially the same discussion of Lemma 2.5 in <cit.>. 
We fix h ∈^+(W)⊗_L^2_q-1,δ satisfying 0≠ h ∈ (d^+_A+sdπ _A) with ||h||_L^2=1. The unique continuation theorems: Proposition 8.6 (ii) of <cit.> for the equation (d^+_A(-)+dπ^+_A)^*(-)=0 on Y× (-∞,-1] and Section 3 of <cit.> for the equation (d^+_A)^*(-)=0 on W[0,∞] imply h|_Y× [-2,0] ∪_Y W_0≠ 0.
Then we choose x_h in Y× [-2,0] ∪_Y W_0 so that h(x_h)≠0 holds.
Since A is the irreducible connection, 

    (A,x_h)={_l(A) ∈ SU(2)  |l : loop based at  x_h }

 is a dense subset of SU(2). 
So we can choose the loops l^h_i based at x_h satisfying 

    {e_i=χ(_l^h(A))}_i generates .

 For a small neighborhood U_x_h of x_h, we can write h by

    h|_U_x_h= ∑_1≤ i ≤ 3h_i ⊗ e_i.

By using a smoothing of δ function, we have

    <h|U_x_h,∑_1≤ i ≤ 3μ^+_i(h)⊗χ(Hol_l_i(A))>_L^2(U_x_h)≠ 0,

where μ^+_i(h) are three self dual 2-forms supported on U_x_h. For a fixed generator {h^1,… ,h^u} of (d^+_A+sdπ^+_A), we get the points {x_h^j}⊂ Y× [-2,0] ∪_Y W_0, small neighborhoods {U_x_h^j}, loops {l^h^j_i} and self dual 2-forms {μ^+_i(h^j)} satisfying (<ref>) for all h^j. We extend the maps l^h^j_i:S^1 W to embeddings S^1× D^3  W. We can choose U_x_h^j satisfying U_x_h^j⊂ l^h^j.  We set 

    Ψ(A):= (l^h^j_i,μ^+_i(h^j))_i,j∈Λ(W),

which satisfies the statement of Step <ref>.


For j≥0 satisfying cs_Y,π(w_j)=0, we show:

For an element b ∈R^*(Y)_π(resp. b=θ)  satisfying cs_Y,π(b)=0, there exists a perturbation Ψ^b such that the operator

    dℱ_π,Ψ^b|_(A,0):T_A^W(b)_δ×^d^+(W)⊗_L^2_q-1,δ


    (resp.  dℱ_π,Ψ^b|_(A,0):T_A^W(b)_(δ,δ)×^d^+(W)⊗_L^2_q-1,(δ,δ))

is surjective for A ∈ (ℱ_π,Ψ^b)^-1(0) ∩ (^W(a)_δ×{ 0}) (resp. A ∈ (ℱ_π,Ψ^b)^-1(0) ∩ (^W(a)_(δ,δ)×{ 0})).

First we show that M^W(b)_π,δ is compact. Let {[A_n]} be any sequence in M^W(b)_π,δ. By the second hypothesis, we have 

    1/8π^2sup_n ∈||F(A_n)||^2_L^2(W[0,∞])≤1/8π^2sup_n ∈||F(A_n)+sπ(A_n)||^2_L^2(W)<min{1 ,Q^2l_Y+3_X}.

By Theorem <ref>, there exist a large positive number N and the gauge transformations {g_n} over W[N,∞] such that {g_n^*A_n}  converges  over W[N,∞] for small δ after taking a subsequence. Note that Y× (-∞,0] ∪_Y W[0,N+1] is a cylindrical end manifold and we can apply the general theory developed on Section 5 of <cit.>. In particular, there exist gauge transformations {h_n} on Y× (-∞,0] ∪_Y W[0,N+1] such that {h_n^*A_n|_Y× (-∞,0] ∪_Y W[0,N+1]} has a chain convergent subsequence in the sense in Section 5 in <cit.> because the bubble phenomenon does occur under the first hypothesis

    1/8π^2sup_n ∈||F(A_n)+sπ(A_n)||^2_L^2(W)<1.

By gluing {g_n} and {h_n}, we obtain a chain convergent subsequence

    [A_n_j]  ([C^1],…,[C^N], [A^0]) ∈ M(b=c_1,c_2)_π×…× M(c_v,c_v+1)_π× M^W(c_v+1)_π,δ

 with c_i ∈R(Y)_π .  Suppose that [A_n_j]  [A^0] ∈ M(b)_π,δ does not hold. We get cs_Y,π(c_v+1)<0 because the moduli spaces 
 
    M(b,c_1)_π,⋯ , M(c_v,c_v+1)_π

  are non-empty sets. However this contradicts to the assumption of M(c)_π,δ=∅ for c∈R(Y)_π with cs_Y,π(c)<0.
  
  
When b is an irreducible connection, the compactness of M(b)_π,δ, Step <ref> and the openness of surjective operators imply Step <ref>. When b is equal to θ, the second hypothesis implies Step <ref>.


For the inductive step, we show:

Suppose there is a perturbation 

    Ψ^w-1=(l^w-1_i, μ^w-1_i)_i∈Λ(W)

 such that the operators 
    dℱ_π,Ψ^w-1|_(A,0):T_A^W(a_j)_δ^+(W)_L^2_q-1,δ

 is surjective for (A,0) ∈ (ℱ_π,Ψ^w-1)^-1∩ (^W(a_j) ×{0}) and j ∈{1, ⋯, w-1}.
Then the space

    K_w:={A∈ M^W(a_w)_π,δ | 0≠ (dℱ_π,Ψ^w-1|_(A,0)) ⊂Ω^+(W)_L^2_q-1,δ}

is compact.

Let {[A_n]} be a sequence in K_w.
By the similar estimate in Step <ref> and Theorem <ref>, we get a chain convergent subsequence

    [A_n_j]  ([B^1],…,[B^N], [A^0]) ∈ M(a_w=b_1,b_2)_π×…× M(b_v,b_v+1)_π× M^W(b_v+1)_π,δ

 with b_i ∈R(Y)_π. Suppose that [A_n_j]  [A^0] ∈ M(a_w)_π,δ does not hold.  In this case, the operators d^+_B^i+dπ_B^i on Y× and the operators dℱ_π,Ψ^w-1|_(A^0,0) on W are surjective in the suitable functional spaces by the assumption of π and the induction. For large j, the operator dℱ_π,Ψ^w-1|_(A^n_j,0) can be approximated by the gluing of the operators d_B^i^++dπ_B^i, dℱ_π,Ψ^w-1|_(A^0,0). By gluing the right inverses of them as in Theorem 7.7 of <cit.>, dℱ_π,Ψ^w-1|_(A^n_j,0) also has a right inverse for sufficiently large j. This is a contradiction and we have the conclusion of Step <ref>.
 
 
For induction, we need to show:

There exists the perturbation Ψ^w satisfying the surjectivity of the operator

    dℱ_π,Ψ^w:^W(a_w)_δ×^d^+(W)⊗_L^2_q-1,δ

for any point in  (ℱ_π,Ψ^w)^-1(0)∩ ((a_w)_δ×{ 0} ).  

We take the perturbation Ψ_A=((l^j_A),(μ^+_j(A))) for each A∈ K_w in Step 1. Because K_w is compact and surjectivity of the operators is open condition, there exist {A_1,⋯ ,A_k}⊂ K_w and a perturbation Ψ^w such that

    dℱ_π,Ψ^w|_(A,0):^W(a_w)_δ×^d^+(W)⊗_L^2_q-1,δ

is surjective for all (A,0) ∈ (ℱ_π,Ψ^w)^-1(0) ∩ (^W(a)_δ×{ 0}).
Here Ψ^w is defined by 

    Ψ^w:=((l^j_A_1⋯ l^j_A_k, l^w-1_i) ,(μ^+_j(A_1),⋯,μ^+_j(A_k), μ^w-1_i))

which satisfies the property in Step <ref>.



Second, we show that the operator dℱ_π,Ψ^w is surjective for any point in  ℱ_π,Ψ^w^-1(0)∩ (^W(a)_δ×  D^d(η) ). Suppose there is no η such that the statement holds. Then there is a sequence {(A_n,ϵ_n)} in M^W(a)_δ,π,Ψ,ϵ_n which satisfies that ϵ_n  0 as n∞ and dℱ_π,Ψ^w|_(A_n,ϵ_n) is not surjective for all n∈. Because the bubble does occur, {A_n} has a chain convergent subsequence to 

    ([B^1],… , [B^N],[A^0]) ∈ M(b_0,b_1)_π×…× M(b_v-1,b_v)_π× M^W(b_v)_δ,π,Ψ,0

 for some b_i ∈R(Y)_π. Since π is a regular perturbation and dℱ_π,Ψ^w|_(A^0,0) is surjective, there exist the right inverses of d^+_B^1+dπ^+_B^1, …,d^+_B^N+dπ^+_B^N and dℱ_π,Ψ^w|_(A^0,0) for suitable functional spaces. By the gluing of the right inverses as in Step <ref>, dℱ_π,Ψ^w|_(A_N,ϵ_N) also has the right inverse for large N. This is a contradiction and this completes the proof.

 For a given data (δ,π,a) in Lemma <ref>, there exist η>0, a perturbation Ψ and a dense subset of R ⊂ B^d(η) ⊂^d such that (Ψ,b) is a regular perturbation for b ∈ R.


This is a conclusion of Lemma <ref>, the argument in Section 3 of <cit.> and the Sard-Smale theorem. 



Applying the implicit function theorem, we get a structure of manifold of M^W(a)_δ,π,Ψ,b. Its dimension coincides with the Floer index (a) of a by Proposition <ref>. Therefore we have:

For given data (δ,π,a) in Lemma <ref>, there exist η>0, a perturbation Ψ and a dense subset of R ⊂ B^d(η) ⊂^d such that M^W(a)_δ,π,Ψ,b has a structure of manifold of dimension (a).




 §.§ Orientation

In <cit.>, Donaldson showed the orientability of the instanton moduli spaces for closed oriented 4-manifolds. In this subsection, we
deal with the case for non-compact 4-manifold W by generalizing Donaldson's argument.  More explicitly, we show that the moduli space M^W(a)_δ,π is orientable. We also follow Fredholm and moduli theory in <cit.> to formulate the configuration space for SU(l)-connections for l≥ 2.


Let Z be a compact oriented 4-manifold which satisfies ∂ Z=Y and H_1(Z)≅ 0. We set Z^+:=(-Z)∪_Y W[0,∞] and Ẑ:= (-Z)∪_Y Y× [0,∞). Fix a Riemannian metric g_Z^+ on Z^+ with g_Z^+|_W[0,∞]=g_W|_W[0,∞] and Riemannian metric g_Ẑ with g_Ẑ|_Y× [0,∞)= g_Y × g^stan_. First, we introduce the configuration spaces for SU(l)-connections on W and Z^+ for l≥ 2 and SU(2)-configuration space for Ẑ.
Fix a positive integer q ≥3. For an irreducible SU(2)-connection a on Y, we define

    ^W(a)_(δ,δ),l:= {A_a+c  | c ∈^1(W)⊗𝔰𝔲(l)_L^2_q,(δ,δ)},


    ^Z^+_δ,l:= {θ+c  | c ∈^1(Z^+)⊗𝔰𝔲(l)_L^2_q,δ},

and

    ^Ẑ(a):= {B_a+c  | c ∈^1(Ẑ)⊗𝔰𝔲(2)_L^2_q},

where 


  * A_a is an SU(l)-connection on W with A_a|_Y× (-∞,-1]=^* (a⊕θ), A_a|_W[0,∞] =θ 

  * B_a is an SU(2)-connection on Ẑ with B_a|_Y× [0,∞)=^*a.

  * L^2_q,(δ,δ)(W)-norm is defined by

    ||f||^2_L^2_q,(δ,δ)(W):= ∑_0≤ i ≤ q∫_W e^τ' δ  |∇_A_a^i f |^2dvol ,

where τ' is defined in Definition <ref> and f is an element in ^1(W)⊗ with compact support.

  * L^2_q,δ(Z^+)-norm is defined by

    ||f||^2_L^2_q,δ(Z^+):= ∑_0≤ i ≤ q∫_Z^+ e^τ”δ  |∇^i_θ f |^2dvol,

where τ”:Z^+ [0,1] is a smooth function satisfying τ”|_W[0,∞]=τ defined in Definition <ref> and  f is an element in ^1(Z^+)⊗ with compact support.

  * L^2_q(Ẑ)-norm is defined by

    ||f||^2_L^2_q(Ẑ):= ∑_0≤ i ≤ q∫_Ẑ  |∇_B_a^i f |^2dvol ,

where  f is an element in ^1(Ẑ)⊗ with compact support.

We also define the SU(l) configuration spaces ^W (a)_δ,δ^l, ^Z^+_δ,l and SU(2)-configuration spaces ^Ẑ(a) by

    ^W (a)_(δ,δ),l:=^W(a)_(δ,δ),l /^W(a)_l, ^Z^+_δ,l:=^Z^+_δ,l /^Z^+_l

and

    ^Ẑ(a):=^Ẑ(a) /^Ẑ(a)

where ^W(a)_l, ^Z^+_l and ^Ẑ(a) are given by 

    ^W(a)_l:={ g∈(W× SU(l)) ⊂(ℂ^l)_L^2_q+1,loc| ∇_A_a(g) ∈ L^2_q,(δ,δ)(W)} ,


    ^Z^+_l:={ g∈(Z^+× SU(l)) ⊂(ℂ^l)_L^2_q+1,loc| d(g) ∈ L^2_q,δ(Z^+) }

and

    ^Ẑ(a):= { g∈(Ẑ× SU(l)) ⊂(ℂ^2)_L^2_q+1,loc| ∇_B_a(g) ∈ L^2_q,δ(Ẑ) }.
 The action of ^W(a)_l(resp. ^Z^+_l, ^Ẑ(a)) on ^W(a)_(δ,δ).l(resp. ^Z^+_δ,l, ^Ẑ(a))  is the pull-backs of connections.
We define the reduced gauge group by 

    ^W(a)_l:= { g ∈^W(a)_l  |lim_t  -∞ g|_Y× t =id } ,


    ^W,fr(a)_l:={ g ∈^W(a)_l | lim_n ∞ g|_W_n id }

and

    ^Z^+_l:= { g ∈^Z^+_l  | lim_n ∞ g|_W_n id  }.

Then we define

    ^W (a)_(δ,δ),l :=(a)^W_(δ,δ)/ ^W(a)_l, ^W,fr (a)_(δ,δ),l :=(a)^W_(δ,δ)/ ^W,fr(a)_l

and

    ^Z^+_δ,l:=^Z^+_δ,l/ ^Z^+_l.


The group ^W,fr(a)_l (resp. ^Z^+_l) has a structure of Banach Lie sub group of ^W(a)_l (resp. ^Z^+_l). By the construction of them, there are exact sequences 

    ^W(a)_l ^W(a)_l (a⊕θ), ^W,fr(a)_l^W(a)_l  SU(l)

and

    ^Z^+_l^Z^+_l  SU(l)

of Lie groups.
The group ^W(a)_l(resp. ^Z^+_l) acts on ^W (a)_(δ,δ),l(resp. ^Z^+_δ,l) freely.


For l≥ 3 and an SU(2)-flat connection a, there exists a positive number δ' such that for a positive real number δ less than δ' the following properties hold. 


  * ^W (a)_(δ,δ),l is simply connected.

  * ^Z^+_δ,l is simply connected.


 We will show only the first property. The second one is shown in a similar way to the first case. We use the condition H_1(Z,)≅ 0 for the second property.


 Since π_i(SU(l))=0 for i=0,1, 
 
    π_1(^W(a)_(δ,δ),l ) is isomorphic to π_1(^W,fr (a)_(δ,δ),l ).

Therefore, we will show π_1(^W,fr (a)_(δ,δ),l)=0.
There exists δ'>0 such that for 0< δ<δ', 

    ^l(a)_(δ,δ)^f ^W (a)_(δ,δ),l^W,fr (a)_(δ,δ),l

is a fibration since (<ref>) has a local slice due to Fredholm and moduli theory in <cit.>. Let W^* be the one point compactification of W. Using (<ref>), we obtain 

    π_1(^W,fr (a)_(δ,δ),l) ≅π_0(^l(a)_(δ,δ)^f ) ≅ [W^*,SU(l)].

Since π_i(SU(l)) vanishes for i=0,1,2,4, the obstruction for an element of [W^*,SU(l)] to be homotopic to the constant map lives in H^3(W^*,π_3(SU(l))) ≅ H^3_comp(W,π_3(SU(l))) ≅ H_1(W,π_3(SU(l)))=0 where the second isomorphism is the Poincaré duality.

This implies 

    π_1(^W,fr (a)_(δ,δ),l)≅ 0.



We now define the determinant line bundles. 
For simplify, we impose Assumption <ref> on Y.
Let π be an element in ∏(Y)^flat and (Ψ,ϵ) be a perturbation in Subsection <ref> and fix an element a ∈R(Y). 
For c ∈^W(a)_(δ,δ),l (^W(a)_(δ,δ),l), we have the following bounded operator 

    d(ℱ_π,Ψ)_c+d^*_L^2_(δ,δ)_c:^1(W) ⊗su(l)_L^2_q,(δ,δ)^0(W) ⊗su(l) ⊕^+(W) ⊗su(l))_L^2_q-1,(δ,δ).

The operators d(ℱ_π,Ψ)_c+d^*_L^2_(δ,δ)_c are the Fredholm operators for small δ. Fix such a δ.
We set

    λ^W (a,l,c):= Λ^max (d(ℱ_π,Ψ)_c) ⊗Λ^max (d(ℱ_π,Ψ)_c)^*.

The determinant line bundles are defined by

    λ^W (a,l):=⋃_c ∈^W(a)_(δ,δ),lλ (a,l,c) ^W(a)_(δ,δ),l

and
 
    λ̂^W (a,l):=⋃_c ∈^W(a)_(δ,δ),lλ (a,l,c) ^W(a)_(δ,δ),l.

We also define 

    λ^Z^+ (l)^Z^+_δ,l and λ^Ẑ(a) ^Ẑ(a)

in a similar way with respect to the operators

    d^+_c+d^*_L^2_δ_c :^1(Z^+) ⊗su(l)_L^2_q,δ^0(Z^+) ⊗su(l) ⊕^+(Z^+) ⊗su(l))_L^2_q-1,δ

for c ∈^Z^+_δ,l and

    d^+_c+d^*_c :^1(Ẑ) ⊗_L^2_q^0(Ẑ) ⊗⊕^+(Ẑ) ⊗)_L^2_q-1

for c ∈^Ẑ(a).


For a given data (a,δ,l) in Proposition <ref>, 
the bundles λ^Z^+ (a,l)^Z^+_δ,l and λ^W(a,l)^W(a)_(δ,δ),l are trivial.

Since the determinant line bundle is a real line bundle, the triviality of λ^Z^+ (a,l)^Z^+_δ,l is a consequence of Proposition <ref>. Therefore we show the triviality of  λ^W(a,l)^W(a)_(δ,δ),l.
We have a fibration 

    (a⊕θ) ^W(a)_(δ,δ),lj^W(a)_(δ,δ),l.

We also have an isomorphism j^*λ^W(a,l) ≅λ̂^W(a,l) for j in (<ref>). 
λ̂^W(a,l) is the trivial bundle for l>2 from Proposition <ref>. So if the fiber (a⊕θ) of (<ref>) is connected, λ^W(a,l) is also trivial. 
The possibilities of (a⊕θ) are SU(l), U(1)× U(l-1), S(U(2)× U(l-2)) and 

    {(z,A) ∈ U(1)× U(l-2) | z^2 detA=1}.

Since these groups are connected, λ^W(a,l) is the trivial bundle.

Suppose that Y satisfies Assumption <ref> and a is an element in R^*(Y).
Let i_1:^W(a)_(δ,δ),2^W(a)_(δ,δ),3 and i_2:^Z^+_δ,2^Z^+_δ,3 be the maps induced by the product with the product connection. 
 There exists a positive number δ' such that for a positive real number δ less than δ', i_1^*λ^W(a,3) ≅λ^W(a,2) and i_2^*λ^Z^+(a,3) ≅λ^Z^+(a,2) hold.


Under  Assumption <ref> on Y, the isomorphism class of these line bundles are independent of the choices of the perturbations π and (Ψ,ϵ) by considering a 1-parameter family of perturbations π_t:=(f,th) and (Ψ,tϵ) for t ∈ [0,1].
So Lemma (5.4.4) in <cit.> implies the conclusion.

Suppose that Y satisfies Assumption <ref> and a is an element in R^*(Y). Let π be an element in ∏(Y)^flat and (Ψ, ϵ) be a regular perturbation for a ∈R^*(Y). For sufficiently small δ, M^W(a)_π,Ψ,ϵ,δ is orientable. Furthermore the orientation of M^W(a)_π,Ψ,ϵ,δ is induced by the orientation of λ^W(a,2).


Using the exponential decay estimate in Proposition 4.3 of <cit.>, we have a inclusion i:M^W(a)_π,Ψ,ϵ,δ B^W(a)_(δ,δ),2 for small δ as a set. From this inclusion i, we regard M^W(a)_π,Ψ,ϵ,δ as a subset in B^W(a)_(δ,δ),2. Applying result of convergence in Corollary 5.2 of <cit.>, we can show that the topology of  M^W(a)_π,Ψ,ϵ,δ in B^W(a)_δ coincides with the topology of M^W(a)_π,Ψ,ϵ,δ in B^W(a)_(δ,δ),2
Also using exponential decay estimate for solutions to the linearized equation in Lemma 3.3 of <cit.> , λ^W(a,2)|_M^W(a)_π,Ψ,ϵ,δ M^W(a)_π,Ψ,ϵ,δ is canonically isomorphic to Λ^max M^W(a)_π,Ψ,ϵ,δ. 



From Theorem <ref>, an orientation of M^W(a)_π,Ψ,ϵ,δ is characterized by the trivialization of λ^W(a,2). On the other hand, to formulate the instanton Floer homology of Y with  coefficient, Donaldson introduced the line bundle λ(a):=λ^(-Ẑ)(a) ⊗λ_(-Ẑ)^* ^(-Ẑ)(a), where λ_(-Ẑ) is given by

    Λ^max (H^0_DR(-Ẑ)⊕ H^1_DR(-Ẑ) ⊕ H^+_DR(-Ẑ))

in Subsection 5.4 of <cit.>.
The orientation of λ(a) is essentially independent of the choice of Z.

We set

    λ_W:= Λ^max (H^0_DR(W)⊕ H^1_DR(W) ⊕ H^+_DR(W)),

and
λ^W(a):= λ^W(a,2) ⊗λ_W ^W (a)_(δ,δ),2.


Suppose that Y satisfies Assumption <ref>. For an irreducible flat connection a, there is a canonical identification between the orientations of λ^W(a ) and the orientations of λ(a).


It suffices to construct an isomorphism λ^W(a)≅λ(a) which is canonical up to homotopy.
First we fix two elements [A] ∈^W (a)_(δ,δ) and [B] ∈^Ẑ(a) which have representative A and B satisfying A_Y× (-∞,-1]=^*a and B|_Y× [1,∞)=^*a.
For such two connections, we obtain an element A# B ∈^Z^+_δ by gluing of connections.
This map induces an isomorphism

    #:det( d_A^*+d^+_A)⊗ det( d_B^*+d^+_B) det( d_A# B^*+d^+_A# B)

from the similar argument of Proposition 3.9 in <cit.>.
Therefore we have identification

    #:λ^W(a)|_[A]⊗λ^Z^+(a)|_[B]λ_Z^+|_[A#B].

If we choose a path from [θ] to [A#B] in ^Z^+_δ, then we have an identification between λ_Z^+|_[θ] and λ_Z^+|_[A#B]. The line bundle λ_Z^+|_[θ] is naturally isomorphic to 

    Λ^max (H^0_DR(Z^+)⊕ H^1_DR(Z^+) ⊕ H^+_DR(Z^+))

 by using Proposition 5.1 in <cit.>. This cohomology group is isomorphic to

    Λ^max (H^0_DR(Z)⊕ H^1_DR(Z) ⊕ H^+_DR(Z))⊗Λ^max (H^0_DR(W)⊕ H^1_DR(W) ⊕ H^+_DR(W))

by using the Mayer-Vietoris sequence.

Therefore

    λ^W(a)|_[A]⊗λ_W^* ≅ (λ^(-Z)(a)|_[B]⊗λ_(-Z)^*)^*

holds. Because λ^Z^+(a,2)^Z^+_δ,2 is orientable by Lemma <ref> and Proposition <ref> , the homotopy class of this identification does not depend on choices of the path, A, B, the bump functions of the gluing map.
We also have the following canonical isomorphism 

    (λ^(-Z)(a)|_[B]⊗λ_(-Z)^*)^* ≅λ^Z(a)|_[B]⊗λ_Z^*.

by the gluing Z and -Z as above discussion and the Mayer-Vietoris sequence.
This completes the proof.

Combining Theorem <ref> and Lemma <ref>, we have:
Under the assumption of Theorem <ref>, an orientation λ(a) and an orientation of λ_W give an orientation of M^W(a)_π,Ψ,ϵ,δ.




§ PROOF OF MAIN THEOREM




Let Y, l_Y and X be as in Section <ref>. 
Take a Riemannian metric g_Y on Y. Fix a non-negative real number r ∈Λ_Y smaller than Q^2l_Y+3_X. Suppose that there is an embedding f of Y into X satisfying f_*[Y]= 1 ∈ H_3(X,). Then we obtain the oriented homology cobordism from Y to -Y by cutting open along Y. Recall that W is a non-compact oriented Riemann 4-manifold W with both of cylindrical end and periodic end which is formulated at the beginning of Subsection <ref>. 


We fix a holonomy perturbation π∈∏(Y) satisfying the following conditions.


  * π is a ϵ-perturbation in Subsection <ref>.

  * π is a regular perturbation in the end of Subsection <ref>.

  * π is an element of ∏(Y)^flat in Definition <ref>.  

  * For a∈R(Y) with 0≤ cs_Y(a) < min{1,Q^2l_Y+3_X} and A ∈ M^W(a)_π,δ,

    1/8π^2sup_n ∈ ||F(A)+sπ(A)||^2_L^2(W) <min{1,Q^2l_Y+3_X}

holds.

  * d^+_θ+  sdπ_θ:^W(θ)_(δ,δ)×^d Ω^+(W)⊗_L^2_q-1,(δ,δ) is surjective.

  * For c ∈R(Y)_π satisfying cs_Y(c)<0, M^W(c)_π,δ is the empty set.

Assumption <ref> and the proof of Thereom 8.4 (ii) of <cit.> implies the existence of the perturbation ssatisfying the third condition. The first, forth, fifth and sixth conditions follow from choosing small h∈ C^l'(SU(2)^d',)_ad of π=(f,h).  

Next we also fix a holomomy perturbation (Ψ,ϵ) satisfying the following conditions.


  * (Ψ,ϵ) is a regular for [b] ∈R(Y) with 0≤ cs_Y(b) ≤ cs_Y(a). 

  * For a∈R(Y) with 0≤ cs_Y(a) < min{1,Q^2l_Y+3_X},

    1/8π^2sup_n ∈ ||F(A)+sπ(A)+σ_Ψ(A,ϵ)||^2_L^2(W) <min{1,Q^2l_Y+3_X}

hold.

To get the first condition, we use Lemma <ref>. The second condition satisfied when we take ϵ sufficiently small.

 In order to formulate the instanton Floer homology of Y with  coefficient, we fix an orientation of fix an orientation of λ(a) for each a ∈ R(Y). The orientation of Y induce an orientation of λ_W. To determine the orientation of M^W(a)_π,Ψ,ϵ,δ, we fix a compact oriented manifold Z with H_1(Z,)≅ 0 as in Subsection <ref>.
 The relation between λ_Z,a and λ(a) is given by
 
    λ^Z(a) ⊗λ_Z≅λ(a).



 Let a be a flat connection satisfying cs_Y(a)<r≤min{Q_X^2l_Y+3,1} and (a)=1. We consider the moduli space M^W(a)_π,Ψ,ϵ,δ. From the choice of these perturbation data and Corollary <ref>, M^W(a)_π,Ψ,ϵ,δ has a structure of 1-dimensional manifold for small δ. From Theorem <ref>, we obtain an orientation of  M^W(a)_π,Ψ,ϵ,δ induced by the orientation of λ_Z,a.  
 
 Let (A,B) be a limit point of M^W(a)_π,Ψ,ϵ,δ. Using Theorem <ref> and the standard dimension counting argument, the limit points of M^W(a)_π,Ψ,ϵ,δ correspond to two cases:


  * (A,B) ∈⋃_b ∈R^*(Y), cs_Y(b)<r,(b)=0 M(a,b)_π× M^W(b)_π,Ψ,ϵ,δ

  * (A,B) ∈ M(a,θ)_π× M^W(θ)_π,Ψ,ϵ,(δ,δ).

For the second case, we use the exponential decay estimate to show B ∈ M^W(θ)_π,Ψ,ϵ,(δ,δ).
Here M(a,b)_π and M(a,θ)_π,δ have a structure of 1-dimensional manifold. The quotient spaces M(a,b)_π/ and M(a,θ)_π,δ/ have a structure of compact oriented 0-dimensional manifold whose orientation induced by the orientation of λ_Z and  action by the translation as in Subsection 5.4 of <cit.>. Corollary <ref> and Theorem <ref> imply that M(b)_π,Ψ,ϵ,δ has a structure of compact oriented 0-manifold whose orientation induced by the orientation of λ_b and λ_W for small δ. Since the formal dimension of M(θ)_π,Ψ,ϵ,(δ,δ) is -3 from Proposition <ref> and there is no reducible solution except θ for a regular perutrbation (Ψ,ϵ), M(θ)_π,Ψ,ϵ,δ consists of just one point. By the gluing theory as in Theorem 4.17 and Subsection 4.4.1 of <cit.>, there is the following diffeomorphism onto its image:

    𝒥:(⋃_b ∈R^*(Y), cs_Y(b)<r, (b)=0(M(a,b)_π/× M(b)_π,Ψ,ϵ,δ) ∪ M(a,θ)_π/)× [T,∞)


    M^W(a)_π,Ψ,ϵ,δ.

By the definition of the orientation of M(a,b)_π/ and M(a,θ)_π/, we can construct 𝒥 as an orientation preserving map. Furthermore, the complement of 𝒥 is compact. Therefore we can construct the compactification of M^W(a)_π,Ψ,ϵ,δ by adding the finitely many points 

    ⋃_b ∈R^*(Y), cs_Y(b)<r,(b)=0  (M(a,b)_π/× M(b)_π,Ψ,ϵ,δ) ∪  M(a,θ)_π/,

which has a structure of compact oriented  1-manifold.
By counting of boundary points of the compactification, we obtain the relation 

    δ^r( n)(a)+ θ^r(a)=0,

where n ∈ CF^0_r(Y) is defined by n(b):= # M(b)_π,Ψ,ϵ,δ. This implies θ^r is a coboundary. Therefore we have 0=[θ^r] ∈ HF^1_r(Y) for 0≤ r ≤min{Q^2l_Y+3_X,1}.

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