Topological degree for equivariant gradient perturbations of an unbounded self-adjoint operator in Hilbert space
Piotr Bartłomiejczyk, Bartosz Kamedulski, Piotr Nowak-Przygodzki
TTOPOLOGICAL DEGREE FOR EQUIVARIANT GRADIENTPERTURBATIONS OF AN UNBOUNDED SELF-ADJOINTOPERATOR IN HILBERT SPACE
PIOTR BARTŁOMIEJCZYK, BARTOSZ KAMEDULSKI,AND PIOTR NOWAK-PRZYGODZKIAbstract. We present a version of the equivariant gradient de-gree defined for equivariant gradient perturbations of an equivari-ant unbounded self-adjoint operator with purely discrete spectrumin Hilbert space. Two possible applications are discussed.
IntroductionTo obtain new bifurcation results, N. Dancer [5] introduced in 1985a new topological invariant for S -equivariant gradient maps, whichprovides more information than the usual equivariant one. In 1994S. Rybicki [14, 16] developed the complete degree theory for S -equi-variant gradient maps and 3 years later K. Gęba extended this theoryto an arbitrary compact Lie group. In 2001 S. Rybicki [15] definedthe degree for S -equivariant strongly indefinite functionals in Hilbertspace. 10 years later A. Gołębiewska and S. Rybicki [8] generalizedthis degree to compact Lie groups. The relation between equivariantand equivariant gradient degree theories were studied in [1, 2, 7].The main goal of this paper is to present a construction and proper-ties of a new degree-type topological invariant Deg ∇ G , which is definedfor equivariant gradient perturbations of a equivariant unboundedself-adjoint Hilbert operator with a purely discrete spectrum (in thegeneral case a compact Lie group). As far as we know, the idea of theconstruction of such an invariant should be attributed to K. Gęba.It is worth pointing out that equivariant gradient perturbations ofan equivariant unbounded self-adjoint operator with a purely discretespectrum appear naturally in a variety of problems in nonlinear anal-ysis, such as the search for periodic solutions of Hamiltonian systems Date : December 24, 2018.2010
Mathematics Subject Classification.
Primary: 47H11; Secondary: 55P91.
Key words and phrases.
Topological degree, unbounded self-adjoint operator,equivariant gradient map. a r X i v : . [ m a t h . A T ] D ec P. BARTŁOMIEJCZYK, B. KAMEDULSKI, AND P. NOWAK-PRZYGODZKI or the study of Seiberg-Witten equations for three dimensional man-ifolds. The purpose of our work is to provide a topological tool thatallows us to solve problems similar to the above mentioned ones.The paper is organized as follows. Section 1 contains some prelimi-naries. In Section 2 we present the construction that leads to the defi-nition of the degree Deg ∇ G . The correctness of this definition is provedin Section 3. The properties of the degree Deg ∇ G are examined in Sec-tion 4. Finally, in Section 5 we discuss two examples of possible appli-cations. 1. PreliminariesThe preliminaries are divided into five brief subsections.1.1. Unbounded self-adjoint operators in Hilbert space.
This sub-section is based on [17]. Let E be a real separable Hilbert space withinner product (cid:104)· | ·(cid:105) and A : D ( A ) ⊂ E → E be a linear operator (notnecessarily bounded) such that its domain D ( A ) is dense in E . Set D ( A ∗ ) = { y ∈ E | ∃ u ∈ E ∀ x ∈ D ( A ) (cid:104) Ax | y (cid:105) = (cid:104) x | u (cid:105) } .Since D ( A ) is dense in E , the vector u ∈ E is uniquely determined by y .Therefore by setting A ∗ y = u we obtain a well-defined linear oper-ator from D ( A ∗ ) to E . The operator A ∗ is called the adjoint operatorof A . We say that A is self-adjoint if A = A ∗ . By the Hellinger-Toeplitztheorem, if A is self-adjoint and D ( A ) = E then A is bounded.It is easy to see that (cid:104) x | y (cid:105) = (cid:104) x | y (cid:105) + (cid:104) Ax | Ay (cid:105) defines an inner product on the domain D ( A ) . Under this product D ( A ) becomes a Hilbert space, which will be denoted by E . Thus D ( A ) and E are equal as sets but equipped with different inner prod-ucts. Note that A treated as an operator from E to E is bounded.We say that a self-adjoint operator A has a purely discrete spectrum ifits spectrum consists only of isolated eigenvalues of finite multiplicity.If E is an infinite dimensional Hilbert space then following conditionsare equivalent:(1) A has a purely discrete spectrum.(2) There is a real sequence { λ n } and an orthonormal basis { e n } such that lim | λ n | = ∞ and Ae n = λ n for n ∈ N .(3) The embedding ı : E → E is compact. OPOLOGICAL DEGREE 3
Local maps in Hilbert space.
Let • E be a real Hilbert orthogonal representation of a compact Liegroup G , • A : D ( A ) ⊂ E → E be an unbounded self-adjoint operatorwith a purely discrete spectrum, • D ( A ) be invariant and A equivariant. Definition 1.1.
We write f ∈ G G ( E ) if • f : D f ⊂ E → E , where D f is an open invariant subset of E , • f ( x ) = Ax − ∇ ϕ ( x ) , where ϕ : E → R is C and invariant, • f − ( ) is compact.Elements of G G ( E ) will be called local maps .1.3. Otopies in Hilbert space.
Let I = [
0, 1 ] . Assume that G actstrivially on I . A map h : Λ ⊂ I × E → E is called an otopy if • Λ is an open invariant subset of I × E , • h ( t , · ) ∈ G G ( E ) for each t ∈ I , • h − ( ) is compact.Given an otopy h : Λ ⊂ I × E → E we can define for each t ∈ I : • sets Λ t = { x ∈ E | ( t , x ) ∈ Λ } , • maps h t : Λ t → E with h t ( x ) = h ( t , x ) .If h is an otopy, we say that h and h are otopic . The relation of beingotopic is an equivalence relation in G G ( E ) .Observe that if f is a local map and U is an open subset of D f suchthat f − ( ) ⊂ U , then f and f (cid:22) U are otopic. This property of localmaps is called the restriction property . In particular, if f − ( ) = ∅ then f is otopic to the empty map.1.4. Euler-tom Dieck ring.
Recall the notion of the
Euler-tom Dieckring following [19]. For a compact Lie group G let U( G ) denote theset of equivalence classes of finite G -CW-complexes. Two complexes X and Y are identified if the quotients X H /WH and Y H /WH have thesame Euler characteristic for all closed subgroups H of G . Recall that X H stands here for the H -fixed point set of X , i.e. X H := { x ∈ X | hx = x for all h ∈ H } and WH for the Weyl group of H , i.e. WH = NH/H .Addition and multiplication in U( G ) are induced by disjoint union andcartesian product with diagonal G -action, i.e. [ X ] + [ Y ] = [ X (cid:116) Y ] , [ X ] · [ Y ] = [ X × Y ] ,where the square brackets stand for an equivalence class of finite G -CW-complexes. In this way U( G ) becomes a commutative ring withunit and is called the Euler-tom Dieck ring of G . P. BARTŁOMIEJCZYK, B. KAMEDULSKI, AND P. NOWAK-PRZYGODZKI
Additively, U( G ) is a free abelian group with basis elements [ G/H ] ,where H is a closed subgroup of G . In consequence, each element of U( G ) can be uniquely written as a finite sum (cid:80) d ( H ) [ G/H ] , where d ( H ) is an integer, which depends only on the conjugacy class of H . The ringunit is [ G/G ] .1.5. Finite dimensional equivariant gradient degree deg ∇ G . Assumethat V is a real finite dimensional orthogonal representation of a com-pact Lie group G . We write f ∈ G G ( V ) if f is an equivariant gradientmap from an open invariant subset of V to V and f − ( ) is compact. Inthe papers [1, 2, 6, 16] the authors defined the equivariant gradientdegree deg ∇ G : G G ( V ) → U( G ) and proved that the degree has the following properties: additivity,otopy invariance, existence and normalization. The product propertyformulated below was proved in [6] and [9]. Theorem 1.2 (Product property) . Let V and W be real finite dimen-sional orthogonal representations of a compact Lie group G . If f ∈ G G ( V ) and f (cid:48) ∈ G G ( W ) , then f × f (cid:48) ∈ G G ( V ⊕ W ) and deg ∇ G ( f × f (cid:48) ) = deg ∇ G ( f ) · deg ∇ G ( f (cid:48) ) in U( G ) . In the next section we will make use of the following result, whichcan be found in [8, Cor. 2.1].
Theorem 1.3.
Let V be a real finite dimensional orthogonal represen-tation of a compact Lie group G . If B is an equivariant self-adjoint iso-morphism of V then deg ∇ G ( B ) is invertible in U( G ) .Remark . Note that Theorem 1.3 holds even if V is trivial. In thiscase deg ∇ G ( B ) is equal to the unit of U( G ) .2. Definition of degreeIn this section we present the construction of the degree Deg ∇ G usingfinite dimensional approximations.2.1. Finite dimensional approximations.
Let us start with some no-tations: • for λ ∈ σ ( A ) denote by V ( λ ) the corresponding eigenspace; • for n ∈ N write V n = ⊕ | λ | (cid:54) n V ( λ ) , V n = ⊕ n − < | λ | (cid:54) n V ( λ ) and A n = A (cid:22) V n ; hence V n = V n − ⊕ V n ; • let P n : E → V n denote the orthogonal projection. OPOLOGICAL DEGREE 5
Assume that U is an open bounded invariant subset of D f such that f − ( ) ⊂ U ⊂ cl U ⊂ D f .Set U n = U ∩ V n . Finally, let f n : U n → V n be given by f n ( x ) = Ax − P n F ( x ) ,where F ( x ) = ∇ ϕ ( x ) .The following two lemmas are needed to prove Lemma 2.3, whichis crucial for the definition of Deg ∇ G . Lemma 2.1.
There is (cid:15) > such that | f ( x ) | (cid:62) (cid:15) for all x ∈ ∂U .Proof. The fact F is compact and ∂U is closed and bounded implies ourclaim. (cid:3) Let us introduce an auxiliary map (cid:101) f n : D f → E given by (cid:101) f n ( x ) = Ax − P n F ( x ) . By definition, (cid:101) f n (cid:22) U n = f n . Lemma 2.2.
There is N such that for n (cid:62) N we have (1) | f ( x ) − (cid:101) f n ( x ) | < (cid:15) for x ∈ cl U , (2) | (cid:101) f n ( x ) | > (cid:15) for x ∈ ∂U .Proof. Since F is compact, F is close to P n F , which gives (1). In turn(2) follows from (1) and Lemma 2.1. (cid:3) Lemma 2.3.
For n (cid:62) N we have f n ∈ G G ( V n ) and, in consequence, deg ∇ G ( f n ) ∈ U( G ) is well-defined.Proof. Since f n is obviously gradient, it is enough to check that f − n ( ) is compact. Note that (cid:101) f n can be considered as an extension of f n oncl U n . By (2) from Lemma 2.2, (cid:101) f n does not have zeroes in ∂U n ⊂ ∂U ,which implies that f − n ( ) = (cid:101) f − n ( ) ∩ U n is compact. (cid:3) Degree definition.
Observe that A n is an equivariant self-adjointisomorphism for n (cid:62)
1. By Theorem 1.3, elements a n := deg ∇ G ( A n ) are invertible in U( G ) . Set m n := a − · a − · · · · · a − n . Definition 2.4.
Let Deg ∇ G : G G ( E ) → U( G ) be defined byDeg ∇ G ( f ) := m n · deg ∇ G ( f n ) for n (cid:62) N .An alternative definition of Deg ∇ G in terms of the direct limit is givenin Appendix A. P. BARTŁOMIEJCZYK, B. KAMEDULSKI, AND P. NOWAK-PRZYGODZKI
3. Correctness of the definitionWe have to prove that our definition does not depend on the choiceof n and the neighbourhood U .3.1. Independence from the choice of n . To show this we will needthe following lemma.
Lemma 3.1.
For n large enough f n + is otopic to f n × A n + in G G ( V n + ) and hence deg ∇ G ( f n + ) = deg ∇ G ( f n × A n + ) . Proof.
First observe there is an open W ⊂ U and natural number N such that • f − ( ) ⊂ W ⊂ U , • P n ( cl W ) ⊂ U n for all n (cid:62) N .Define h n + : I × cl W n + → V n + by h n + ( t , x ) = ( − t ) f n + ( x ) + t ( f n × A n + )( x ) .We set n sufficiently large. One can show that h n + ( t , x ) (cid:54) = t ∈ I and x ∈ ∂W n + . In consequence, h n + (cid:22) I × W n + is a finite dimensionalequivariant gradient otopy between f n + (cid:22) W n + and f n × A n + (cid:22) W n + (otherwise there would be a point x ∈ ∂W such that f ( x ) = f n + and f n × A n + are otopic to their restrictions to W n + , which completesthe proof. (cid:3) From Lemma 3.1 and Theorem 1.2 we can easily conclude thatdeg ∇ G ( f n + ) = deg ∇ G ( f n × A n + ) = deg ∇ G ( f n ) · deg ∇ G ( A n + ) = a n + · deg ∇ G ( f n ) .This gives m n + · deg ∇ G ( f n + ) = m n + · a n + · deg ∇ G ( f n ) = m n · deg ∇ G ( f n ) ,which shows that Deg ∇ G ( f ) does not depend on the choice of n largeenough.3.2. Independence from the choice of U . According to our defini-tion Deg ∇ G ( f ) = Deg ∇ G ( f (cid:22) U ) . Now we will prove that in fact Deg ∇ G ( f ) isindependent from the choice of the neighbourhood U . Lemma 3.2.
Let W and U be open bounded sets such that f − ( ) ⊂ W ⊂ U ⊂ cl U ⊂ D f . Then
Deg ∇ G ( f (cid:22) W ) = Deg ∇ G ( f (cid:22) U ) . OPOLOGICAL DEGREE 7
Proof.
By the analogue of Lemma 2.1 (with ∂U replaced by cl U \ W ), | f ( x ) | (cid:62) (cid:15) for x ∈ cl U \ W and by Lemma 2.2, | f ( x ) − (cid:101) f n ( x ) | < (cid:15) for x ∈ cl U . Hence (cid:101) f n ( x ) (cid:54) = x ∈ cl U \ W . In consequence, f n ( x ) (cid:54) = x ∈ cl U n \ W n . ThereforeDeg ∇ G ( f (cid:22) U ) = m n · deg ∇ G ( f n (cid:22) U n ) = m n · deg ∇ G ( f n (cid:22) W n ) = Deg ∇ G ( f (cid:22) W ) . (cid:3) Corollary 3.3.
Let U and U (cid:48) be open bounded subsets of D f such that f − ( ) ⊂ U ∩ U (cid:48) ⊂ cl ( U ∪ U (cid:48) ) ⊂ D f . Then
Deg ∇ G ( f (cid:22) U ) = Deg ∇ G ( f (cid:22) U ∩ U (cid:48) ) = Deg ∇ G ( f (cid:22) U (cid:48) ) . In this way we have proved that Deg ∇ G ( f ) does not depend on thechoice of admissible U .4. Degree propertiesIn this section we prove that our degree Deg ∇ G : G G ( E ) → U( G ) hasall properties analogous to the well-known properties of the finite di-mensional equivariant gradient degree deg ∇ G . Additivity property. If f , f (cid:48) ∈ G G ( E ) and D f ∩ D f (cid:48) = ∅ then Deg ∇ G ( f (cid:116) f (cid:48) ) = Deg ∇ G ( f ) + Deg ∇ G ( f (cid:48) ) . Otopy invariance property.
Let f , f (cid:48) ∈ G G ( E ) . If f is otopic to f (cid:48) then Deg ∇ G ( f ) = Deg ∇ G ( f (cid:48) ) . Existence property. If Deg ∇ G ( f ) (cid:54) = then f ( x ) = for some x ∈ D f . Normalization property.
Deg ∇ G ( A + P ) = [ G/G ] = U( G ) , where P : E → V = ker A is the orthogonal projection. Product property.
Let E and E (cid:48) be real Hilbert orthogonal representa-tions of a compact Lie group G . If f ∈ G G ( E ) and f (cid:48) ∈ G G ( E (cid:48) ) , then f × f (cid:48) ∈ G G ( E ⊕ E (cid:48) ) and Deg ∇ G ( f × f (cid:48) ) = Deg ∇ G ( f ) · Deg ∇ G ( f (cid:48) ) , where the dot here denotes the multiplication in U( G ) .Proof. Additivity.
Immediately from the additivity of deg ∇ G we obtainDeg ∇ G ( f (cid:116) f (cid:48) ) = m n · deg ∇ G ( f n (cid:116) f (cid:48) n ) = m n · ( deg ∇ G ( f n ) + deg ∇ G ( f (cid:48) n )) = Deg ∇ G ( f ) + Deg ∇ G ( f (cid:48) ) . P. BARTŁOMIEJCZYK, B. KAMEDULSKI, AND P. NOWAK-PRZYGODZKI
Otopy invariance.
Let the map h : Λ ⊂ I × E → E given by h ( t , x ) = Ax − F ( t , x ) be an otopy. We introduce the following notation: Λ t = { x ∈ E | ( t , x ) ∈ Λ } , h t : Λ t → E , h t ( x ) = h ( t , x ) , Λ n = Λ ∩ ( I × V n ) , h n : Λ n → V n , h n ( t , x ) = Ax − P n F ( t , x ) , Λ tn = Λ t ∩ V n , h tn : Λ tn → V n , h tn ( x ) = h n ( t , x ) .Note that for the needs of this subsection the time parameter t of theotopy is a superscript, not a subscript. According to the above notationwe have to show that Deg ∇ G ( h ) = Deg ∇ G ( h ) . Since h − ( ) is compact,there is an open bounded set W ⊂ I × E such that h − ( ) ⊂ W ⊂ cl W ⊂ Λ .Hence for i =
0, 1 we have ( h i ) − ( ) ⊂ W i ⊂ cl W i ⊂ Λ i ,where W i = { x ∈ E | ( i , x ) ∈ W } . Similarly as in Lemma 2.1, there is (cid:15) > | h ( z ) | (cid:62) (cid:15) for z ∈ ∂W . On the other hand, similarlyas in Lemma 2.2, there is N such that (cid:12)(cid:12) h ( z ) − (cid:101) h n ( z ) (cid:12)(cid:12) < (cid:15) for z ∈ cl W and n (cid:62) N , where (cid:101) h n : Λ → E is given by (cid:101) h n ( t , x ) = Ax − P n F ( t , x ) .Therefore | h n ( z ) | (cid:62) (cid:15) for z ∈ ∂W n ⊂ ∂W . From the above: • h n (cid:22) W n is a finite dimensional equivariant gradient otopy, • Deg ∇ G ( h i ) = m n · deg ∇ G ( h in (cid:22) W in ) ,which, by the otopy invariance of deg ∇ G , givesDeg ∇ G ( h ) = m n · deg ∇ G ( h n (cid:22) W n ) = m n · deg ∇ G ( h n (cid:22) W n ) = Deg ∇ G ( h ) . Existence. If f − ( ) = ∅ then f is otopic with the empty map. HenceDeg ∇ G ( f ) = Deg ∇ G ( ∅ ) = Normalization.
Observe that A + P is an injection anddeg ∇ G (( A + P ) n ) = deg ∇ G ( Id (cid:22) V ) · deg ∇ G ( A ) · . . . · deg ∇ G ( A n ) = m − n for any n (cid:62)
1. HenceDeg ∇ G ( A + P ) = m n · deg ∇ G (( A + P ) n ) = [ G/G ] . OPOLOGICAL DEGREE 9
Product formula.
Let f ( x ) = Ax − F ( x ) and f (cid:48) ( x ) = A (cid:48) x − F (cid:48) ( x ) .Observe that, by Theorem 1.2, if f n ∈ G G ( V n ) and f (cid:48) n ∈ G G ( V (cid:48) n ) then f n × f (cid:48) n ∈ G G ( V n ⊕ V (cid:48) n ) anddeg ∇ G ( f n × f (cid:48) n ) = deg ∇ G ( f n ) · deg ∇ G ( f (cid:48) n ) .Moreover, for n large enoughDeg ∇ G ( f ) = m n · deg ∇ G ( f n ) ,Deg ∇ G ( f (cid:48) ) = m (cid:48) n · deg ∇ G ( f (cid:48) n ) .Since for any i (cid:62) ∇ G (( A × A (cid:48) ) i ) = deg ∇ G ( A i × A (cid:48) i ) = deg ∇ G ( A i ) · deg ∇ G ( A (cid:48) i ) ,we haveDeg ∇ G ( f × f (cid:48) ) = m n · m (cid:48) n · deg ∇ G ( f n × f (cid:48) n ) = m n · m (cid:48) n · deg ∇ G ( f n ) · deg ∇ G ( f (cid:48) n ) = Deg ∇ G ( f ) · Deg ∇ G ( f (cid:48) ) . (cid:3) Remark . The normalization property can be formulated more gen-erally, but the proof of this fact will appear elsewhere. Namely, let x ∈ V n and, in consequence, Gx ⊂ V n . Define U = { x + y + z | x ∈ Gx , y ∈ (cid:0) T x ( Gx ) (cid:1) ⊥ ⊂ V n , | y | < (cid:15) , z ∈ (cid:0) V n (cid:1) ⊥ ⊂ E } and f : U → E by f ( x + y + z ) = ( A + P )( y + z ) .Then Deg ∇ G ( f ) = [ G/G x ] .5. Possible applicationsWe should emphasize that this section contains not real applicationsof the theory but only two exemplary situations illustrating potentialapplications.5.1. Applications to Hamiltonian systems.
The search for periodicsolutions in Hamiltonian systems is one of the fundamental problemsin nonlinear analysis (see for instance [3, 12, 13, 20]). Consider theHamiltonian system of ODE dpdt = − H q , dqdt = H p , where H ∈ C ( R n , R ) and p , q ∈ R n or equivalently dzdt = J H z ,where z = ( p , q ) and J = (cid:18) − II (cid:19) .The function H is called the hamiltonian or energy.Rewrite the Hamiltonian system as( ∗ ) ˙ z = J ∇ H ( z ) , z ∈ R n or equivalently − J ˙ z − ∇ H ( z ) = z ∈ H T of the equation ( ∗ ), where H T ( T >
0) denotes the completion of the set of smooth T -periodicfunctions from R to R n in the norm associated to the inner product ( u | v ) H T = (cid:82) T uv dt + (cid:82) T ˙ u ˙ v dt . For this purpose we apply the methodof the topological degree Deg ∇ S . Namely, let E = L ( S , R n ) and E = H ( S , R n ) . Moreover, denote by D the set E equipped with the innerproduct from E .Observe that • E and E are Hilbert spaces and orthogonal representations ofthe group SO ( ) = S with the S -action given by the shift intime, • A : D → E given by Az = − J ˙ z is an equivariant unboundedself-adjoint operator with a purely discrete spectrum, • ∇ H ( z ) is a gradient of the invariant functional ϕ : E → R de-fined by ϕ ( z ) = (cid:82) π H ( z ( t )) dt , • ∇ H ◦ ı : E → E is a compact map by the compactness of the in-clusion ı : E → E .We can now formulate the main result of this subsection. Theorem 5.1.
Assume that λ > and the set of zeros of the map f λ ( z ) =− J ˙ z − λ ∇ H ( z ) is compact. If Deg ∇ S ( f λ ) (cid:54) = then the equation ( ∗ ) hasa solution in H πλ .Proof. First note that if f − λ ( ) is compact then f λ is an element of G S ( E ) . By the existence property, Deg ∇ S ( f λ ) (cid:54) = f λ ( z ) = z ∈ E . Hence a lift (cid:101) z ∈ H πλ of z given by (cid:101) z ( t ) = z ( ρ ( t )) ,where ρ : R → S is the standard covering projection, is a solutionof ( ∗ ), which is our claim. (cid:3) OPOLOGICAL DEGREE 11
Applications to the Seiberg-Witten equations.
The descriptionof the Seiberg-Witten equations presented here is necessarily sketchy(for more details we refer the reader to [4, 10, 11, 18]). Let M be aclosed oriented Riemannian 3-manifold. A Spin c -structure on M con-sists of rank two Hermitian vector bundle S → M called the spinorbundle . We write Ω ( M , i R ) for the space of smooth imaginary-valued1-forms on M and Γ ( S ) for the space of smooth cross-sections of thespinor bundle S → M . For each a ∈ Ω ( M , i R ) there is an associated Dirac operator D a : Γ ( S ) → Γ ( S ) .Recall that, in what follows, d stands for the exterior derivative and ∗ denotes the Hodge star. For a pair ( a , ϕ ) ∈ Ω ( M , i R ) ⊕ Γ ( S ) the Seiberg-Witten equations are (cid:14) D a ϕ = ∗ da = Q ( ϕ ) ,where Q ( ϕ ) ∈ Ω ( M , i R ) is a certain quadratic form (nonlinear partof the equations). The solutions of Seiberg-Witten equations are zerosof the Seiberg-Witten map SW : Ω ( M , i R ) ⊕ Γ ( S ) → Ω ( M , i R ) ⊕ Γ ( S ) given by SW ( a , ϕ ) = ( ∗ da − Q ( ϕ ) , − D a ϕ ) .After suitable Sobolev completion the Seiberg-Witten map SW can bewritten in the form A − F , where A = ( ∗ da , − D a ϕ ) is an unboundedself-adjoint operator with a purely discrete spectrum and F is a gra-dient map. Moreover, the Seiberg-Witten map is equivariant for theaction of the group S , which acts trivially on the component aris-ing from the differential forms and as complex multiplication on thespinor component. It suggests that the SW map should fit to our ab-stract setting of the degree Deg ∇ S . Unfortunately, the set of zeros ofthe SW map is not compact. However, we hope that it is possible toreduce our problem to some subspace of Ω ( M , i R ) in such a way thatthe reduced SW map will have a compact set of zeros, which will becontained in the set of zeros of the original SW map. Verifying thisclaim is, however, still in progress.Appendix A.Definition 2.4 may be seen as a simple particular case of a moregeneral construction called the direct limit of a direct system of groups . Namely, for i =
0, 1, . . . let G i denote an abelian group and α i : G i → G i + a group homomorphism. With this notation we get the sequence G α −→ G α −→ G α −→ G → · · · Let (cid:101) G := (cid:96) ∞ i = G i denote a disjoint union, i.e. (cid:101) G = { ( i , m ) | i ∈ N , m ∈ G i } .We introduce in (cid:101) G an equivalence relation. For i > j we write ( i , m ) ∼ ( j , l ) if α i − ◦ · · · ◦ α j + ◦ α j ( l ) = m .The direct limit of groups is the set of equivalence classes of the aboverelation, denoted by lim −→ G i = (cid:101) G/ ∼ .Let lim −→ U( G ) denote a direct limit of groups, where • G i = U( G ) for all i , • α i is multiplication by an element a i = deg ∇ G ( A i , V i ) ∈ U( G ) .With this notation we can alternatively define our degree as a functionDeg ∇ G : G G ( E ) → lim −→ U( G ) ≈ U( G ) given byDeg ∇ G ( f ) := [( n , deg ∇ G ( f n , U n ))] for n large enough. References [1] P. Bartłomiejczyk, K. Gęba, M. Izydorek, Otopy classes of equivariant local maps ,J. Fixed Point Theory Appl. 7(1) (2010), 145–160.[2] P. Bartłomiejczyk, P. Nowak-Przygodzki,
The Hopf type theorem for equivariantgradient local maps , J. Fixed Point Theory Appl. 19(4) (2017), 2733–2753.[3] T. Bartsch, A. Szulkin,
Hamiltonian systems: periodic and homoclinic solutionsby variational methods.
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