Topological Invariants for Equivariant Flows: Conley Index and Degree

Home , Equivariant topology

Institute of Mathematics Polish Academy of Sciences

Marcin Styborski

Topological invariants for equivariant ﬂows: Conley index and degree

PhD Thesis

Supervisor: Dr hab. Marek Izydorek Gdańsk University of Technology

Gdańsk 2009 Contents

Contentsi

0 Introduction1 0.1 Overview of the results...... 2

1 Classical Conley’s theory5 1.1 Morse–Conley–Zehnder equation...... 5 1.2 Continuation to a gradient...... 9 1.3 Euler characteristic of the index...... 10 1.4 Conley index and the Brouwer degree...... 10

2 LS -index 13 2.1 LS -ﬂows and the index...... 13 2.1.1 Cohomological LS -Conley index...... 16 2.2 LS -index and the Leray–Schauder degree...... 18 2.2.1 Alternative approach in the particular case...... 21

3 Equivariant theory 25 3.1 Basic equivariant topology...... 26 3.1.1 Orbit types...... 27 3.1.2 Slices...... 28 3.1.3 G-complexes...... 29 3.1.4 Euler ring U(G) ...... 30 3.2 Degree for G-equivariant gradient maps...... 32 3.3 Equivariant Conley index...... 34 3.4 Equivariant Morse–Conley–Zehnder equation...... 37 3.4.1 Poincaré polynomial for an isolated orbit...... 41 3.4.2 Some multiplicity results...... 45 3.5 Continuation of equivariant maps to a gradient...... 51

4 On the invertibility in U(G) 54 4.1 Technicalities...... 54 4.2 Self-invertibility of SV ...... 55

5 Appendix 58

Bibliografy 62 Chapter 0

Introduction

About forty years have passed since Charles Conley defined the homotopy index. Thereby, he generalized the ideas that go back to the calculus of variations work of Marston Morse. Within this long time the Conley index has proved to be a valuable tool in nonlinear analysis and dynamical systems. A significant development of applied methods has been observed. Later, the index theory has evolved to cover such areas as discrete dynamical systems, or analysis of flows defined on locally noncompact spaces cf. LS -index. Using the Conley theory, one is interested in the behavior of the particular sets of solutions, called isolated invariant sets, of differential equations. The index of an isolated invariant set S is a homotopy type (or, in case of an LS -index, a stable homotopy type) of the quotient X/A of a certain pair, called the index pair. It will be denoted by h(S). Since the homotopy types cannot be lined up (like, for instance, the real numbers) and they are often very difficult to distinguish, they are fairly hard to work with. Thus, the cohomological index H∗(X/A) has been found to be more accessible to the applications. It it easier to compare this index with other algebraic topological characteristics of the dynamical systems. Probably the most important feature, among others, of the Conley index is the invariance with respect to small perturbations of the initial differential equation. A large collection of tools, called the homotopy invariants, has this special property. They include: the index of a zero of a vector field, topological degree, intersection number, Lefschetz number etc. Herein, we are focused on the topological degree and some of its extensions. The over- all aim of this thesis is to study the relationship between the degree of a vector field and the Conley index of the induced flow. A large part of the thesis is devoted to the equivariant version of the Morse type inequalities (called equivariant Morse–Conley–Zehnder equation). The equivariant Morse inequalities have been used to compare the G-Conley index with the gradient equivariant degree. It was actually my primary intention. How- ever, this Morse–Conley–Zehnder equation seems to be very useful in the critical point theory. Therefore, I decided to place some simple multiplicity results for critical orbits of invariant (with respect to the Lie group action) functions. They are rather well known to many mathematicians. My intention was only to indicate possible directions in which one can go using these methods. Still, it would be interesting to extend these methods to infinite-dimensional domains, i.e., to apply the G-equivariant MCZ equation to critical point theory of strongly indefinite functionals on Hilbert 0.1 Overview of the results 2 spaces. From this point of view, it would allow us to investigate the Hamil- tonian systems using the variational treatment. As it is shown in [24], this problem is naturally equipped with symmetries of a group SO(2) (and its subgroups).

0.1 Overview of the results.

The celebrated Poincaré-Hopf Theorem establishes a relationship between the local invariants of a vector field at its zeroes and the global invariants of the compact manifold where it is defined. The theses contained in this dissertation are another results of this kind. Here the compact manifold is replaced by a so-called index pair, and the topological degree plays a role of the local invariant of a vector field. We treat separately three cases. 1. Firstly, we provide a known comparison of the classical Conley index and the Brouwer degree (Theorem (1.17)). Namely, if φv is a (local) flow of the differential equation x˙ = −v(x), and S is an isolated φv-invariant set with an isolateing neighborhood N (cf. Section 1.1), then

χ(h(S)) = deg(v, N).

The above formula has been first proved by McCord in [33]. Earlier Dancer in [9] proved this kind of relation for considerably smaller class of isolated invariant sets, precisely for degenerate critical points. A simple proof can be found in the book by Rybakowski [41] (See Chapter 3, Theorem 3.8). We present an elegant proof of this fact given by Razvan and Fotouhi in [37], based on Morse inequalities and Reineck continuation theorem [38]. 2. The LS -Conley index, the extension of the Conley’s invariant, is presented, and the relations to the Leray–Schauder degree are studied. The extension of the classical Conley’s theory was introduced by Gęba, Izydorek and Pruszko in [16]. They considered so-called LS -vector fields in a Hilbert space, i.e., completely continuous perturbations of a bounded linear operator L: H → H, and defined the index for flows induced by such maps. There is a particular property that makes the LS -index applicable to many variational problems. Namely, an operator L can be strongly indefinite, i.e., both positive and negative eigenspaces of L can be infinite-dimensional. Further development of this homotopy invariant was presented by Izydorek in [23]. He defined a cohomological LS -index and using this index gave existence results for various strongly indefinite problems. We briefly sketch out this definition. A cohomological version of the LS -index allows us to define the Betti numbers and the Euler characteristic of the LS -index in the most natural way. Let H be a real, infinite-dimensional Hilbert space. With a locally Lips- chitz vector field f : H → H, which is a completely continuous perturbation, say K, of the bounded (invertible) linear operator L, one can associate a 0.1 Overview of the results 3

t local ﬂow φf satisfying d φt = −f ◦ φt , φ0 = id. dt f f f Under certain assumptions we prove the formula (cf. Theorem (2.13)) ˆ χ(hLS (S)) = degLS (f, N). The right-hand side of the above equality stands for the standard Leray– Schauder degree with respect to a bounded set N. The map fˆ is deﬁned by fˆ(x) = x+L−1K(x). On the other hand we have the Euler characteristic (cf.

Definition (2.11)); hLS (S) stands for the LS -index of an isolated invariant t set S = inv(N) of the flow φf . The proof is based on finite-dimensional formula mentioned above. Similar result was obtained by Kryszewski and Szulkin in [31] for S being a critical point of a smooth strongly indefinite functional. We also give an alternative method of proving Theorem (2.13), at least in a particular case, see Theorem (2.22). The isolated invariant set is the origin of a Hilbert space, and a map L is of the form Lx := x+ − x−, where x = x+ + x− ∈ H = H+ ⊕ H−. 3. At last, the G-equivariant Conley index and the G-equivariant gradient degree is studied. We placed the most emphasis on this case. With this end in view we proved an equivariant version of Morse–Conley–Zehnder equation, see Theorem (3.47). The key point to obtain this result was to accurately define the Poincaré polynomial of the G-index hG(S) of an isolated invariant G-set. Having in mind the form of the elements of the Euler ring (cf. Proposition (3.16)), we define

∞ X X q (H) >(H) (H) q G PG(t, hG(S)) := rank H (X /G, (X ∪ A )/G) t u(H), (H)∈Φ(G) q=0 where (X,A) is an arbitrary G-index pair for S. Beside the fact that this polynomial has an unfriendly form, and causes some technical diﬃculties, the proof of the Morse–Conley–Zehnder equation is a consequence of standard cohomological arguments. With the help of the prepared tools, and using approximation techniques for gradient G-equivariant mappings provided by Gęba (see Theorem (3.24)), we prove that (vide Theorem (3.61))

∇ u(hG(S)) = degG(f, Ω), where u stands for, roughly speaking, an equivariant Euler characteristic (taking values in the Euler ring U(G)). In addition to the comparison of the Conley index and the degree, one can also find two, unconnected at the first sight, results. The first one is concerned with the G-flows and asserts that an isolated invariant set of a 0.1 Overview of the results 4 G-flow can be linked by a special kind of homotopy to the isolated invariant set of a gradient G-flow (cf. Theorem (3.74)). This is the G-equivariant counterpart of the Reineck continuation theorem [38]. Using it I proved that the G-equivariant Conley index is a homotopy type of a finite G-complex (cf. Corollary (3.79)). The second one is rather purely algebraic. Gołębiewska and Rybicki showed in [18] that if V is a finite dimensional orthogonal G- representation, then u(SV ) is an invertible element in the ring U(G) (see Definition (3.15)). By strengthening the assumptions I give a very simple, geometric proof of this result. Namely, I assume that G is finite and abelian. At the same time I show that u(SV ) is self-invertible, i.e., u(SV )−1 = u(SV ). As already mentioned, some multiplicity results are also given (cf. Propo- sitions (3.66), (3.68) and (3.73)). These results provide an estimation from below of the number of critical orbits of G-invariant functions, for G = Zp (p being a prime number), and G = SO(2). The case of the most general Z2-action on an inner product space of finite dimension is presented. The main argument is the equivariant Morse–Conley–Zehnder equation.

Acknowledgements.

First and foremost I would like to express my gratitude to my advisor, prof. Marek Izydorek for his professional guidance and constant encourage- ment which made this work possible. I am very grateful for his suggestions and many hours of enlightening discussions. I would like to thank prof. Kazimierz Gęba for always having his door open to answer my questions. I also express my gratitude to all the partic- ipants of the seminar on Topological Methods in Nonlinear Analysis which takes place at Gdańsk University of Technology for friendly criticism of my presentations. The work was supported in part by MNiSW grant N N201 273235. Chapter 1

Classical Conley’s theory

The purpose of this chapter is to provide the rudiments of Conley’s index theory for flows on locally compact metric spaces. The reader will find here basic definitions which will be needed in the further parts of the thesis. The material is presented in such a way that we will be able to give a complete proof of the formula joining the Conley index and the topological degree.

1.1 Morse–Conley–Zehnder equation.

Let X be a locally compact metric space. Recall, that a continuous map φ: D → X is called a local ﬂow on X if the following properties are satisﬁed:

• D is an open neighborhood of {0} × X in R × X;

• for each x ∈ X there exist αx, ωx ∈ R ∪ {±∞} such that (αx, ωx) = {t ∈ R;(t, x) ∈ D}; • φ(0, x) = x and φ(s, φ(t, x)) = φ(s + t, x) for all x ∈ X and s, t ∈ (αx, ωx) such that s + t ∈ (αx, ωx).

In the case of D = R × X, we call φ the flow on X. We will interchangably t 0 t+s t s use φ (x) and φ(t, x). Thus, we have φ = idX , and φ = φ ◦ φ . The main objects of this theory are isolated invariant sets and associated with them isolating neighborhoods. Let φt be a flow on X. A subset S of X S t is called an invariant set, if S = t∈R φ (S). For N ⊂ X we define the maximal invariant set contained in N:

n t o inv(N) := x ∈ N; φ (x) ∈ N, t ∈ (αx, ωx) .

If N is compact and inv(N) ⊂ int(N), then N is called an isolating neighborhood, and S = inv(N) is an isolated invariant set. Let N be a compact subset of X. We say that L ⊂ N is positively invariant relative to N if for any x ∈ L the inclusion φ[0,t](x) ⊂ N implies that φ[0,t](x) ⊂ L. (1.1) Deﬁnition (Index pair). A compact pair (N,L) is called an index pair for S, if: • N \ L is a neighborhood of S and S = inv N \ L ;

• L positively invariant relative to N; 1.1 Morse–Conley–Zehnder equation 6 • if x ∈ N and there exists t > 0, such that φt(x) 6∈ N, then there exists s ∈ [0, t], such that φs(x) ∈ L. The next two theorems are crucial in the deﬁnition of homotopy Conley index. The proofs can be found in Salamon’s paper [45] (1.2) Theorem. Every isolated invariant set S admits an index pair (N,L). If (N,L) is a pair of spaces, L ⊂ N, then the quotient N/L is obtained from N by collapsing L to a single point denoted by [L], the base point of N/L. A set X ⊂ N/L is open if either X is open in N and X ∩ L = ∅ or the set (X ∩ N \ L) ∪ L is open in N. Recall that f :(X, x0) → (Y, y0) is a homotopy equivalence if there exists a map g :(Y, y0) → (X, x0) such that g ◦ f is homotopic to id|X rel. x0 and f ◦ g is homotopic to id|Y rel. y0. If there is a homotopy equivalence f :(X, x0) → (Y, y0) we say that the pairs (X, x0) and (Y, y0) are homotopy equivalent or they have the same homotopy type. The homotopy type of (X, x0) is denoted by [X, x0].

(1.3) Theorem. Let (N0,L0) and (N1,L1) be two index pairs for the isolated invariant set S. Then the pointed topological spaces N0/L0 and N1/L1 are homotopy equivalent. (1.4) Definition. If (N,L) is any index pair for the isolated invariant set S, then the homotopy type h(S, φt) = [N/L] is said to be the Conley (homotopy) index of S. When the flow is clear from context, we just write h(S) for short. Theorem (1.3) says that h(S) is independent of the choice of an index pair. Let us illustrate the concept of Conley index by the following simple example. (1.5) Example. Let Ω ⊂ Rn be an open and bounded set and f : Rn → R be a smooth function such that (∇f)−1(0) ∩ ∂Ω = ∅. The smoothness of f implies that ∇f is a locally Lipschtz continuous map, and hence by the d theorem of Picard-Lindelöf the equation dt u(t) = ∇f(u(t)) defines a local t flow on Ω: φf (x) = u(t), where u:(αx, ωx) → Ω is a solution curve of the above equation passing through x at t = 0, and defined on its maximal t interval of existence (αx, ωx). The rest points of φf are the critical points of f. They are hyperbolic if f is a Morse function, i.e., the Hessian of f is nonsingular at every x ∈ Crit(f), where Crit(f) = {x ∈ Rn; ∇f(x) = 0}. In this case the number

2 indf (x) = #{negative eigenvalues of the Hessian ∇ f(x)} is well defined. The Conley index of an isolated invariant set S = {x}, where x ∈ Crit(f), is the homotopy type of a pointed k-sphere, where k = n − indf (x). This example shows that the Conley index and the Morse index give us the same qualitative information about the flow near the critical point, 1.1 Morse–Conley–Zehnder equation 7 whenever the latter is defined. Below, we state the basic properties of the Conley index. Let 0 denote the homotopy type of a pointed one point space. (1.6) Proposition (cf. [5]). Let φt : X → X be a local flow on a locally compact metric space and let N be an isolating neighborhood for φt with S = inv N. The Conley index h(S) has the following properties: Nontriviality If h(S) 6= 0, then S 6= ∅;

Summation formula If S = S1 ∪ S2 is a disjoint union of isolated invariant sets, then h(S) = h(S1) ∨ h(S2) (wedge sum; for the deﬁnition see page 15);

Multiplication formula If Si is an isolated invariant set of a local flow t φi : R × Xi → Xi, i = 1, 2 then S = S1 × S2 is a isolated invariant set t t t t t of φ = φ1 × φ2 : R × X1 × X2 → X1 × X2 and h(S, φ ) = h(S1, φ1) ∧ t h(S2, φ2) (smash product; for the definition see page 15). In what follows we restrict ourselves to consider flows instead of local flows. If φ is a local flow on X defined by a vector field (integral curves of differential equation), then one can replace the initial vector field by the compactly supported one, such that the initial and the new map coincide on isolating neighborhood. Let φt be a flow on X. For x ∈ X define its α-limit and ω-limit sets as follows: α(x) := \ φ(−∞,−t](x), ω(x) := \ φ[t,+∞)(x). t≥0 t≥0 (1.7) Definition. A Morse decomposition of an isolated invariant set S is a finite collection M (S) = {Mi; 1 ≤ i ≤ l} of subsets Mi ⊂ S, which are disjoint, compact and invariant, and which can be ordered (M1,M2,...,Ml) S so that for every x ∈ S \ 1≤j≤l Mj there are indices i < j such that

ω(x) ⊂ Mi, α(x) ⊂ Mj.

Notice that in the previous example, the set Crit(f) of all critical points of f forms a Morse decomposition of inv(Ω). Indeed, we can arrange the critical points {x1, . . . , xm} in the following manner: i < j whenever f(xi) > f(xj).

Assuming that all groups Hq(A, B) have a ﬁnite rank for all q ≥ 0, deﬁne the formal power series

∞ X P(t, A, B) = rank Hq(A, B) · tq q=0 called the Poincaré series of pair (A, B). If the pair (X,A) is of ﬁnite type, q i.e., H (A, B) = 0 for q ≥ q0, then we say that P(t, A, B) is a Poincaré 1.1 Morse–Conley–Zehnder equation 8 polynomial of (A, B). One can prove that for an isolated invariant set there is an index pair (N,L) for which the isomorphism H∗(N,L) ∼= H∗(N/L) holds. Such an index pair is called regular (cf. [36, 45]). We can therefore deﬁne the Poincaré polynomial1 of h(S) as

P(t, h(S, φt)) := P(t, N, L) where (N,L) is any regular index pair for S. The following theorem generalizes the classical Morse inequalities that give the lower bounds for the number of critical points of a smooth function on a compact oriented closed manifold M (cf. [6, 42]). (1.8) Theorem (Morse–Conley–Zehnder equation, cf. [6, 42]). If S is an isolated invariant set with a Morse decomposition M (S) = {Mi; 1 ≤ i ≤ l}, then there is a polynomial Q with nonnegative coeﬃcients such that

l X (1.9) P(t, h(Mi)) = P(t, h(S)) + (1 + t)Q(t). i=1 (1.10) Example (cf. Definition 1.1, Example 1.8 and Theorem 1.15 in [7]). The Lyusternik–Schnirelmann category of a space X is the smallest cardinality of an open covering of X by contractible subsets. For the torus T 2 = R2/Z2, it equals 3. It is the lower bound of the number of critical points that a smooth real–valued function on a torus could possess. One can check that the function F (x, y) = sin πx sin πy sin π(x + y) has exactly three critical points. The standard example with a hight function on a torus shows that there is a function with four nondegenerate critical points (cf. [34]). The question is: Does a Morse function f : T 2 → R with precisely three critical points exist? The answer is negative and comes immediately from Theorem (1.8). To see this, assume that f : T 2 → R is a Morse function with three critical points. The possible indices that occur in this situation are: (i) 2, 2, 0 or (ii) 2, 0, 0 or (iii) 2, 1, 0. Let us consider only the third case. The argument for both (i) and (ii) is exactly the same. Consider the negative gradient flow of f. The critical points of f form a Morse decomposition of the torus and, as we have seen in example (1.5) the Conley indices are homotopy types of pointed spheres of dimension 2, 1, 0, respectively. Hence the left hand side of (1.9) is t2 + t + 1. Obviously, the pair (T 2, ∅) is an index pair for T 2, and P(t, h(T 2)) = t2 + 2t + 1. Applying (1.9), we obtain the equality t2 + t + 1 = t2 + 2t + 1 + (1 + t)Q(t), which cannot be true. 1We will see later (cf. Proposition (1.15) and Corollary (1.16)) that h(S) is a homotopy type of a finite CW -complex. Therefore, in view of isomorphism H∗(N,L) =∼ H∗(N/L), the pair (N,L) is of finite type. 1.2 Continuation to a gradient 9 1.2 Continuation to a gradient.

Let φ: R × X × [0, 1] → X be a continuous family of ﬂows on X, i.e., t φλ := φ(t, · , λ): X → X is a ﬂow on X. Suppose that N ⊂ X is compact t and Si = inv(N, φi), i = 0, 1. We say that two isolated invariant sets S0 and t S1 are related by continuation, or S0 continues to S1, if for all φλ λ ∈ [0, 1], N is an isolating neighborhood. The notion of continuation is essential in the Conley index theory, as demonstrated by the following theorem.

(1.11) Theorem ([5]). If S0 and S1 are related by continuation, then their Conley indices coincide. Recall that a Morse–Smale gradient flow satisfies the following (i) all bounded orbits are either critical points of the potential function or orbits connecting two critical points; (ii) stable and unstable manifolds of the rest points intersect transversally. Let Ω ⊂ Rn be an open set, F :Ω → Rn a smooth vector field and let t φF :Ω → Ω be a flow generated by x˙(t) = −F (x(t)). Assume that N is an isolating neighborhood and S = inv(N). (1.12) Theorem (Reineck [38]). Set S can be continued to an isolated invariant set of a positive gradient flow of certain function f defined on the open set U containing N and without changing F on Ω \ N. Moreover, this can be done in such a way that the new flow is Morse–Smale. Remarks. The fact that such function f exists has been proved by Robbin and Salamon in [39]. They showed that for an isolated invariant set S = inv N there exists a smooth function f : U → R defined on a neighborhood of N such that2 • f(x) = 0 iff x ∈ S and

d t • dt |t=0f(φ (x)) < 0 for all x ∈ Ω \ S. The function which satisfies this conditions is called the Lyaponov function. In general, one cannot expect that for an isolated invariant set its Lyaponov function would have only nondegenerate critical points, i.e., the rest points of gradient flow are hyperbolic. By the Kupka–Smale theorem (cf. Theorem 6.6 in [2]) this can be obtained via arbitrary small perturbation of ∇f in the C1-topology. Hence, without loss of generality, we can assume that the gradient flow is Morse–Smale. Following Reineck, one can explicitly write the homotopy connecting −F and ∇f. Define h:Ω × [0, 1] → Rn by setting (1.13) h(x, λ) = ρ(x)[λ ∇f(x) + (λ − 1)F (x)] + (ρ(x) − 1)F (x), where ρ:Ω → [0, 1] is a smooth function which equals 1 on a compact neighborhood of S (denoted M, M ⊂ int(N)); and ρ is zero on Ω \ N.

2Cf. Lemma (3.75) on page 52 1.3 Euler characteristic of the index 10 We have adopted the Reineck theorem to the equivariant case. The proof is presented in Section 3.5.

1.3 Euler characteristic of the index.

Recall that the Euler characteristic of a topological pair (X,A) is defined as ∞ (1.14) χ(X,A) = X(−1)q rank Hq(X,A), q=0 provided that pair (X,A) is of a finite type. Notice that χ(X,A) is equal to P(−1,X,A). If both Hq(X) and Hq(A) are finitely generated (e.g. if X and A are CW-complexes), the integer χ(X,A) is well defined. In particular, if A is a point in X (that is, X is a pointed space), then we have P∞ q ˜ q ˜ q χ(X, {pt}) = q=1(−1) rank H (X), where H (X) stands for the reduced cohomology. In particular, the Euler characteristic is well defined for the Conley index of an isolated invariant set. The next proposition, due to Gęba (cf. [14], Proposition 5.6.), says that one can always choose a nice space from the homotopy class, namely a finite CW-complex. (1.15) Proposition. Let N be an isolating neighborhood for a gradient Morse–Smale flow φt on Rn. Then h(inv(N), φt) is a homotopy type of finite CW-complex. (1.16) Corollary. Let N be an isolating neighborhood for a flow φt on Rn generated by x˙ = −F (x). Then h(inv(N), φt) is a homotopy type of finite CW-complex. Proof. Since inv(N) is related by continuation to some isolated invariant set of gradient Morse–Smale flow, the result follows from Proposition (1.15).

1.4 Conley index and the Brouwer degree.

The connection between the Conley index and the topological degree are noticeable at first glance. The homotopy invariance of the Brouwer degree corresponds to the continuation property of the Conley index. The existence axiom refers to nontriviality property which says that nontrivial index implies nonempty isolated invariant set. The next common feature of both invariants is that they are determined by a behaviour of a vector field (flow) on a boundary of the set under investigation. The first quite general result concerned with the relationship between homotopy invariant (local indices of zeros) of vector field and the Conley index has been proved by McCord in [33]. Earlier Dancer in [9] proved the t formula χ(h({x} , φf )) = deg(f, Ux), where x is a degenerate rest point of a gradient flow of −f and Ux stands for its neighborhood. He thoroughly 1.4 Conley index and the Brouwer degree 11

t discussed the Betti numbers of index h({x} , φf ). We shall present the proof of a more general fact, where the degenerate critical point in Dancer’s formula is replaced by an isolated invariant set and the flow is not necessarily gradient. We will briefly recall the notion of the degree of a map. Let Ω ⊂ Rn be an open and bounded set. If f : Ω → Rn is a continuous map and does not vanish on the boundary ∂Ω, then it is well known that there is an integer deg(f, Ω) ∈ Z, called the Brouwer degree (cf. for instance [32, 44]). It satisfies the following axioms:

Nontriviality If 0 ∈ Ω then deg(I, Ω) = 1, where I is an identity map;

Existence If deg(f, Ω) 6= 0 then f −1(0) ∩ Ω is nonempty;

Additivity If Ω1, Ω2 are open, disjoint subsets of Ω and there is no zeros of f in the complement Ω \ (Ω1 ∪ Ω2), then

deg(f, Ω) = deg(f, Ω1) + deg(f, Ω2);

Homotopy invariance If h: Ω × [0, 1] → Rn is a continuous map such that h(x, t) 6= 0 for all (x, t) ∈ ∂Ω × [0, 1], then

deg(h( · , 0), Ω) = deg(h( · , 1), Ω)

There is a generic situation, when the degree is easy to calculate. If ϕ: Ω → R is a Morse function such that deg(∇ϕ, Ω) is deﬁned, then X deg(∇ϕ, Ω) = (−1)indϕ(x). x∈(∇ϕ)−1(0)∩Ω

(1.17) Theorem (cf. [37]). Let F :Ω → Rn be a locally Lipschitz map and t t denote by φF the local ﬂow generated by x˙ = −F (x). If N is an φF -isolating neighborhood and S = inv(N) then

(1.18) χ(h(S)) = deg(F, int(N)).

In what follows we will use deg(F,N) instead of deg(F, int(N)). Proof. By the Reineck continuation theorem, S continues to an isolated t invariant set of a Morse–Smale gradient ﬂow φf , that consists of only nondegenerate critical points of f and of connecting orbits between them. De- note this set by S0. By the continuation property of the Conley index 0 h(S) = h(S ). The set of critical points {x1, . . . , xm} forms a Morse de- 0 t composition of S , and by Example (1.5) one has that h({xi}, φf ) is the homotopy type of a pointed k-sphere, where k = n − indf (xi). Hence, the t Poincaré polynomial of h({xi}, φf ) is of the form

t n−indf (xi) (1.19) P(t, h({xi}, φf )) = t . 1.4 Conley index and the Brouwer degree 12 Applying Theorem (1.8) one obtains

χ(h(S)) = χ(h(S0)) = P(−1, h(S0)) (1.20) m m X t n X indf (xi) = P(−1, h({xi}, φf )) = (−1) (−1) i=1 i=1

For 1 ≤ i ≤ m, let Ωi be a neighborhood of xi in N such that Ωi ∩ Ωj = ∅. Using the homotopy invariance of the Brouwer degree and the additivity property leads to

m X (1.21) deg(−F,N) = deg(∇f, N) = deg(∇f, Ωi). i=1

Now it is easy to compute deg(∇f, Ωi). Since f is a Morse function, the 2 hessian ∇ f(xi) is a non-degenerate linear operator. The degree of ∇f µ with respect to Ωi is (−1) , where µ is the number of negative eigenvalues 2 indf (xi) of ∇ f(xi). That is, deg(∇f, Ωi) = (−1) . By (1.21) we obtain

m X (1.22) deg(F,N) = (−1)n deg(−F,N) = (−1)n (−1)indf (xi) i=1 Combining (1.20) and (1.22) we obtain formula (1.18). (1.23) Example. The simplest example for Theorem (1.17) is given by the equation x˙ = x on Rn. Hence the vector ﬁeld is −id: Rn → Rn, and its degree with respect to the unit ball depends on the dimension n, and equals (−1)n. The origin is an isolated equilibrium with an index pair (Dn,Sn−1). The Euler characteristic of an index is obviously χ(Dn/Sn−1) = (−1)n. (1.24) Example. The map F : R2 → R2 F (x, y) := (−x − y + x(x2 + y2), x − y + y(x2 + y2)) gives us a little bit more reﬁned illustration. The annulus

n 2 2 2 o A = (x, y) ∈ R ; r ≤ x + y ≤ R 0 < r < 1 < R is an isolating neighborhood. Indeed, the inner product hF (x, y), (x, y)i = (x2 +y2)2 −(x2 +y2) shows that for x2 +y2 < 1 the vector ﬁeld points inside the annulus, while for x2 + y2 > 1 the vectors point outside of it. The exit set is a disjoint union of the boundary circles. The index is a homotopy type of a wedge sum S2 ∨ S1. It is easily seen that S2 ∨ S1 is composed of 0-,1-, and 2-dimensional cells. Hence the Euler characteristic modulo a basepoint equals zero. The additivity property of the Brouwer degree implies quickly that deg(F,A) = 0. Chapter 2

LS -index

The homotopy invariant called the LS -index is a generalization of the classical homotopy index introduced by Conley. The construction of the LS -index presended in [16], based on the Galerkin-type approximation, reminds the way the Leray–Schauder degree extends the classical Brouwer degree. This extension originated from the application of the index to the Hamiltonian dynamics. Searching for periodic solutions of Hamiltonian systems is converted into a problem of finding critical points of certain action functional Φ: H → R defined on infinite dimensional real Hilbert space. Moreover, Φ turns out to be strongly indefinite, i.e., the gradient flow of Φ has an infinite dimensional both stable and unstable manifolds, so the classical Morse theory approach cannot be used. It is worthwhile to mention that the LS -index has been also successfully applied by Izydorek and Rybakowski in the study of strongly indefinite elliptic systems cf. [26, 28]. The purpose of this chapter is to provide the basic facts about the LS -index and to prove the formula relating the index to the Leray–Schauder degree. In Subsection 2.1.1 we give the definition of the Betti numbers and Euler characteristic of an LS -index. Theorems (2.13) and (2.15) are crucial in this chapter.

2.1 LS -ﬂows and the index.

Let H be a real, separable Hilbert space, and L: H → H be a linear bounded operator which satisﬁes the following assumptions:

L∞ (L.1) L gives a splitting H = n=0 Hn onto ﬁnite dimensional, mutually orthogonal L-invariant subspaces;

(L.2) L(Hn) = Hn for n > 0 and L(H0) ⊂ H0, where H0 is a subspace corresponding to the part of spectrum on imaginary axis, i.e., σ0(L) :=

σ(L|H0 ) = σ(L) ∩ iR;

(L.3) σ0(L) is isolated in σ(L).

It is possible that dim H± = ∞, where H− (resp. H+) is an invariant subspace corresponding to those parts of spectrum of L which lie on the left (resp. right) half complex plane. Operators with the above property are called strongly indefinite. 2.1 LS -flows and the index 14 Let Λ be a compact metric space. A family of flows indexed by Λ is a continuous map φ: R × H × Λ → H such that φλ : R × H → H defined by t φλ(t, x) = φ(t, x, λ) is a flow on H. As before we write φ (x, λ) instead of φ(t, x, λ). If X ⊂ H and φ is a family of flows indexed by Λ then we define

n t o inv(X × Λ) = inv(X × Λ, φ) := (x, λ) ∈ X × Λ; φ (x, λ) ∈ X, t ∈ R . (2.1) Definition. A family of flows φt : H × Λ → H is called a family of LS -flows if φt(x, λ) = etLx + U(t, x, λ), where U : R × H × Λ → H is completely continuous. Recall, that a map is completely continuous if it is continuous and maps bounded sets to relatively compact sets. (2.2) Definition. We say that a map f : H × Λ → H is a family of LS - vector fields, if f is of the form f(x) = Lx + K(x, λ), (x, λ) ∈ H × Λ, where K : H × Λ → H is completely continuous and locally Lipschitz map.

If in the above definitions Λ = {λ0}, we drop the parameter space out from notation, and we say that f is an LS -flow or an LS -vector field. Suppose that f : H → H is an LS -vector field, f(x) = Lx + K(x). We say that f is subquadratic if |hK(x), xi| ≤ a kxk2 + b for some a, b > 0. One can prove that if f is subquadratic then f generates an LS -flow (cf. [23]). That is for all x ∈ H, there exists a C1-curve (·) φ (x): R → H satisfying d φt(x) = −f ◦ φt(x), φ0(x) = x, dt and is of the form φt(x) = e−tLx + U(t, x), where U : R × H → H is completely continuous. Without loss of generality we will restrict our consideration to subquadratic LS -vector fields (cf. [16, 23]). An isolating neighborhood for a flow φt on infinite dimensional space is defined similarly to finite dimensional case. The difference lies in the fact that we cannot expect compactness of that set. (2.3) Definition. A bounded and closed set N is an isolating neighborhood for a flow φt if and only if inv(N) ⊂ int(N). The isolating neighborhoods are stable with respect to small perturbation of the flow. The sense of this concept is given by the following. (2.4) Proposition (Gęba et al. [16]). Let φ: R × H × Λ → H be a family of LS -flows. For any bounded and closed N ⊂ H the set

Λ(N) = {λ ∈ Λ; inv(N, φλ) ⊂ int(N)} is open in Λ. 2.1 LS -flows and the index 15 The key feature of the LS -flows is the following compactness property. (2.5) Proposition (Gęba et al. [16]). Let Λ be a compact metric space and let φ: R × H × Λ → H be a family of LS -flows. If N is a closed and bounded, then S := inv(N × Λ) is a compact subset of N × Λ. We are going to work in the category of compact metrizable spaces with a base point. The notion f :(X, x0) → (Y, y0) means that f is a continuous map preserving base points, i.e., f(x0) = y0. The Cartesian product is defined in this category by (X, x0) × (Y, y0) = (X × Y, (x0, y0)). The wedge of two pointed spaces, i.e., the space X ∨ Y = X × {y0} ∪ {x0} × Y is closed in X × Y . Hence, the smash product X ∧ Y = (X × Y )/(X ∨ Y ) is also an object in that category. In addition, if f : X → Y and g : X0 → Y 0 then f ∧ g : X ∧ X0 → Y ∧ Y 0 is defined. Consider the circle as the unit interval modulo its end points S1 = [0, 1]/{0, 1}. The suspension functor is defined to be the smash product SX := S1 ∧ X. For any m ∈ N we define SmX := S(Sm−1X). ∞ Let ν : N ∪ {0} → N ∪ {0} be a fixed map and suppose that (En)n=n(E) is ν(n) ∞ a sequence of spaces and (εn : S En → En+1)n=n(E) is a sequence of maps. ∞ ∞ (2.6) Definition. We say that a pair E = ((En)n=n(E), (εn)n=n(E)) is a ν(n) spectrum if there exists n0 ≥ n(E) such that εn : S En → En+1 is a homotopy equivalence for all n ≥ n0. One can define the notion of maps of spectra, homotopy of spectra, their homotopy type etc. For us it is sufficient to know that a homotopy type [E] of a spectrum E is uniquely determined by a homotopy type of a pointed space En for n sufficiently large. Moreover, in order to define the ∞ ν(n) homotopy type [E] one only needs a sequence (En)n=n(E) such that S En is homotopy equivalent to En+1 for n sufficiently large. Assume that f : H → H is an LS -vector field, f(x) = Lx + K(x). Let φt : H → H be the LS -flow generated by f and assume that N ⊂ H is t an isolating neighborhood for φ . Denote by Pn : H → H the orthogonal n Ln − + projection onto H = i=1 Hi. Set Hn := H− ∩ Hn and Hn := H+ ∩ Hn and define n n fn : H → H , fn(x) = Lx + PnK(x). t Let φn : H → H be a flow induced by fn. The definition of LS -Conley index is based on the following. n (2.7) Lemma (Gęba et al. [16]). There exists n0 ∈ N such that N = n t N ∩ H is an isolating neighborhood for a flow φn provided that n ≥ n0. n t By the above lemma the set Sn := inv(N , φn) is an isolated and invariant (by definition) and thus admits an index pair (Yn,Zn) by Theorem 1.2. The Conley index of Sn is the homotopy type [Yn/Zn]. Fix a map − ν : N ∪ {0} → N ∪ {0} by setting ν(n) := dim Hn+1. Using the continuation property of the Conley index one can prove that the pointed space Yn+1/Zn+1 is in fact homotopy equivalent to the ν(n)-fold suspension of 2.1 LS -flows and the index 16

Yn/Zn, that is ν(n) [Yn+1/Zn+1] = [S (Yn/Zn)] ∞ ∞ for all n ≥ n0. The sequence (En)n=n0 = (Yn/Zn)n=n0 represents the spectrum, say E and uniquely determines its homotopy type [E]. This leads us to the definition. (2.8) Definition. Let φt be an LS -flow generated by an LS -vector field. If N is an isolating neighborhood for φt and S := inv(N, φt), then the homotopy type of spectrum t hLS (S, φ ) := [E] is well defined and we call it the LS -Conley index of S with respect to φt. When the flow is clear from context we just write hLS (S). Let 0 represents the homotopy type of spectrum such that for all n ≥ 0 En consists of a distinguished point and εn maps the point of En into the point in En+1. (2.9) Proposition (Gęba et al. [16]). The LS -Conley index has the following properties: Nontriviality Let φt : H → H be an LS -flow and N ⊂ H be an isolating t neighborhood for φ with S := inv(N). If hLS (S) 6= 0, then S 6= ∅; Continuation Let Λ be a compact, connected and locally contractible metric space. Assume that φt : H × Λ → H is a family of LS -flows. t Let N be an isolating neighborhood for a flow φλ for some λ ∈ Λ and t Sλ := inv(N, φλ). Then there is a compact neighborhood Uλ ⊂ Λ such that

hLS (Sµ) = hLS (Sν)

for all µ, ν ∈ Uλ.

2.1.1 Cohomological LS -Conley index. The main reference for this section is [23]. Now and subsequently H∗ denotes the Alexander–Spanier ∞ cohomology functor. Let E = (En, εn)n=n(E) be a spectrum. Define ρ: N ∪ Pn−1 {0} → N ∪ {0} by setting ρ(0) = 0 and ρ(n) = i=0 ν(i) for n ≥ 1. For a fixed q ∈ Z consider a sequence of cohomology groups q+ρ(n) H (En), n ≥ n(E). Denote by S∗ : Hq(X) → Hq+1(SX) the suspension isomorphism. Define a sequence of homomorphisms {hn}n≥0 such that the following diagram commutes q+ρ(n+1) hn - q+ρ(n) H (En+1) H (En) -

) ε q+ n n ρ ν( (n − +1) ∗ ) - (S q+ρ(n+1) ν(n) H (S En) 2.1 LS -ﬂows and the index 17

q+ρ(n) Thus we see that {H (En), hn} forms an inverse system and we are ready to make the following definition. (2.10) Definition. The qth cohomology group of a spectrum E is defined to be q q+ρ(n) CH (E) := lim{H (En), hn}. ←− ν(n) Since En+1 is homotopically equivalent to S En for n ≥ n0, we see that q+ρ(n+1) q+ρ(n) hn : H (En+1) → H (En) q+ρ(n) is an isomorphism for n ≥ n0 and the sequence of groups H (En) stabilizes. This simple observation implies that:

q ∼ q+ρ(n) • CH (E) = H (En) for n ≥ n0;

∗ ∗ • the graded group CH (E) is ﬁnitely generated if H (En0 ) is ﬁnitely generated;

• the spectrum E is of finite type if the space En0 is of finite type. These groups may be nonzero for both positive and negative q’s (cf. [23] or Example (2.25)). Now we are able to define the Betti numbers and the Euler characteristic of an LS -Conley index represented by the spectrum E in the obvious way. (2.11) Definition. Let E be a fixed spectrum. The qth Betti number of E is defined as q βq(E) := rank CH (E), and the Euler characteristic is given by

X q χ(E) := (−1) βq(E). q∈Z

(2.12) Remark. There exist n0 such that for all n ≥ n0 we have χ(E) = ρ(n) (−1) χ(En). q ∼ q+ρ(n) Proof. Since CH (E) = H (En) for n ≥ n0 we have

ρ(n) ρ(n) X q (−1) χ(E) = (−1) (−1) βq(E) q∈Z X q+ρ(n) = (−1) βq+ρ(n)(En) = χ(En). q∈Z 2.2 LS -index and the Leray–Schauder degree 18 2.2 LS -index and the Leray–Schauder degree.

In this section we shall prove an infinite dimensional counterpart of The- orem (1.17). The main idea is to use a finite dimensional approximation and the proof is in fact based on the finite dimensional formula (1.18). For better clarity we divide the result into two separate statements. Firstly, we will present a proof for maps being completely continuous perturbations of an isomorphism L: H → H, and next we show how to weaken the assumption about the linear part. Namely, we assume that L is merely a selfadjoint operator. The theorem generalizes the result of Chang (Theorem 3.3. in [4]) as well as the result of Kryszewski and Szulkin (Theorem 6.1. in [31]). They considered a gradient mappings on Hilbert spaces (manifolds) satisfying a so-called Palais–Smale condition, and compared the LS -degree with the Euler characteristic of a Gromoll–Mayer pair of the critical point. The last two authors generalized Chang’s result for a strongly indefinite functionals. In the next theorem we do not restrict our attention to the gradients, nor to critical points of functionals as isolated invariant sets.

Let U be an open and bounded subset of H. Denote by degLS (f, U) the Leray–Schauder degree, defined for completely continuous perturbations of an identity, i.e., f(x) = x+F (x), where F is completely continuous such that F (x) 6= x on ∂U. For more details about the degree theory we refer to the book by Lloyd [32]. Consider an LS -vector field f in H, f(x) = Lx+K(x), where L is strongly indefinite linear bounded and invertible operator, and K is a completely continuous map. Suppose that f does not vanish on ∂U. We will define the degree for the class of such maps in the following manner:

−1 degL(f, U) := degLS (I + L K,U). Since the zero sets for both f and I + L−1K are the same, and L−1K is completely continuous, the above deﬁnition works. The degL inherits all the properties of the Leray–Schauder degree. In particular one has:

Nontriviality If 0 ∈ U then degL(L, U) = 1;

Existence If degL(f, U) 6= 0 then f has a zero inside U;

Additivity If U1,U2 are open, disjoint subsets of U and there are no zeros of f in the completion U \ (U1 ∪ U2), then

degL(f, U) = degL(f, U1) + degL(f, U2);

Homotopy invariance If h: H × [0, 1] → H is an LS -vector ﬁeld for all t ∈ [0, 1] such that h(x, t) 6= 0 for all (x, t) ∈ ∂U × [0, 1], then

degL(h( · , t),U)) is independent of t ∈ [0, 1]. The following theorem is the main part of the author’s paper [47]. 2.2 LS -index and the Leray–Schauder degree 19 (2.13) Theorem. Assume that f : H → H is an LS -vector ﬁeld, f(x) = Lx + K(x), L: H → H an isomorphism and φt : H → H is an LS -ﬂow generated by f. Let N be an isolating neighborhood for φt and S := inv(N). Then the following equality holds true

(2.14) χ(hLS (S)) = degL(f, N).

Proof. Let hLS (S) = (En, εn)n≥n(E) and assume that n0 is chosen such that ρ(n) χ(hLS (S)) = (−1) χ(En) (cf. Remark (2.12)) and

−1 −1 n degLS (I + L K,N) = deg(I + PnL K,N ) for all n ≥ n0. According to the ﬁnite dimensional formula (1.18) one has

ρ(n) ρ(n) n (−1) χ(En) = (−1) deg(L + PnK,N ). Thus

ρ(n) ρ(n) n χ(hLS (S)) = (−1) χ(En) = (−1) deg(L + PnK,N ) ρ(n) −1 n = (−1) deg L|Hn · deg(I + PnL K,N ) −1 n −1 = deg(I + PnL K,N ) = degLS (I + L K,N) = degL(f, N),

ν since the degree of the linear isomorphism L|Hn with respect to 0 is (−1) , where ν is the number of negative eigenvalues of L. But in this case it is n Pn − Pn−1 exactly dim H− = i=1 dim Hi = i=0 ν(i) = ρ(n). This completes the proof. Now consider a weaker assumption about the operator L: H → H. We would like to admit the case when L is not an invertible operator, but is selfadjoint, i.e., hLx, yi = hx, Lyi for all x, y ∈ H. Let P0 : H → H denote the orthogonal projection onto H0, the kernel of L. Deﬁne Lb : H → H by Lxb := Lx + P0x. Since the kernel of L is orthogonal to the image of L, we see that Lb is an isomorphism. In particular, if L is invertible, then Lb = L. If f is a vector ﬁled of the form Lx+K(x), where K is completely continuous, we can write it equivalently as

f(x) = Lxb + Kc(x), where Kc(x) = K(x) − P0x. Note that Kc is completely continuous as well, since dim H0 < ∞. As before for an open bounded subset U ⊂ H and LS -vector ﬁeld f = L + K, such that 0 6∈ f(∂U), we set

−1 degL(f, U) := degLS (I + Lb K,Uc ). (2.15) Theorem. Assume that f : H → H is an LS -vector ﬁeld with a selfadjoint linear part L: H → H and φt : H → H is an LS -ﬂow generated by f. Let N be an isolating neighborhood for φt and S := inv(N). Then

(2.16) χ(hLS (S)) = degL(f, N). 2.2 LS -index and the Leray–Schauder degree 20 Proof. If L is selfadjoint then

n n deg(L + PnK,N ) = deg(Lb + PnK,Nc ),

L∞ since PnP0 = P0 and L preserves the splitting of H = n=1 Hn. Next

n −1 n deg(Lb + PnK,Nc ) = deg Lb|Hn · deg(I + PnLb K,Nc ).

ρ(n) Observe that deg Lb|Hn = (−1) . Indeed, the number of negative eigenvalues of L and Lb coincide, because Lb diﬀers from L only on the kernel of L by the identity. That is there are only the λ = 1 of multiplicity dim H0 −1 n added to spectrum of L. The deg(I + PnLb K,Nc ) stabilizes for large n −1 and represents degLS (I + Lb K,Nc ) = degL(f, N). The result follows by (2.14). In fact, this theorem can be formulated for much larger class of operators L. It is easy to see that L is admissible if H = Ker L⊕im L, where ⊕ denotes a direct sum (not orthogonal). This condition allows us to deﬁne the degL in the above way. As a corollary we are going to formulate some properties of the numbers

χ(hLS (S)) which are immediate consequence of the properties of the Leray– Schauder degree. (2.17) Corollary. Suppose S is an isolated invariant set of an LS -ﬂow generated by an LS -vector ﬁeld f. Then the number χ(hLS (S)) has the following properties:

Existence If χ(hLS (S)) 6= 0, then S contains a rest point of the ﬂow;

Additivity If S1,S2 are the isolated invariant subsets of S and all zeros of f are contained in S1 ∪ S2, then χ(hLS (S)) = χ(hLS (S1)) + χ(hLS (S2));

Homotopy invariance Let U ⊂ H be an open set and N1, N2 be a bounded closed sets contained in U. If S1 = inv(N1) and S2 = inv(N2) are isolated invariant sets of the LS -flows generated by the homotopic1 0 00 LS -vector fields, then χ(hLS (S )) = χ(hLS (S )). As it was pointed out by McCord in [33], the additivity property follows from the Morse inequalities [23]. Although, while the Morse inequalities give us more information about the dynamics, the additivity in the above Corollary does not assume that the collection of sets S1 and S2 fulfills the admissibility condition (admissible ordering), which is essential in the Morse inequalities approach.

1It is understood that the homotopy is assumed to be admissible from the degree theory point of view. That is, the LS -vector fields f0 and f1 are homotopic if there exists a homotopy ht connecting f1 and f2 and such that ht(x) 6= 0 for x ∈ ∂U and all t ∈ [0, 1]. 2.2 LS -index and the Leray–Schauder degree 21 2.2.1 Alternative approach in the particular case. In this section the equality (2.14) will be obtained via direct calculation, in the case when L = (−I,I): H− ⊕ H+ → H− ⊕ H+ and S being an isolated zero of a given vector field. ∞ (2.18) Definition. We say, that a sequence {Pn}n=1,Pn : H → H is strongly convergent to the identity I : H → H, if lim Pnx = x for all x ∈ H. n→∞

(2.19) Lemma. If K : H → H is a compact operator and Pn : H → H, n = 1, 2,... is a sequence of orthogonal projections onto Hn that is strongly convergent to the identity, then

(1) lim kPnK − Kk = 0; n→∞

(2) lim kPnKPn − Kk = 0; n→∞

(3) lim kQnKk = 0, where Qn : H → H is the orthogonal projection onto n→∞ Hn. Proof. Statement (1) is a well known from the Riesz–Schauder theory. Since

kPnKPn − Kk ≤ kPnKPn − PnKk + kPnK − Kk and since PnK is compact, in order to prove (2) it is enough to show that for any compact A one has lim kAPn − Ak = 0. If A is compact, then the n ∗ adjoint operator A is compact as well and we may write kAPn − Ak = ∗ ∗ ∗ k(APn − A) k = kPnA − A k → 0. Finally, we have an estimation

∞ 0 ≤ kQ Kk ≤ X Q K = k(I − P )Kk < ε n i=n i n−1 provided n ≥ n0. This proves (3). (2.20) Deﬁnition. We say that A ∈ B(H) is hyperbolic, if

dist(σ(A), iR) := inf |x − λ| > 0. λ∈σ(A), x∈iR

The set of all hyperbolic operators will be denoted by Bhip(H). Recall, that the multivalued map B(H) 3 A 7→ σ(A) ⊂ C is upper semi continuous, that is for all A ∈ B(H) and > 0, there exists δ > 0, such that inequality kA − Bk < δ implies sup dist(λ, σ(A)) < . λ∈σ(B)

(2.21) Lemma. Bhip(H) is an open subset of B(H).

Proof. Set ρ := dist(σ(A), iR). There exists δ > 0 such that for all B in δ-neighborhood of A

sup dist(λ, σ(A)) < ρ/2. λ∈σ(B) 2.2 LS -index and the Leray–Schauder degree 22 Thus, the triangle inequality gives us the following estimation

dist(σ(B), iR) = inf |µ − x| ≥ inf (|x − λ| − |λ − µ|) µ∈σ(B), x∈iR µ∈σ(B), λ∈σ(A), x∈iR ρ ρ ≥ inf |x − λ| − sup ( inf |λ − µ|) > ρ − = > 0, λ∈σ(A), x∈iR µ∈σ(B) λ∈σ(A) 2 2 which completes the proof.

(2.22) Theorem. Assume that f(x) = Lx + K(x) is an LS -vector ﬁeld on H and L: H → H is such that

2 2 hLx, xi = kx+k − kx−k , where x = (x−, x+) ∈ H− ⊕ H+, both H± are of inﬁnite dimension. Let t f(0) = 0, Df(0) ∈ Bhip(H) and φ is an LS -ﬂow generated by f. Then S = {0} is an isolated invariant set for φt and there exists ρ > 0, such that

(2.23) χ(hLS (S)) = degL(f, B%).

Here B% stands for the open ball in H of radius %

Proof. The assumption f(0) = 0 and Df(0) ∈ Bhip(H) guarantees that −1 S = {0} is an isolated invariant set and x0 = 0 is isolated in the set f (0) (cf. Remark 1.11 in [1]). In order to compute the index on the left-hand side of (2.23) consider a sequence of ﬁnite dimensional approximations

n n fn : H → H , fn(x) = Lx + PnK(x).

Let A := DK(0) and notice that A is a compact linear map. Since the derivative Df(0) = L + A is a hyperbolic operator, then by Lemmas (2.19) and (2.21) there exists n0 ∈ N such that Dfn(0) = L + PnA is hyperbolic, provided n ≥ n0. n n The closure of B% := B% ∩ H is an isolating neighborhood for the n t invariant set Sn := {0} ⊂ H for the ﬂow φn generated by fn for n ≥ n1 (cf. Lemma (2.7)). Assume that n0 is chosen such that n0 ≥ n1. One has a n0 n0 n0 n0 n0 splitting H = Hc− ⊕Hc+ where Hc− (resp. Hc+ ) stands for unstable (resp. stable) subspace of the linear equation x˙ = −Dfn0 (0)x. In the hyperbolic case, the Conley index is exactly the homotopy type of a pointed sphere: n0 dim Hb− h(Sn0 ) = [S , ∗]. n0 dim Hb− Denote by En0 the space that is homotopy equivalent to (S , ∗). In order to establish the relation between En0 and En0+1, we have to compute n0+1 n0+1 the index of the ﬂow generated by fn0+1 : H → H . Note that the n0 derivative Dfn0+1(0) = L + Pn0+1A preserves the splitting H ⊕ Hn0+1. It is easily seen if we write it in the following way

n0 n0 L n0 + P A + L + Q A: H ⊕ H → H ⊕ H . |H n0 |Hn0+1 n0+1 n0+1 n0+1 2.2 LS -index and the Leray–Schauder degree 23

t Hence φn0+1 is a product flow and one has h(Sn0+1) = h(Sn0 ) ∧ h({0}, η), where h({0}, η) is an index of {0} ⊂ Hn0+1 with respect to the flow generated by x˙ = −L x − Q Ax. |Hn0+1 n0+1 Since kQ DK(0)k → 0, the maps L and L + Q A are n |Hn0+1 |Hn0+1 n0+1 homotopic for sufficiently large n0 and the index h({0}, η) is determined by the dimension of the unstable subspace of the linear equation

x˙ = −L x. |Hn0+1

− + − + Set Hn0+1 = Hn0+1 ⊕ Hn0+1, where Hn0+1 (resp. Hn0+1) is the unstable ∗ ∗ − (resp. stable) subspace of L and deﬁne ν : N → N by ν(n) = dim Hn+1. One has

n0 ν(n0) ν(n0) dim Hb− h(Sn0+1) = h(Sn0 ) ∧ [S , ∗] = [S S , ∗]

(2.24) Corollary. En+1 is the ν(n)-fold suspension of En, provided that n is sufficiently large. ∗ ∗ Pn−1 Define ρ: N → N by ρ(0) = 0 and ρ(n) = i=0 ν(i). According to definition of cohomological Conley index one has an isomorphism

q ∼ q+ρ(n) CH (hLS (S)) = H (h(Sn)), n ≥ n0.

n q ∼ q+ρ(n) dim Hb− ∼ n It follows that CH (hLS (S)) = H (S , ∗) = Z for q = dim Hc− − ρ(n) and hence

n dim Hb−−ρ(n) χ(hLS (S)) = (−1) , n ≥ n0.

n0 dim Hb− −ρ(n0) In particular we have χ(hLS (S)) = (−1) . By the stability property of the Leray–Schauder degree

−1 −1 n degL(f, B%) = degLS (I + L K,B%) = deg(I + L PnK,B% ) for n ≥ n0.

n ρ(n) From the fact that deg(L|Hn ,B% ) = (−1) and

n n dim Hb− deg(L + PnK,B% ) = (−1) we conclude that

−1 n n n −1 deg(I + PnL K,B ) = deg(L + PnK,B% ) · [deg(L|Hn ,B% )] dim Hn −ρ(n) = (−1) b− for n ≥ n0 and the proof of (2.23) is completed.

2 ∞ (2.25) Example. Let H = ` with the standard basis (en)n=1 and decom- pose H as follows: H = H1 ⊕ H2, where

n 2 o n 2 o H1 = x ∈ ` : x2n−1 = 0, n ∈ N ,H2 = x ∈ ` : x2n = 0, n ∈ N . 2.2 LS -index and the Leray–Schauder degree 24 Let L(x, y) = (−x, y). It can be written as an inﬁnite diagonal matrix

−1 0 0 ···    0 1 0 ··· L =    0 0 −1 ···  . . . .  . . . ..

Notice that both stable and unstable subspaces of L are inﬁnite dimensional. Deﬁne two linear compact operators K1,2 : H → H by the formula 5 K (e ) := (−1)i e . i n 2n n

Let B% := {x ∈ H; kxk < %} and let f : H → H be an LS -vector ﬁeld such that f(x) = Lx + K1(x) inside the ball Br and f(x) = Lx + K2(x) outside the ball BR for some 0 < r < R < ∞. It is easy to check that

degL(f, Br, 0) = degL(f, BR, 0) = −1.

The origin is an isolated invariant set of the ﬂow induced by x˙ = −f(x). In order to compute its LS -Conley index let us introduce the subspaces

n 2 o Hn = x ∈ ` : xi = 0, i 6= 2n − 1, 2n .

Then each of these subspaces is two dimensional, Hn’s are mutually or- ∞ thogonal L-invariant and H = ⊕n=1Hn. Each Hn can be represented as − + − + Hn ⊕ Hn , where Hn (resp. Hn ) is repelling (attracting) subspace of L|Hn . − Here we have ν(n) = dim Hn+1 = 1. Set Si := inv Bi, i = r, R. One can easily check that hLS (Sr) is a homotopy type of a spectrum E such that k+1 Ek = S , a pointed (k + 1)-sphere for k ≥ 1 while hLS (SR) is a homotopy 0 0 type of a spectrum E , where Ek is a pointed sphere of dimension k − 1 for k ≥ 1. The computations of cohomology give us

( , q = 1; ( , q = −1; Hq(E) = Z Hq(E0) = Z 0, else. 0, else. and the Euler characteristic of both hLS (Sr) and hLS (SR) is equal −1. The conclusion is that Sr 6= SR and it can not be captured via the degree theory. Chapter 3

Equivariant theory

So far we have investigated the homotopy invariants for flows and vector fields having in mind a close relationship between them. We have estab- lished a formula relating the degree of a compact field f : H → H to the Conley index of an isolated invariant set of a flow generated by f. Since the topological degree as well as the Conley index have equivariant counterparts, the natural task is to investigate the relation joining these topological tools. Various definitions and properties of the equivariant degree have been inten- sively studied by many authors. Let us only mention the following: Dancer [8], Gęba, Krawcewicz, Wu [17], Ize, Massabo, Vignoli [21, 22] and Rybicki [43]. In this thesis we are going to deal with the degree for equivariant gradient mappings introduced by Gęba in [14]. Although the restriction to the class of the potential maps is appreciable, this degree became a very pow- erful tool for the variational approach to many problems (cf. [18, 40, 44]). It is worth to noticing that this restriction (gradient mappings) does not take any contributions when the symmetries of finite groups are taken into consideration. This is related to the result originally due to Parusiński [35] which says that gradient maps are homotopic iff they are gradient homotopic. The Z2-equivariant version of this result is presented in the paper by Janczewska and the author [29]. The equivariant Conley index has been introduced by Floer in [12] in order to study hyperbolic invariant sets and some bifurcation questions for Hamiltonian systems in the joint paper (Floer and Zehnder [13]). An extension of this index to infinite dimensional spaces, motivated by [16], has been defined by Izydorek in [24]. This thesis is concerned, after all, only with the finite dimensional version of the G-index. Nevertheless the author is aware of the fact that the results contained herein may have a natural generalization to the infinite dimensional G-flows and equivariant vector (compact) fields. The work is in progress. In the presence of a group action the role of the Euler characteristic of a G-space X plays the universal additive invariant u(X), the element of the ring U(G) (called the Euler ring) associated with a compact Lie group G. The universal additive invariant has the same properties as the usual Euler number. In particular, if G is a trivial group, u(X) coincides with χ(X). In Subsection 3.1.4 we describe the ring U(G), and give some basic facts about u(X). In this chapter we will repeat some definitions that can be found in the previous part of the thesis in order to put the known notions in the context of 3.1 Basic equivariant topology 26 G-equivariant theory. The main result of this part is the equivariant Morse– Conley–Zehnder equation in Section 3.4. This tool allows us to capture the relationship between equivariant Conley index and degree for equivariant gradient maps. Then, in Subsection 3.4.2, this equation is used to prove simple multiplicity results for critical orbits of an invariant function. Finally in Section 3.5 we shall extend the Reineck continuation theorem to the equivariant setting.

3.1 Basic equivariant topology.

(3.1) Deﬁnition. A compact Lie group is a topological group G such that

• G has a structure of a smooth compact manifold;

• the composition map G × G → G; (g, h) 7→ gh−1 is smooth.

Throughout this chapter G stands for a compact Lie group. One can always think of G to be a ﬁnite group or a subgroup of Aut(RN ) the group of all linear isomorphism of RN . (3.2) Deﬁnition. A linear representation V (G-representation) of a group G is a pair (Rn, ρ) consisting of the Euclidean vector space and a homomorphism

n (3.3) ρ: G → Aut(R ). The representation is called orthogonal if the image of (3.3) is contained in the group O(n) of all linear isomorphisms of Rn preserving the inner product, i.e., hρ(g)x, ρ(g)yi = hx, yi for all g ∈ G and vectors x, y ∈ Rn. Two linear representations V = (Rn, ρ) and W = (Rn, ς) are said to be isomorphic iﬀ there exists an invertible linear mapping `: V → W such that for g ∈ G `ρ(g) = ς(g)`. The following basic result can be found for example in [3]. (3.4) Theorem. Each linear G-representation is isomorphic to an orthogonal representation. It allows us to focus, without any loss of generality, our attention on the orthogonal G-representations. Unless we comment to the contrary, we shall always assume that V is an orthogonal representation. An action of a group G (G-action) on a topological space X is a (continuous) map G × X → X sending (g, x) to gx and satisfying the following properties:

• ex = x for all x ∈ X, where e stands for the identity of G;

• h(gx) = (hg)x for all g, h ∈ G and x ∈ X. 3.1 Basic equivariant topology 27 A G-space is a pair consisting of an underlying space X with a given G- action. If V is a G-representation there is a natural linear action of a group G on Rn given by (g, x) 7→ ρ(g)x. For an abbreviation we will write gx instead of ρ(g)x. A subset Ω of a G-space X is called a G-invariant (a G-set) provided that x ∈ Ω and g ∈ G imply gx ∈ Ω. If X and Y are G-spaces, then a continuous map f : X → Y is called a G-equivariant map (a G-map) if the relation f(gx) = gf(x) holds for all x ∈ X and g ∈ G.

3.1.1 Orbit types. Let G be a compact Lie group. The subgroup H ⊂ G is called conjugate to a subgroup K ⊂ G if there is g ∈ G such that H = g−1Kg. The conjugacy deﬁnes an equivalence relation, and we will write (H) for a conjugacy class of H. The set of all conjugacy classes of closed subgroups of G will be denoted by Φ(G). The set Φ(G) is partially ordered. We write

(3.5) (H) ≤ (K) if gHg−1 ⊂ K for some g ∈ G.

Assume that X is a G-space. The isotropy group at a point x ∈ X is a subgroup of G deﬁned by

Gx := {g ∈ G; gx = x} .

That is, this is the set of those elements of G that leave point x ﬁxed. The isotropy group at x measures somehow the symmetry of point x. The most symmetric are those, whose isotropy group is the whole G, e.g. the origin of a representation. For each x ∈ X, the group Gx is closed in G. Given A ⊂ X, the G-orbit of A is deﬁned by GA = {ga; g ∈ G, a ∈ A}. In particular the set Gx = {gx; g ∈ G} is the G-orbit through x (G-orbit of x). Isotropy groups of points on the same G-orbit are conjugate subgroups −1 of G, precisely Ggx = g Gxg.

(3.6) Definition. Points x, y ∈ X have the same orbit type if Gx and Gy are conjugate subgroups of G. Hence the points on the same orbit have the same orbit type. Since the orbit type is determined by a conjugacy class, the set Φ(G) of all conjugacy classes will be called the set of orbit types1. (3.7) Theorem ([30], Corollary 4.25). The set of orbit types of a finite dimensional representation of a Lie group is finite.

1In [3], an orbit type is called an isotropy type. The notion of orbit type is used for equivalence class of G-orbits under equivariant homeomorphism. However each orbit type contains a coset space G/H and the type of G/K equals the type G/K iff (H) = (K). The main difference lies in the partial ordering of the set of orbit types. If P ' G/H and Q ' G/K then type(P ) ≥ type(Q) iff (H) ≤ (K). 3.1 Basic equivariant topology 28 Let H be a closed subgroup of G. Throughout the rest of this chapter we will use the following notation:

H X = {x ∈ X; H ⊂ Gx} = {x ∈ X; hx = x, h ∈ H} (H) H X = GX = {x ∈ X;(H) = (K) for K ⊂ Gx}

XH = {x ∈ X; Gx = H}

X(H) = GXH = {x ∈ X;(Gx) = (H)} >(H) (H) [ (K) X = X \ X(H) = X . (K)>(H)

(3.8) Definition. An orbit Gx and its orbit type (Gx) are called principal if Gx has a G-invariant open neighborhood that contains no orbit of smaller orbit type with respect to the partial order (3.5). The significance of the notion of principal orbits asserts the following (3.9) Theorem ([3], Theorem 3.1). Let (H) be a principal orbit type. Then the union V(H) of orbits of principal type is open and dense in V . If V is a G-representation, then V H is a linear subspace of V , not necessarily a G-subrepresentation. Although on V H we have an induced action of the normalizer NH := {g ∈ G; g−1Hg = H}. Indeed for x ∈ V H and n ∈ NH one has nx = nhx = h0nx for some h, h0 ∈ H. Hence nx is fixed by the action of H. If H is a closed subgroup of a compact Lie group G, then there is a natural action of G on the left coset space G/H by left translations, i.e., G×G/H → G/H, (g0, gH) 7→ g0gH. It is a typical example of a homogenous space. Moreover it is a classical result that G/H is a smooth compact G- manifold (cf. for instance [30]). The next proposition is a statement of the fundamental facts concerning actions of a compact Lie groups on G-manifolds. (3.10) Proposition ([11], p. 63). Let M be a G-manifold. Then

• G-orbits are compact G-invariant submanifolds of M;

• if x ∈ M, Gx is equivariantly diﬀeomorphic to G/Gx via the map sending gx to the coset gGx; • if G acts freely on M, then M/G has a unique smooth structure with respect to which the canonical surjection p: M → M/G is the principal G-bundle.

3.1.2 Slices. Given a closed subgroup H ⊂ G and a H-space X one can deﬁne an action H × G × X → G × X setting (h, (g, x)) 7→ (gh−1, gx). The orbit space of this action is denoted by G×H X and is called twisted product of G and X. For an abbreviation we denote the equivalence classes just by its representatives (g, x). 3.1 Basic equivariant topology 29

0 The space G×H X carries an action of a group G deﬁned by (g , (g, x)) 7→ (g0g, x). If f : X → Y is a H-map, there exists an induced G-map

G ×H f : G ×H X → G ×H Y given by (G ×H f)(g, x) := (g, f(x)). If V is an H-representation one can π show that G×H V is a differentiable manifold and the projection G×H V −→ G/H is a differentiable G-fibre bundle with π−1(gH) ∼= V (that is, is a G- vector bundle). For details, see [10]. Recall that the surjection π : E → M is a G-vector bundle if E and M are G-spaces, π is a G-map and g : Ex → Egx is a linear isomorphism. Of course the condition of local triviality is needed. (3.11) Theorem (The Slice theorem, [30] p. 184). Let G be a compact Lie group and M a smooth G-manifold. For every x ∈ M, the orbit Gx is a G-invariant submanifold of M. Let N denote the normal G-vector bundle of Gx in M. Then the fibre Nx over x of N is a representation space of the isotropy group Gx so that N is isomorphic to

G ×Gx Nx → G/Gx as a smooth G-vector bundles. Moreover there exists a G-invariant open neighborhood U of Gx in M and a G-diﬀeomorphism f : G ×Gx Nx → U such that the restriction of f to the zero section gives the G-diﬀeomorphism from G/Gx onto Gx. If we take the G-invariant Riemannian metric on M then the Gx-action on Nx is given by the Gx-action on the orthogonal complement of TxGx in TxM.

3.1.3 G-complexes. The object of our interest, the Conley index, is a homotopy type of a pointed space which supports the structure of CW- complex. Although the notion of CW-complex is well known in topology, we present here some basic deﬁnitions, since the G-equivariant Conley index joins the notion of CW-complex and G-space. The deﬁnitions are borrowed from the paper by Gęba and Rybicki [15]. We use the standard notation Sn−1 = {x ∈ Rn; kxk = 1} and Dn = {x ∈ Rn; kxk ≤ 1} for the unit (n − 1)-sphere and the unit n-ball in Rn respectively. In what follows we assume that Dn carries the trivial G-action, i.e., gx = x for all x ∈ Dn and g ∈ G. We set Bn = Dn \ Sn−1.

(3.12) Deﬁnition. Let (X,A) be a compact pair of G-spaces and {Hj}, j = 1, 2, . . . , q be a family of closed subgroups of G. We say that X is obtained from A by simultaneously attaching the family of equivariant k- cells of orbit type {(Hj); j = 1, . . . , q} if there exists a G-map q G k ϕ: D × G/Hj → X j=1

q F k k which maps B × G/Hj homeomorphically onto X \ A. We call ϕ(D × j=1 G/Hj) a closed k-dimensional cell of orbit type (Hj). 3.1 Basic equivariant topology 30 (3.13) Deﬁnition. Let X be a compact G-space. A ﬁnite equivariant CW- decomposition of X consists of an increasing family of G-subsets X0 ⊂ X1 ⊂ n n S ... ⊂ X = X and a family {Hj,k; j = 1, . . . , q(k)} of closed subgroups k=0 of G such that q(0) 0 F • X = G/Hj,0; j=1 • the space Xk is obtained from Xk−1 by simultaneously attaching the family of equivariant k-cells of orbit type {(Hj,k); j = 1, . . . , q(k)} for each 1 ≤ k ≤ n.

A pointed G-space is a pair (X, x0) where X is a G-space with a distinguished point x0 called the base point and such that the action of G leaves the base point ﬁxed. The pointed G-spaces are the objects of the category whose morphisms are G-maps preserving the base point. If X is a G-space without base point, then the superscript plus X+ means that X is considered as a pointed space with a separate base point added.

(3.14) Deﬁnition. Let (X, x0) be a pointed compact G-space. A pointed ﬁnite equivariant CW-decomposition of (X, x0) consists of an increasing family of G-subsets X−1 ⊂ X0 ⊂ X1 ⊂ ... ⊂ Xn = X and a family n S {Hj,k; j = 1, . . . , q(k)} of closed subgroups of G such that k=0 −1 • X = {x0};

q(0) 0 F • X = {x0} t G/Hj,0; j=1 • the space Xk is obtained from Xk−1 by simultaneously attaching the family of equivariant k-cells of orbit type {(Hj,k); j = 1, . . . , q(k)} for each 1 ≤ k ≤ n. n S The family {Hj,k; j = 1, . . . , q(k)} is called the orbit type of the de- k=0 composition of X. For short we use the term G-complex (pointed G- complex) for a (pointed) G-space if there exists a (pointed) ﬁnite equivariant CW-decomposition of X (resp. (X, x0)).

3.1.4 Euler ring U(G). If (X, x0) and (Y, y0) are pointed G-spaces (gx0 = x0 and gy0 = y0 for all g ∈ G) then we say that (X, x0) and (Y, y0) have the same G-homotopy type iﬀ there exist a pair of G-maps f :(X, x0) → (Y, y0) and g :(Y, y0) → (X, x0) such that gf ∼G id(X,x0) and fg ∼G id(Y,y0). The symbol ∼G means that if Ht is a homotopy joining two G-equivariant maps, then for all t ∈ [0, 1] the map Ht is a G-map as well. Of course, the relation ∼G is an equivalence and the equivalence class under relation ∼G is denoted by [X]G. We say that [X]G is the G-homotopy type of X. 3.1 Basic equivariant topology 31

Let us introduce the symbol F (G) for the category whose objects are pointed G-complexes and F [G] for the set of all G-homotopy types of pointed G-complexes. For (X, x0), (Y, y0) ∈ F (G) we deﬁne its wedge sum to be X ∨ Y := (X × {y0} ∪ {x0} × Y, (x0, y0)) ∈ F (G). and its smash product

X ∧ Y := X × Y/X ∨ Y.

Of course we also have X ∧ Y ∈ F (G). Let F = Z[F [G]] be the free abelian group generated by the G-homotopy classes of pointed G-complexes and let N be the subgroup of F generated by all elements [A]G −[X]G +[X/A]G, where A is a pointed G-subcomplex of X. Define U(G) := F/N. The class of [X]G ∈ F [G] under this identification will be denoted by u(X). Directly from the definition of U(G) we see that the addition can be obtained via the wedge sum

u(X) + u(Y ) = u(X ∨ Y ).

Moreover the assignment (X,Y ) 7→ X ∧ Y induces the multiplication in U(G) (cf. [10]), that is

u(X)u(Y ) = u(X ∧ Y ).

(3.15) Deﬁnition. The set U(G) with the composition laws deﬁned as above is called the Euler ring of the group G. As we have mentioned earlier the coset space G/H of a compact Lie group over the closed subgroup H is a smooth compact G-manifold and hence, due to theorem of Illman in [20], is a G-complex. Therefore G/H+ ∈ F (G) and we can consider the element u(G/H+) ∈ U(G). In what follows we will G + write u(H) instead of u(G/H ). The abelian group structure of the ring U(G) is fairly easy and its description is given in the following statement. (3.16) Proposition ([10]). As a group U(G) is the free abelian group with G basis u(H), where (H) ∈ Φ(G). If X ∈ F (G), then

X (H) >(H) G (3.17) u(X) = χ(X /G, X /G)u(H). (H)∈Φ(G)

Here χ stands for the Euler characteristic of the pair of CW-complexes. As G a ring U(G) is commutative with the unit u(G). The Euler characteristic of a cell complex K can be expressed as an ∞ P k alternating sum χ(K) = (−1) sk, where sk is the number of k-cells in k=0 the complex K. This formula holds in an equivariant setting as well and we have a nice tool for computations. 3.2 Degree for G-equivariant gradient maps 32 n S (3.18) Proposition ([15]). Let X ∈ F (G) and let {Hj,k; j = 1, . . . , q(k)} k=0 be an orbit type of the decomposition of X. Then X G (3.19) u(X) = n(H)(X)u(H), (H)∈Φ(G) n P k where n(H)(X) = (−1) ν((H), k) and ν((H), k) is the number of equiv- k=0 ariant k-cells of orbit type (H).

3.2 Degree for G-equivariant gradient maps.

In this section we briefly recall definition of the degree for gradient G- maps presented in [14]. Paper [14] is the main reference for this section, where the reader can find proofs of theorems discussed below. We say that a function ϕ: V → R is G-invariant if ϕ is constant on the orbits of G, i.e., ϕ(gx) = f(x) for x ∈ V and g ∈ G. If f : V → V is a gradient of some continuously differentiable G-invariant function f = ∇ϕ, then we call it G-equivariant gradient map. As an immediate consequence of the above definition and the chain rule we get the property that f(gx) = gf(x) for all x ∈ V and g ∈ G. In the same manner we define the homotopy joining two equivariant gradient maps. That is the map h : V × [0, 1] → V is a gradient G- homotopy if there exists a G-invariant function q : V × [0, 1] → R of class C1 (q(gx, t) = gq(x, t)) such that h(x, t) = ∇q(x, t) for all t ∈ [0, 1]. The gradient is taken with respect to the x variable. Fix an open bounded and G-invariant subset Ω ⊂ V and f : V → V a gradient G-map. We say that a pair (f, Ω) is ∇G-admissible provided that f(x) 6= 0 for x ∈ ∂Ω. In other words an ∇G-admissible pair is an equivariant map of pairs f :(V, ∂Ω) → (V,V \{0}). We say that two ∇G-admissible pairs (f0, Ω) and (f1, Ω) are ∇G-homotopic if there exists a gradient ∇G-homotopy h: V × [0, 1] → V connecting them, i.e., hi = fi, i = 0, 1 and such that the pair (ht, Ω) is ∇G-admissible for t ∈ [0, 1]. From now on f : V → V will always mean an equivariant gradient map. Let x be a fixed point in V of an orbit type (H), i.e., H = Gx. We have an orthogonal splitting

(3.20) V = Tx(Gx) ⊕ Wx ⊕ Nx, where Wx is the orthogonal complement of Tx(Gx) in the tangent space ⊥ −1 Tx(V(H)) and Nx = Tx(V(H)) . Assume that x ∈ f (0) and f is diﬀeren- tiable at x. Then Tx(Gx) ⊂ Ker Df(x) since for all g ∈ G f(gx) = 0. Notice that Tx(V(H)) is an invariant subspace of the linear map Df(x). Therefore with respect to the decomposition (3.20) Df(x) has a form 0 0 0    (3.21) 0 Kf(x) 0  , 0 0 Lf(x) 3.2 Degree for G-equivariant gradient maps 33

i.e., Kf(x) := Df(x)|Wx and Lf(x) := Df(x)|Nx . (3.22) Deﬁnition. An orbit Gx is called a regular zero orbit of f , if f(x) = 0 and Ker Df(x) = Tx(Gx). It means that the map Kf(x) ⊕ Lf(x): Wx ⊕ Nx → Wx ⊕ Nx is an isomorphism. The Morse index of the regular zero orbit Gx is deﬁned to be the number of negative eigenvalues of − k Kf(x), k := dim Wx . We set σ(Gx) := (−1) .

For an open G-set U such that U is a compact subset of V(H) and ε > 0 deﬁne

N(U, ε) := {v ∈ V : v = x + n, x ∈ U, n ∈ Nx, |n| < ε} .

The set N(U, ε) will be called a tubular neighborhood of type (H) provided that the decomposition v = x + n is unique. Let N(U, ε) be a tubular neighborhood of type (H). The gradient equivariant map f is (H)-normal on N(U, ε) if for all v = x + n ∈ N(U, ε)

f(v) = f(x) + n.

(3.23) Deﬁnition (Generic pair). We say that ∇G-admissible pair (f, Ω) is generic if there exists an open G-subset Ω0 ⊂ Ω such that

−1 (Gen.1) f (0) ∩ Ω ⊂ Ω0;

1 (Gen.2) f|Ω0 is of class C ;

−1 (Gen.3) f (0) ∩ Ω0 is composed of regular zero orbits;

−1 (Gen.4) for each H with Z = f (0) ∩ Ω(H) 6= ∅ there exists a tubular neighborhood N(U, ε) of type (H) such that Z ⊂ N(U, ε) ⊂ Ω and f is (H)-normal on N(U, ε).

The next theorem allows us to deﬁne the gradient degree for a ∇G- admissible pair (f, Ω).

(3.24) Theorem (Generic Approximation Theorem, [14]). For any ∇G- admissible pair (f, Ω) there exists a generic pair (f1, Ω) such that (f, Ω) and (f1, Ω) are ∇G-homotopic.

(3.25) Lemma ([14]). If (f, Ω) is a ∇G-admissible pair then there exists a gradient G-map f1 : V → V such that (i) f1(x) = f(x) for x ∈ V \ Ω and (ii) (f1, Ω) is ∇G-admissible and generic.

Proof of Theorem (3.24). Let (f1, Ω) is ∇G-admissible and generic pair from the Lemma (3.25). Deﬁne G-homotopy h: V × [0, 1] → V as h(x, t) = (1 − t)f(x) + tf1(x). Clearly the pair (h( · , t), Ω) is ∇G-admissible for all t ∈ [0, 1]. 3.3 Equivariant Conley index 34

(3.26) Deﬁnition. Let (f, Ω) be a ∇G-admissible pair. The G-equivariant gradient degree of (f, Ω) is an element of the Euler ring U(G) deﬁned as

∇ X G degG(f, Ω) := n(H)u(H), (H)∈Φ(G) where X n(H) := σ(Gxi)

(Gxi )=(H) −1 and Gxi are the disjoint orbits of type (H) in f1 (0) ∩ Ω. Here (f1, Ω) is any generic pair G-homotopic to (f, Ω). The above deﬁnition is correct because of the following

(3.27) Theorem ([14]). If two generic pairs (f0, Ω) and (f1, Ω) are G- homotopic then ∇ ∇ degG(f0, Ω) = degG(f1, Ω). (3.28) Example. Let Ω = {(x, y) ∈ R2; 1/2 < x2 + y2 < 3/2} and the action of G = SO(2) on the real plane is given simply by rotation, i.e., for

"cos θ − sin θ# (3.29) γ = ∈ SO(2), θ ∈ [0, 2π) θ sin θ cos θ

γθ(x, y) = (x cos kθ − y sin kθ, x sin kθ + y cos kθ), k ∈ N. Hence V is a plane R2 with rotations by the angle kθ. Deﬁne ϕ: V → R by the formula: ϕ(x, y) = −(x2 + y2 − 1)2. It is easy to check that ϕ is SO(2)-invariant function, and the pair (∇ϕ, Ω) is a ∇G-admissible. Each 2 point except the origin has an orbit type (Zk), that is V(Zk) = R \{(0, 0)}. The map f = ∇ϕ vanishes at the point (x0, y0) = (1, 0), and consequently the whole orbit G(1, 0) ≈ S1 is the set of zeros of f. The derivative at (1, 0) is a map (u, v) 7→ (−8u, 0) with the kernel Ker Df(1, 0) = span([0, 1]) which is exactly the tangent space T(1,0)G(1, 0). The Morse index of G(1, 0) is 1 and hence σ(G(1, 0)) = −1. Directly from the deﬁnition one obtains deg∇(f, Ω) = −uG . G (Zk)

3.3 Equivariant Conley index.

As we have noticed earlier (cf. Example (1.5)) with a locally Lipschitz vector field v : Rn → Rn one can associate a local flow by integration of a differential equation. More precisely, through each point x ∈ V passes a n maximal integral curve φx :(αx, βx) → R satisfying dφ x (t) = v(φ (t)) dt x φx(0) = x. 3.3 Equivariant Conley index 35

Setting D := {(t, x) ∈ R × V ; t ∈ (αx, βx)} and φ(t, x) := φx(t) we obtain a local flow on V , that is (i) D ⊂ R × V is an open neighborhood of {0} × V and φ: D → Rn is continuous (ii) if (t, x) ∈ D and (s, φ(t, x)) ∈ D then (s + t, x) ∈ D and φ(s, φ(t, x)) = φ(s + t, x) (iii) φ(0, x) = x. Now we are concerned with an equivariant vector fields. It is not sur- prising that the G-equivariant vector fields generate local G-flows. We are going to formulate this fact in the following (3.30) Lemma (cf. Lemma 2.10.5 in [11]). Let V be a representation of a compact Lie group G and v : V → V be a G-equivariant, locally Lipschiz vector field. Then the differential equation

x˙(t) = v(x(t)) defines a local G-flow. That is (i) the set D ⊂ R × V is a G-set, i.e., if (t, x) ∈ D then (t, gx) ∈ D for all g ∈ G; (ii) φ(t, gx) = gφ(t, x) for all (t, x) ∈ D and g ∈ G. Proof. The invariance of D under an action of a group comes from the invariance of V . Let φx :(αx, βx) → V represents an integral curve passing ˜ through a point x. Define φ := gφx :(αx, βx) → V . Since

˜˙ ˙ ˜ φ(t) = gφx(t) = gv(φx(t)) = v(gφx(t)) = v(φ(t)) ˜ ˜ and φ(0) = gx, then φ is a curve through a point gx. Hence (αx, βx) ⊂ (αgx, βgx) and from the uniqueness of the solution φgx = gφx. Replacing in ˜ −1 the above argument x by gx and φ by g φgx we obtain (αgx, βgx) ⊂ (αx, βx) and φgx = gφx for all x ∈ G and x ∈ Ω. This show that φ is a G-flow. From now on we will consider local flows generated by vector fields at least of class C1. Without loss of generality we can assume that differential equation x˙ = v(x) generates a flow, i.e., the set D = R × V . We repeat basic definitions and notions which are necessary for the definition of the Conley index in the presence of an action of a Lie group G. Let φ be a G-flow on V . For a G-set X ⊂ V the maximal invariant subset under the flow φ in X is given by

n t o inv(X) = x ∈ X; φ (x) ∈ X, for all t ∈ R .

Since X is G-invariant so is inv(X). If X is in addition compact and inv(X) ⊂ int X, then X is called an isolating neighborhood and inv(X) is an isolated invariant set. For an isolated invariant set there exists a G- index pair (N,L), i.e., the pair of compact G-invariant subsets of V such that (i) the closure of N \L is an isolating neighborhood; (ii) L is positively invariant rel. N and (iii) if x ∈ N and φ[0,t](x) 6⊂ N for some t > 0, then φs(x) ∈ L for some s ∈ [0, t]. For the existence of a G-index pair we refer to [12, 13, 14]. 3.3 Equivariant Conley index 36 The G-homotopy type of the quotient N/L does not depend on the particular choice of the index pair. Recall that N/L is obtained from N by collapsing all points in L to the point [L] which is distinguished in N/L. The action of G on N/L is induced from the action on N and g[L] = [L] for all g ∈ G. Assume X ⊂ V is an isolated neighborhood of a flow φ. (3.31) Definition. The G-equivariant Conley index of S := inv(X) denoted by hG(S) (or sometimes hG(X, φ), to indicate the isolating neighborhood and the flow) is defined to be a G-homotopy type of a pointed G-space N/L, where (N,L) is an arbitrary G-index pair for S. That is hG(X, φ) := [N/L]G The equivariant Conley index has the same properties as the ordinary one. In particular the continuation property holds. We say that φ: R × V × [0, 1] → V is a continuous family of G-flows on V if φλ : R × V → V is a G-flow on V for all λ ∈ [0, 1], where φλ(t, x) = φ(t, x, λ). Notice that we do not restrict the class of flows to the gradient one if it is not specified otherwise. (3.32) Proposition. Suppose that X is a compact G-subset of V and φ is a continuous family of G-flows on V . If X is an isolating neighborhood for λ 0 1 φ , λ ∈ [0, 1], then hG(X, φ ) = hG(X, φ ). The next proposition asserts that the G-equivariant Conley index carries the structure of a G-complex (recall, that it means a finite G-CW-complex). In fact, the statement can be made stronger by eliminating the assumption that the vector field is a gradient, see Corollary (3.79). We will prove this using the continuation theorem for equivariant flows (cf. Theorem (3.74)).

(3.33) Proposition (Proposition 5.6. in [14]). Let (f, Ω) be a ∇G-admissible pair and let φt denote the G-flow generated by −f. Assume that Ω is an t isolating neighborhood. Then G-index hG(Ω, φ ) is a homotopy type of a finite G-complex (3.34) Example. Let us reexamine the example (3.28) with G = SO(2). Consider the negative gradient G-flow on V = R2 given by − ∇ϕ(x, y) = (4x(x2 + y2 − 1), 4y(x2 + y2 − 1)).

The set N = Ω = {(x, y) ∈ V ; 1/2 ≤ x2 + y2 ≤ 3/2} is an isolating G- invariant neighborhood and inv(N) = {(x, y) ∈ V ; x2 + y2 = 1}. The index pair can be chosen to be (N, ∂N). The Conley index is a G-homotopy type of a G-complex consisting of one 0-cell of orbit type (G) (as a distinguished point with the trivial action) and one 1-cell of orbit type (Zk). According to formula (3.19) we have u(h (N)) = −uG . Notice that we do not take G (Zk) into account the distinguished point. Notice also, that u(hG(N)) coincide ∇ with degG(f, Ω) computed in Example (3.28). 3.4 Equivariant Morse–Conley–Zehnder equation 37 3.4 Equivariant Morse–Conley–Zehnder equation.

The Morse–Conley–Zehnder equation establishes the relationship between the Conley index of an isolated invariant set with the indices of its Morse decomposition, cf. Theorem (1.8). In particular, if the flow is given by the gradient of a Morse function it generalizes the classical Morse inequalities which give the estimation of the number of critical points by the topological invariants of the underlying domain. In this section we shall generalize this result to the G-equivariant setting. We will use the equation to derive some multiplicity results for critical G-orbits and to obtain the relationship between the G-equivariant Conley index and the gradient equivariant degree. The nonequivariant equation is expressed in terms of Poincaré polynomials with integer coefficients being the Betti numbers of certain index pairs. We are going to define the Betti numbers of G-equivariant Conley index and then Poincaré polynomial appropriate for our purposes. Of course, one can expect that in the case G = {e} (the trivial group), the obtained equation will coincide with the classical one. Now and subsequently let H∗ denote the Aleksander-Spanier cohomology with coefficients in some principal ideal domain R. This particular cohomology theory is chosen because it satisfies the following strong excision property: Given two closed pairs (X,A) and (Y,B) in V and a closed continuous map f :(X,A) → (Y,B) such that f induces a bijection of X \ A onto Y \ B one has an isomorphism f ∗ : Hq(Y,B; R) → Hq(X,A; R) for all q ≥ 0. For a more general statement of this fact we refer to the book by Spanier [46], Theorem 6.6.5. It is worthwhile to mention that in our approach to the equivariant theory there is no equivariant cohomology at all. If E is a R-module then we set

rank E = dim(E ⊗R QR), whenever dim(E ⊗R QR) is finite. Otherwise rank E = ∞. Here QR stands for the field of quotients of the ring R. The comparison of the classical Euler characteristic with its equivariant analogue u(X) (defined merely for a homotopy type of a G-complexes, see Proposition (3.16)) being an element of the Euler ring U(G) leads us to the conclusion that the k-th Betti numbers of X ∈ F (G) schould be the collection of the numbers rank Hk(X(H)/G, X>(H)/G), where (H) ∈ Φ(G). Since we are concerned with the G-index which is determined by an arbitrary G-invariant index pair the following definition seems to be reasonable. (3.35) Definition. Let (X,A) be a compact pair of G-invariant subsets of V . The numbers

q q (H) >(H) (H) β(H)(X,A) := rank H (X /G, (X ∪ A )/G), (H) ∈ Φ(G) are called the q-th Betti numbers of the pair (X,A). 3.4 Equivariant Morse–Conley–Zehnder equation 38 For an abbreviation we put

(H) >(H) (H) (XH A) := (X /G, (X ∪ A )/G). Recall that for a compact sets X ⊃ Y ⊃ Z there exists a connecting homomorphism δq : Hq(Y,Z) → Hq+1(X,Y ), and a long exact sequence of a triple (X,Y,Z):

q−1 q q q (3.36) ... −−→δ Hq(X,Y ) −→ı Hq(X,Z) −→ Hq(Y,Z) −→δ ..., where ıq i q are homomorphisms induced by inclusions ı:(X,Z) ,→ (X,Y ) and  :(Y,Z) ,→ (X,Z) respectively. In order to overcome diﬃculties connected with deﬁnition of the Betti numbers we will need a couple of technical results.

(3.37) Lemma. If N2 ⊃ N1 ⊃ N0 is a triple of compact G-sets, (H) ∈ Φ(G), then

 >(H) (H)   >(H) (H) >(H) (H)  ∗ (H) N1 ∪ N0 ∗ N2 ∪ N1 N2 ∪ N0 H N /G,  ∼= H  ,  . 1 G G G

Proof. We are going to use the strong excision property of the Alexander- Spanier cohomology. Firstly we check that the pairs in question are closed. Indeed, for a closed G-subset N ⊂ V and a closed subgroup H ⊂ G one has N H = N ∩ V H . Since V H is a linear subspace of V then V H is closed and so is N H . Further N (H) = GN H is closed, because the action of a compact Lie group is a closed map (Theorem 1.1.2 in [3]). The set of orbit types of a finite dimensional representation is always finite, hence N >(H) is closed as a finite sum of closed sets. Lastly, the set of orbits N/G endowed with the quotient topology is closed since the projection N → N/G taking x into its orbit is closed (Theorem 1.3.1 in [3]). Clearly, the inclusion

 >(H) (H)   >(H) (H) >(H) (H)  (H) N1 ∪ N0 N2 ∪ N1 N2 ∪ N0 e: N /G,  ,→  ,  1 G G G

(H) >(H) is continuous and closed. Moreover for each x ∈ (N1 /G) \ ((N1 ∪ (H) N0 )/G) one has e(x) = x. So the strong excision property applies and the result follows. (3.38) Lemma. Assume that the bottom row of the diagram

∗ (ξ)∗ ∗ ∗ ∗+1 −−−→ı H∗(X,Z) −−−→ H∗(A, B) −−−→δ η H∗+1(X,Y ) −−−→ı x  (3.39) ξ∗=∼  ∗ ∗ ∗ ∗+1 −−−→ı H∗(X,Z) −−−→ H∗(Y,Z) −−−→δ H∗+1(X,Y ) −−−→ı is exact, ξ∗ : H∗(Y,Z) → H∗(A, B) is an isomorphism and η∗ := (ξ∗)−1. Then the upper row is exact. 3.4 Equivariant Morse–Conley–Zehnder equation 39 Proof. Let a ∈ Im ı∗. Then a ∈ Ker ∗ equivalently a ∈ Ker (ξ∗∗) since ξ∗ is an isomorphism. If b lies in Im(ξ∗∗) then b = ξ∗∗a for some a ∈ H∗(X,Z) and ∗a = η∗b which means that η∗b ∈ Ker δ∗ and b ∈ Ker δ∗η∗. This reasoning can be reverted. And at last if c = δ∗η∗b for some b, then c ∈ Im δ∗ so c ∈ Ker ı∗+1. The following lemma is a consequence of well known theorem from linear algebra. The proof can be found for instance in [41]. (3.40) Lemma. If E −→f F −→g G is an exact sequence of homomorphisms of R-modules, then rank F = rank Im f + rank Im g.

q Assuming that the modules H (XH A) are of ﬁnite rank we deﬁne the formal power series taking values in U(G)

 ∞  X X q q G PG(t, X, A) :=  β(H)(X,A)t  u(H). (H)∈Φ(G) q=0 q If β(H)(X,A) = 0 for q suﬃciently large and for all (H) ∈ Φ(G) then we call it Poincaré polynomial of the pair (X,A). Notice that PG( · ,X,A) can be viewed as an element of the polynomial ring U(G)[t].

(3.41) Proposition. If X0 ⊂ X1 ⊂ ... ⊂ Xm is a ﬁltration of compact P P q q G q G-sets, then there exists QG(t) = ( ρ(H)t )u(H) with all ρ(H) ≥ 0 such (H) q that m X (3.42) PG(t, Xj,Xj−1) = PG(t, Xm,X0) + (1 + t)QG(t). j=1 Proof. Fix (H) ∈ Φ(G). By Lemmas (3.37) and (3.38) we have a long exact sequence (3.43) δq−1ηq−1 ıq ξq q (H) (H) q (H) q (H) (H) q ... −−−−−→ H (XjH Xj−1) −−→ H (XjH X0) −−−−−→ H (Xj−1H X0) → .... q Here ı(H) and (H) are suitable inclusions, ξ(H) stands for the isomorphism   X>(H) ∪ X(H) X>(H) ∪ X(H) q ∼ q j j−1 j 0 H (Xj−1 X0) = H  ,  . H G G

q q q q and η(H) is its inverse. Set ρ(H)(Xj,Xj−1,X0) := rank Im δ(H)η(H) . The exactness of (3.43) and Lemma (3.40) imply that q β(H)(Xj−1,X0) q q q = ρ(H)(Xj,Xj−1,X0) + rank Im ξ(H)(H) q q q = ρ(H)(Xj,Xj−1,X0) + β(H)(Xj,X0) − rank Im ı(H) q q q q−1 = ρ(H)(Xj,Xj−1,X0) + β(H)(Xj,X0) − β(H)(Xj,Xj−1) + ρ(H) (Xj,Xj−1,X0). 3.4 Equivariant Morse–Conley–Zehnder equation 40 Consequently one has

q q q q q−1 β(H)(Xj,Xj−1) + β(H)(Xj−1,X0) = β(H)(Xj,X0) + ρ(H) + ρ(H) Multiplying the above equality by tq and summing over q ≥ 0 and (H) ∈ Φ(G) one has

(3.44) PG(t, Xj,Xj−1) + PG(t, Xj−1,X0) ˆ = PG(t, Xj,X0) + (1 + t)QG(t, Xj,Xj−1,X0),

∞ ! ˆ P P q q G where QG(t, Xj,Xj−1,X0) := ρ(H)(Xj,Xj−1,X0)t u(H). Sum- (H)∈Φ(G) q=0 m P ˆ ming (3.44) over 2 ≤ j ≤ m and setting QG(t) = QG(t, Xj,Xj−1,X0) j=2 we obtain desired result. (3.45) Definition. Let X be an isolating neighborhood of a G-flow on V and (M1,...,Mm) be a G-invariant Morse decomposition of S = inv(X) (cf. Definition (1.7)). A G-invariant index filtration is a sequence N0 ⊂ N1 ⊂ ... ⊂ Nm of compact G-invariant subsets of V such that (Nk,Nk−1) is a G-index pair for Mk and (Nm,N0) is an index pair for S. (3.46) Proposition. Every G-invariant Morse decomposition admits a G- invariant index filtration. Proof. Let us forget for awhile that a Morse decomposition has a group symmetry. It is well known that every Morse decomposition admits an index filtration N0 ⊂ ... ⊂ Nm (cf. for instance [36, 37, 45]). Averaging a given filtration over group G we obtain a G-invariant index filtration for a G-invariant Morse decomposition. The compactness of Ni, 0 ≤ i ≤ m survives since G is assumed to be compact (Corollary 1.1.3 in [3]). Let S be an isolated invariant set of a G-equivariant flow φ. Define the Poincaré polynomial for the G-index of S as

PG(t, hG(S)) := PG(t, N, L), where (N,L) is an arbitrary index pair for S. (3.47) Theorem (Equivariant Morse–Conley–Zehnder equation). Let S be an isolated invariant set of a G-equivariant ﬂow φ and let (M1,...,Mm) be P P q q G a Morse decomposition of S. Then there exists QG(t) = ( ρ(H)t )u(H) (H) q q with all integer coeﬃcients ρ(H) ≥ 0 such that

m X (3.48) PG(t, hG(Mj)) = PG(t, hG(S)) + (1 + t)QG(t). j=1 Proof. The proof is a straightforward consequence of the Propositions (3.41) and (3.46). 3.4 Equivariant Morse–Conley–Zehnder equation 41 3.4.1 Poincaré polynomial for an isolated orbit. The aim of this section is to give the relationship between the gradient equivariant degree and the equiavariant Conley index using Theorem (3.47). In order to do this, we shall be interested for awhile in the case where the isolated invariant set S of a G-ﬂow φ is an isolated zero orbit Gx of a gradient G-map f : V → V . A couple lemmas concerning the action of compact Lie groups will be needed. We start with the following proposition whose proof can be found in [3] (Proposition 0.1.9 on page 4). (3.49) Proposition. Let G be a compact group and H is a closed subgroup of G, then gHg−1 = H iﬀ gHg−1 ⊂ H.

(3.50) Corollary. Let x ∈ V , H := Gx and Sx be a slice at x. If Q ⊂ Sx ∩ V(H), then each point of Q is stationary under H.

Proof. Let y ∈ Q. By Theorem (3.11) we see that Gy ⊂ H. Since Gy and −1 H are conjugate, there exists g ∈ G such that gHg = Gy ⊂ H. The Proposition (3.49) implies that Gy = H.

(3.51) Lemma. Let x ∈ V , H := Gx and Sx be a slice at x. If Q ⊂ (H) >(H) Sx ∩ V(H), then (G ×H Q) = G/H × Q and (G ×H Q) = ∅.

Proof. By the deﬁnition G ×H Q is a homogenous space of an action of the −1 group H on G × Q deﬁned by h(g, x) := (gh , hx). Since Q ⊂ Sx ∩ V(H) we have h(g, x) = (gh−1, x) so the quotient space is G/H × Q. We claim that G/H × Q ⊂ V(H). Indeed, both Q and G/H ≈ Gx are contained in V(H), hence h(Hg, q) = (Hgh−1, hq) = (Hg, q) for h ∈ H, that is G/H × Q ⊂ V (H). If there would exist K ! H such that k(Hg, q) = (Hg, q) for all k ∈ K, then Hg, q ∈ V (K) and it will be a contradiction, since G/H and Q ⊂ V(H). The result follows.

(3.52) Deﬁnition. We say that a G-invariant subset X0 of a G-set X is a strong G-deformation retract of X if there exists a G-homotopy r : X × [0, 1] → X such that the following properties holds true: • r(x, 0) = x for all x ∈ X;

• r(x, t) = x for all (x, t) ∈ X0 × [0, 1];

• r(x, 1) ∈ X0. If π : E → M is a G-vector bundle, then M is a strong G-deformation retract of E. Indeed, we identify M with a zero section of a bundle π : E → M, that is M = {(x, v) ∈ E; v = 0 ∈ Ex}. The homotopy is given by the formula r((x, v), t) = (x, (1 − t)v).

(3.53) Proposition. Suppose that (f, Ω) is a ∇G-admissible and generic pair. Let φ denotes the ﬂow generated by −f and Gx0 is an isolated zero orbit of f such that Gx0 = inv(Ω). Then

− dim Wx0 G (3.54) PG(t, hG(Gx0)) = t u(H) ∈ U(G) 3.4 Equivariant Morse–Conley–Zehnder equation 42

where (H) = (Gx0 ). If Gx0 is a principal regular zero orbit then the assump- − tion about genericity can be removed. Recall that Wx0 stands for unstable subspace of −Df(x0).

Proof. Suppose that (f, Ω) is a ∇G-admissible and generic. We shall con- struct an index pair for Gx0 via suitable choice of the index pair in the ﬁber ⊥ of the bundle over an orbit Gx0. Let Ey = (τyGx0) . By the Slice Theorem (3.11) the projection p: E → Gx0, where

E = {(x, v) ∈ Gx0 × V : v ∈ Ex}

is a smooth vector bundle isomorphic to π : G ×H Ex0 → Gx0. Recall that + − Ex0 is a H-representation space. The subspace Ex0 ⊂ V splits into Ex0 ⊕Ex0 , the stable and unstable subspaces corresponding to positive and negative spectrum of Df(x0). Since x0 is nondegenerate critical point, there exists an open H-neighborhood U of zero in Ex0 such that the ﬂow is given in local H-coordinates ψ : U → Ex0 by the system of equations

x˙ + = A1x+ + g1(x) + − (3.55) , x = (x+, x−) ∈ Ex0 ⊕ Ex0 x˙ − = A2x− + g2(x) for |x| = max{|x+| , |x−|} ≤ 2 with g1,2 with Dg1,2 vanishes at zero, i.e., |g1,2(x)| = o(|x|) as |x| → 0. Moreover one can chose the coordinates such that |g1,2(x)| and kDg1,2(x)k can be as small as we want (cf. Appendix). The linear parts are chosen such that there exists λ > 0 for which the following estimations hold

2 hA1x+, x+i ≤ − λ |x+| (3.56) 2 hA2x−, x−i ≥λ |x−| .

− If so, let B := {x ∈ Ex0 ; |x| ≤ 1} and B := {x ∈ B; |x−| = 1}. Then N := −1 −1 − ψ (B) and L := ψ (B ) is an index pair for the system on Ex0 . Finally, the index pair (X,A) for Gx0 is given by X := G ×H N and A := G ×H L. We shall use the assumption that (f, Ω) is generic. It implies that the normal direction for V(H) is attracting. Set B0 := {(x+, x−) ∈ B; |x+| = 0} − − −1 and B0 := {(x+, x−) ∈ B ; |x+| = 0} and next N0 := ψ (B0) and L0 := −1 − ψ (B0 ). There is a strong H-deformation retract of (N,L) onto (N0,L0), hence by the functoriality property of the twisted product the pair X0 := G ×H N0 and A0 := G ×H L0 is a strong G-deformation retract of (X,A). The sets N0 and L0 are contained in V(H) an by the Lemma (3.51) we have q ∼ q ∼ q H (XH A) = H (X0H A0) = H ((G/H × N0)/G, (G/H × L0)/G) ( R, for q = k; ∼= Hq(N ,L ) = , 0 0 0, else.

− since the pair (N0,L0) is a homological pointed k-sphere, where k = dim Wx0 a dimension of a subspace composed by the reppeling directions, that is the k G number of negative eigenvalues of Df(x0). Hence P(t, hG(Gx0)) = t u(H). 3.4 Equivariant Morse–Conley–Zehnder equation 43

Suppose now, that (f, Ω) is not a generic pair, but Gx0 is a regular zero orbit and (H) is a principal orbit type, H := Gx0 . Then one can ﬁnd an open G-subset Ω0 ⊂ Ω0 ⊂ Ω such that Gx0 = inv(Ω0) and Ω0 ⊂ V(H). The result follows by using the same arguments as above.

(3.57) Corollary. Let (f, Ω) be a ∇G-admissible and generic pair, φ is a ﬂow generated by −f and Gx0 is an isolated zero orbit of f such that Gx0 = inv(Ω). Then

G (3.58) u(hG(Gx0)) = σ(Gx0)u(H)

where (H) = (Gx0 ). The formula remains valid if (f, Ω) is a ∇G-admissible pair and Gx0 is a principal regular zero orbit. Proof. By the above Proposition

− dim Wx0 G G u(hG(Gx0)) = P(−1,X,A) = (−1) u(H) = σ(Gx0)u(H).

(3.59) Corollary. Let (f, Ω) be a ∇G-admissible generic pair, φ is a ﬂow generated by −f and Gx0 is an isolated zero orbit of f such that Gx0 = ∇ inv(Ω). Then u(hG(Gx0)) = degG(f, Ω). The formula remains valid if (f, Ω) is a ∇G-admissible pair and Gx0 is a principal regular zero orbit. The G-index of Conley is additive in the following sense. (3.60) Proposition. If S is an isolated invariant G-set, and S is a disjoint union S1 ∪ S2 of isolated invariant G-sets, then

PG(t, hG(S)) = PG(t, hG(S1)) + PG(t, hG(S2)).

Proof. Let (X,A) (resp. (Y,B)) be a G-index pair for S1 (resp. S2). Since S1 and S2 are isolated one can chose those pair to be disjoint. It is clear that (X ∪ Y,A ∪ B) is a G-index pair for S. Since the pairs in question are disjoint and G-invariant one has

q H ((X ∪ Y )H (A ∪ B)) = Hq(X(H)/G ∪ Y (H)/G, (X>(H) ∪ A(H))/G ∪ (Y >(H) ∪ B(H))/G).

Therefore, by the fact that pairs (XH A) and (YH B) are disjoint, we conclude that q ∼ q q H ((X ∪ Y )H (A ∪ B)) = H (XH A) ⊕ H (YH B). The above isomorphism implies that

q q q rank H ((X ∪ Y )H (A ∪ B)) = rank H (XH A) + rank H (YH B). and the result follows (according to the fact that addition in U(G) is by coordinates cf. Proposition (3.16)). 3.4 Equivariant Morse–Conley–Zehnder equation 44

Gęba defined the gradient equivariant degree of a ∇G-admissible pair (f, Ω) as a class in U(G) representing the homotopy type of the G-Conley index hG(Ω, φf ), where φf stands for the flow generated by −f (cf. [14]). The following theorem states that one can justify this definition by comparison of the degree (Definition (3.26)) and the Conley index, and by using the equivariant Morse–Conley–Zehnder equation.

(3.61) Theorem. Let (f, Ω) be a ∇G-admissible pair and let Ω be an isolating G-invariant neighborhood of a ﬂow φf generated by the equation x˙ = −f(x); S := inv(Ω). Then

∇ u(hG(S)) = degG(f, Ω).

Proof. Firstly, we will show that S can be continued to an isolated invariant G-set of the flow given by the generic function. By the compactness of ∂Ω one can choose T > 0 such that for any x ∈ ∂Ω there is t ∈ [−T,T ] and φ(t, x) 6∈ Ω. Define Ω1 := Ω \ φ(∂Ω × [−T,T ]). It is clear that (f, Ω1) is ∇G-admissible. By Lemma (3.25) there is a gradient G-map f1 : V → V satisfying f1(x) = f(x) for all x ∈ V \ Ω1 and the pair (f1, Ω1) is generic. Define the homotopy h: V × [0, 1] → V by the formula h(x, λ) := (1 − λ λ)f(x)+λf1(x) and let φ stands for the flow generated by −h( · , λ). Notice λ that h(x, λ) = f(x) for all x ∈ V \Ω1 that is φ = φ on the set ∂Ω×[−T,T ]. Therefore Ω is an isolating neighborhood for the flow φλ for λ ∈ [0, 1]. The continuation property of the equivariant Conley index applies and one has h (S ) = h (S), where S := inv (Ω). Since the pair (f , Ω) is G 1 G 1 φf1 1 generic the set S1 is composed of regular zero orbits Gx1, . . . , Gxm of the function f1 and flow lines between them. Moreover, the collection of orbits M = (Gx1, . . . , Gxm) forms a Morse decomposition of S1. We choose an ordering of M given by the potential ϕ1 :Ω → R, f1 = ∇ϕ1, i.e., one can order the critical orbits in such a manner that ϕ1(Gxi) < ϕ1(Gxj) whenever i > j. By the equivariant Morse–Conley–Zehnder equation

m X u(hG(S1)) = PG(−1, hG(S1)) = PG(−1, hG(Gxk, φf1 )) k=1

For 1 ≤ k ≤ m take open G-subsets Ωk ⊂ Ω, such that Ωi ∩ Ωj = ∅ and Ωk is an isolating neighborhood for an isolated critical zero orbit Gxk. By ∇ Corollary (3.57) one has PG(−1, hG(Gxk, φf1 )) = degG(f1, Ωk). Hence

m X ∇ ∇ ∇ u(hG(S)) = u(hG(S1)) = degG(f1, Ωk) = degG(f1, Ω) = degG(f, Ω) k=1 by the additivity property and the homotopy invariance of the gradient equivariant degree (cf. [40], Theorem 3.2.). (3.62) Example. As an easy example we will show, following Rybicki (cf. [44], Lemma 4.1.), how to compute the gradient equivariant degree of the 3.4 Equivariant Morse–Conley–Zehnder equation 45 pair (−id,B), where −id: V → V , V is an orthogonal ﬁnite dimensional representation of G := SO(2) = {γθ; 0 ≤ θ < 2π} (γθ is given by equation (3.29)) and B stands for the unit ball in V . Let us introduce the following irreducible representation of G. The notation is borrowed from [44]. For 2 m ∈ N let R[1, m] := (R , ρm), where ρm : G → O(2) is given by

ρm(γθ)(x, y) = (x cos mθ − y sin mθ, x sin mθ + y cos mθ). Lk For k ∈ N set R[k, m] = i=1 R[1, m]. Similarly we define R[k, 0] = Lk i=1 R[1, 0], where R[1, 0] stands for trivial representation on the real line. Each orthogonal finite dimensional representation of G can be represented, Lp up to equivalence, as V = i=0 R[ki, mi], where ki, mi ∈ N for 1 ≤ i ≤ p, k0 ∈ N ∪ {0} and 0 = m0 < m1 < . . . < mp. The multiplicative structure of U(G) is well known and can be expressed explicitly (cf. [44]). For the convenience we denote the trivial subgroup of G as . If a = a uG + P∞ a uG and b = b uG + P∞ b uG then Z1 0 G j=1 j Zj 0 G j=1 j Zj ∞ (3.63) ab = a b uG + X(a b + a b )uG 0 0 G 0 j j 0 Zj j=1 Notice that B is an isolating G-neighborhood for the flow defined by the identity vector field and the sphere S := ∂B is an exit set. Hence, one has to compute u(SV ) = u(B/S) (comp. Section4.4.1). According to Lemma (4.4) and formula (3.63) one has

p p V ⊕p [k ,m ] Y [k ,m ] Y [1,m ] k u(S ) = u(S i=0R i i ) = u(SR i i ) = u(SR i ) i i=0 i=0

Since SR[1,mi] is composed of, for instance, one 0-cell of orbit type G and R[1,mi] G G one 1-cell of orbit type m the equality u(S ) = u − u holds. Also Z i G Zmi R[1,0] G u(S ) = −uG, therefore p p V k0 G Y G G ki k0 G Y G G u(S ) = (−1) u (u − u ) = (−1) u (u − kiu ) G G Zmi G G Zmi i=1 i=1 p k0 G X G = (−1) (u + kiu ) G Zmi i=1

∇ k0 G Pp G By Theorem (3.61) we obtain deg (−id,B) = (−1) (u + kiu ). G G i=1 Zmi

3.4.2 Some multiplicity results. As an application of the equivariant MCZ equation we shall prove a simple multiplicity result in the critical point problem. Before we proceed to the statement of the result we brieﬂy describe some special action of the cyclic group. Let p be a prime number and k1, . . . , kn an integers relatively prime to 2n ∼ n p. Consider an action of Zp on R = C generated by the rotation

2πik1/p 2πikn/p (3.64) ρ(z1, . . . , zn) = (e z1, . . . , e zn). 3.4 Equivariant Morse–Conley–Zehnder equation 46

n This action is free. Any nonzero z ∈ C has a nonzero coordinate zj and 2πiskj /p then e zj 6= zj for 0 < s < p since kj is relatively prime to p. The group 2n−1 acts via isomerties hence the sphere S is a Zp-invariant set. The orbit 2n−1 space S /Zp is called the Lens space, denoted by Lp = Lp(k1, . . . , kn). In particular for p = 2 we have L = RP 2n−1. The above construction can be performed for an arbitrary integer p > 1. We chose p prime to have a structure of a field in the set Zp, as a set of coefficients for a cohomology theory. The cohomology groups of L with Zp coefficients are known and they are (cf. [19], Example 3.41 on page 251) ( ˜ q ∼ 0, for q = 0; H (Lp; Zp) = Zp, for 1 ≤ q ≤ 2n − 1. The reduced cohomology denotes to be the cohomology relative to a basepoint. (3.65) Definition. Let f : V → R be a smooth G-invariant function. • The orbit Gx is called critical orbit of f, if ∇f(x) = 0 (and consequently, for each y ∈ Gx, ∇f(y) = 0). • The critical orbit Gx of f is said to be hyperbolic, if Gx is a regular zero orbit of ∇f (cf. Definition (3.22)).

2n (3.66) Proposition. Assume that V = R is a Zp-representation with the action given by (3.64) and f : V → R is a smooth Zp-invariant function. Suppose that

(1) there exists a Zp-isolating neighborhood X0 such that 0 ∈ S0 := inv(X0) and (t, h (S )) = uZp ; PZp Zp 0 Zp

1 2 (2) f(x) = − 2 |x| + ϕ∞(x), in a neighborhood of the inﬁnity and ∇ϕ∞ is bounded.

If f has only a ﬁnite number of critical orbits, say {Zpx0,..., Zpxm}, and all of them are hyperbolic, then there are at least 2n of them. Moreover ( p) = E (E stands for trivial subgroup) for 1 ≤ k ≤ 2n (2np critical Z xk points), and each number in the set {1,..., 2n} is the Morse index of some critical point.

Proof. Consider the negative gradient Zp-flow φf of x˙ = −f(x). Let Dρ(V ) (resp. Sρ(V )) stands for the disk (sphere) in V of radius ρ > 0. It follows from (2), that DR(V ) is a Zp-isolating neighborhood for sufficiently large R, and (X,A) = (DR(V ),SR(V )) is a Zp-index pair for this flow. Notice E that X /Zp ≈ (Lp × [0, 1])/(Lp × {1}) =: CLp is a cone over Lp and Zp E ˆ (X ∪ A )/Zp ≈ Lp × {0}∪˙ [Lp × {1}] =: CLp is a disjoint union of the bottom and the top of the cone. One has ( q ˆ ∼ q 1 ∼ 0, for q = 0; H (CLp, CLp; Zp) = H (S(Lp) ∨ S , {pt}; Zp) = Zp, for 1 ≤ q ≤ 2n. 3.4 Equivariant Morse–Conley–Zehnder equation 47

q Zp >Zp Zp ∼ q 0 ∼ It is easily seen, that H (X /Zp, (X ∪A )/Zp) = H (S , {pt}) = Zp for q = 0 and is zero otherwise. Thus the Poincaré polynomial of the Zp-Conley index of S := inv(X) is

(t, h (S)) = uZp + (t2n + t2n−1 + ... + t)uZp . PZp Zp Zp E

Since all equilibria are hyperbolic they form together with S0 a Morse decomposition (S0,M1,...,Mm) of S. All nonzero orbits are principal, hence q Zp by Proposition (3.53), the Poincaré polynomial of hZp (Mi) is t uE provided that q is the Morse index of Mi. Denote by ck the number of critical orbits of index k. By the MCZ equation (3.48) there are nonnegative integers a0, a1,... such that

2n 2n 2n X k X k X k ckt = t + a0 + (ak−1 + ak)t . k=0 k=1 k=1 That is 2n 2n X k X k c0 + ckt = a0 + (ak−1 + ak + 1)t . k=1 k=1

Since a0 might be zero we have no information about c0, but ck ≥ 1 for k = 1,..., 2n. (3.67) Remark. The assumption (1) of the above Proposition can be 1 2 achieved by the following: f(x) = 2 |x| + ϕ0(x), in a neighborhood of zero and |∇ϕ0(x)| = o(|x|) as x → 0. Indeed, such condition implies that the origin is a critical point of f, and S0 = {0} is an isolated invariant set. The Zp-index pair for S0 is given by (Dr(V ), ∅), where r is suﬃciently small. E Zp The pair (Dr /Zp,Dr /Zp) is homotopy equivalent to the pointed one point space and Hq(DZp / , ∅) ∼ only for q = 0. Hence (t, h (S )) = uZp . r Zp = Zp PZp Zp 0 Zp In the next Proposition, let G := SO(2). ∼ (3.68) Proposition. Let V = R[n + 1, 1] be a G-representation. Assume that f : V → R is a smooth G-invariant function and

(1) there exists a G-isolating neighborhood X0 such that 0 ∈ S0 := inv(X0) G and PG(t, hG(S0)) = uG;

1 2 (2) f(x) = − 2 |x| + ϕ∞(x), in a neighborhood of inﬁnity and ∇ϕ∞ is bounded.

If f has only a ﬁnite number of critical orbits, say {Gx0, . . . , Gxm}, and all of them are hyperbolic, then there is at least n + 1 of them. Moreover Gxk = E for 1 ≤ k ≤ n + 1, and each number in the set {2k − 1; 1 ≤ k ≤ n + 1} is a Morse index of some critical orbit.

Proof. As in the preceding proof we take the pair (X,A) = (DR(V ),SR(V )) as a G-index pair for the G ﬂow of − ∇f. Here we have XE/G ≈ (CP n × 3.4 Equivariant Morse–Conley–Zehnder equation 48

[0, 1])/(CP n × {1}) =: CCP n, a cone over the complex projective space CP n and (XG ∪ AE)/G ≈ CP n × {0}∪˙ [CP n × {1}] =: CˆCP n is a disjoint union of the bottom and the top of the cone. Now, we are going to use the cohomology with integer coeﬃcient. Thus

q n n ∼ q n 1 H (CCP , CˆCP ; Z) = H (S(CP ) ∨ S , {pt}; Z) ( , for q ≤ 2n + 1 odd; ∼= Z 0, for q even.

q G >G G ∼ Moreover, H (X /G, (X ∪ A )/G) = Z for q = 0 and is zero otherwise. Therefore

G 2n+1 2n−1 3 G PG(t, hG(X)) = uG + (t + t + ... + t + t)uE.

Applying the MCZ equation one obtains the equality

n+1 n+1 X 2k−1 X 2k (3.69) c0 + c2k−1t + c2kt = k=1 k=1 n+1 n+1 X 2k−1 X 2k a0 + (a2k−2 + a2k−1 + 1)t + (a2k−1 + a2k)t , k=1 k=1 where cj is the number of critical orbits of index j and a0, a1,... are nonnegative integers. From (3.69) we read oﬀ that c0 ≥ 0, c2k ≥ 0 and c2k−1 ≥ 1 for 1 ≤ k ≤ n + 1.

We turn now to the case of the most general Z2-representation. Let Rt (resp. Ra) be a one-dimensional Z2-representation with the trivial (resp. antipodal) action. Let V be an orthogonal representation of a group Z2 ` k isomorphic to Rt ⊕ Ra for k ≥ 1. Notice that a Z2-equivariant isomorphism ` ` k k A : V → V is of the form At ⊕ Aa, where At : R → R and Aa : R → R . Assume that f : V → R is an asymptotically quadratic Z2-invariant smooth function, i.e., there exist two symmetric linear Z2-maps A0,A∞ : V → V such that

1 (1f ) f(x) = − 2 hA0x, xi + ϕ0(x) and ∇ϕ0(x) = o(|x|), as x → 0; 1 (2f ) f(x) = − 2 hA∞x, xi + ϕ∞(x) and ∇ϕ∞(x) = o(|x|), as x → ∞. Clearly, if f is asymptotically quadratic, then the map ∇f is asymptotically linear. Moreover, assume that

(3f ) f is nonresonance at zero and inﬁnity, i.e., both maps A0 and A∞ are isomorphisms, and

(4f ) f has only a ﬁnite number of critical Z2-orbits, {x1, gx1, . . . , xn, gxn}, and all of them are hyperbolic. 3.4 Equivariant Morse–Conley–Zehnder equation 49

Consider a Z2-flow φf generated by − ∇f. It follows, from the assumptions above, that the origin is an isolated invariant set for φf and there is another, maximal isolated invariant Z2-set T such that {0} ∈ T . There is a + − + − decomposition V = V0 ⊕ V0 (resp. V = V∞ ⊕ V∞ ) corresponding to the positive and negative spectrum of A0 (resp. A∞). Denote by Dρ(V ) (resp. Sρ(V )) the disk (resp. sphere) in V of radius ρ. It is clear, that the pair + (Dr(V ),Sr(V0 )) is a Z2-index pair for {0} for r sufficiently small. Similarly, + the Z2-pair (DR(V ),SR(V∞ )), for R sufficiently large, is an index pair for T . To proceed further we will calculate the cohomology groups of the pair (D(V )ES(V )) using Z2 coefficients, that is the groups

q E Z2 E H (D(V ) /Z2, (D(V ) ∪ S(V ) )/Z2; Z2), q ≥ 0. In order to visualize the geometry we need the concept of the join of two topological spaces. Since we are dealing with quite friendly spaces, as disks and spheres, the task is much simpler than it might be possible in general. Given two topological spaces X and Y , the join X ∗ Y is the quotient space X × Y × [0, 1]/ ∼, where the equivalence ∼ is given by (x1, y, 0) ∼ (x2, y, 0) for x1, x2 ∈ X and y ∈ Y and (x, y1, 1) ∼ (x, y2, 1) for all x ∈ X and y1, y2 ∈ Y . We shall list some properties of the join which will be needed later. (i) The join of X and a 0-sphere is homeomorphic to the (unreduced) suspension of X: S0 ∗ X ' SX; (ii) Sk ∗ S` ' Sk+`+1. Since the join is associative it follows by induction that Sk ∗ X ' Sk+1X, the (k + 1)-folded suspension of X. The property (ii) implies in particular that if V and W are two ﬁnite dimensional orthogonal G-representations, and S(V ) denotes the sphere {x ∈ V ; |x| = 1} then

S(V ⊕ W ) ' S(V ) ∗ S(W ).

V Notice that S := D(V )/S(V ) ' S(V ⊕ Rt) (cf. Lemma (4.2)). The disk D(V )E is (k + `)-dimensional and contains the `-disk D(V )Z2 on which the group acts trivially. After collapsing the sphere S(V ) in D(V ) the `-disk D(V )Z2 becomes an `-sphere contained as a meridian in a sphere V `+1 k `+1 k S ' S(V ⊕ Rt) = S(Rt ⊕ Ra) ' S(Rt ) ∗ S(Ra). The group acts on `+1 k the join S(Rt ) ∗ S(Ra) as follows: g(x, y, t) = (gx, gy, t) = (x, −y, t) for `+1 k all x ∈ S(Rt ), y ∈ S(Ra) and t ∈ [0, 1]. Factoring out by the action of ` k−1 ` Z2 we obtain S ∗ RP . Collapsing away the circle S (comming from the Z2 disk D(V ) ) one can see that the pair (D(V )ES(V )) is equivalent, up to a homotopy type, to the pair (S` ∗ RP k−1,S`). We will examine the groups q ` k−1 ` H (S ∗ RP ,S ; Z2) using the long exact sequence of a pair. One has an exact sequence of reduced cohomology groups

q−1 ` q ` k−1 ` q k−1 ` q ` ... → H (S ) → H (S ∗ RP ,S ) → H (RP ∗ S ) → H (S ) → ... q ` k−1 ∼ for q ≥ 0. By the suspension isomorphism one obtains H (S ∗RP ) = Z2 for ` + 2 ≤ q ≤ ` + k. Substituting in the above sequence q = ` + i for 3.4 Equivariant Morse–Conley–Zehnder equation 50 i = 2, . . . , k we obtain a short exact sequence

`+i ` k−1 ` =∼ 0 → H (S ∗ RP ,S ) −→ Z2 → 0. If q = ` + 1, then

=∼ `+1 ` k−1 ` 0 → Z2 −→ H (S ∗ RP ,S ) → 0. Hence, for k ≥ 1 and ` ≥ 0 one has

( , q = ` + i for i = 1, 2, . . . , k; (3.70) Hq(D(V ) S(V )) ∼= Z2 E 0, else.

We are also interested in the cohomology of the pair (D(V ⊕ U)ES(V )), where V is as above and U is an arbitrary Z2-representation. By the following lemma one can reduce the task to the previous situation.

(3.71) Lemma. The pair (D(V ⊕ U)ES(V )) is homotopy equivalent to the pair (D(V )ES(V )). Proof. It suﬃces to show that the pairs (D(V ⊕U),S(V )) and (D(V ),S(V )) are Z2-homotopy equivalent. Identify D(V ⊕ U) with D(V ) × D(U) via the natural Z2-homeomorphism. Deﬁne p:(D(V ) × D(U),S(V )) → (D(V ),S(V )) and q :(D(V ),S(V )) → (D(V ) × D(U),S(V )) by setting p(x, y) := x and q(x) := (x, 0). Clearly both p and q are Z2- equivariant, pq = id(D(V ),S(V )) and qp is homotopic with id(D(V )×D(U),S(V )) via Z2-homotopy h(x, y, t) := (x, ty). Let us now go back to the computations of the indices of {0} and T . + `0 k0 + `∞ k∞ Suppose that V0 = Rt ⊕ Ra and V∞ = Rt ⊕ Ra . The above consider- ations show that the Poincaré polynomials of the indices of {0} and T are of the form

(t, h ({0} , φ )) = t`0 uZ2 + (t`0+1 + ... + t`0+k0 )uZ2 and PZ2 Z2 f Z2 E (t, h (T, φ )) = t`∞ uZ2 + (t`∞+1 + ... + t`∞+k∞ )uZ2 . PZ2 Z2 f Z2 E Notice that if x ∈ V is a nondegenerate critical orbit with isotropy group Z2 (i.e., in fact, is a critical point), then

(t, h ({x})) = t`x uZ2 + (t`x+1 + ... + t`x+kx )uZ2 , PZ2 Z2 Z2 E

+ `x where the numbers `x and kx are deﬁned via the equality Vx = Rt ⊕ kx Ra . On the other hand, for a nondegenerate critical orbit {y, gy} with the + dim Vy Z2 isotropy group E one has PZ2 (t, hZ2 ({y, gy})) = t uE (cf. Proposition + + (3.53)). Here Vx (resp. Vy ) is the unstable subspace of a linear map 3.5 Continuation of equivariant maps to a gradient 51

2 2 −∇ f(x) (resp. −∇ f(y)). If (1f )–(4f ) are satisﬁed, then combining all these data with the equation (3.48) one obtains the following equalities: n `0 X `x `∞ (3.72) t + ait i = t + (1 + t)Q1(t) i=1

k0 k∞ X `0+i X `∞+i t + Z (t) = t + (1 + t)Q2(t) i=1 i=1 where Z , Q1, Q2 are some unknown polynomials with nonnegative integer coeﬃcients. The numbers ai for 1 ≤ i ≤ n may be one or zero. Notice that ai = 1 if xi is a critical orbit of orbit type Z2. It may happen that `xi = `xj for i 6= j.

(3.73) Proposition. Suppose that f : V → R is a smooth Z2-invariant function satisfying conditions (1f )–(4f ). If `∞ 6= `0, then f has at least two nonzero critical points x, y ∈ V (two orbits of orbit type Z2). Additionally, + + one has an estimations on the Morse indices: dim Vx ≥ `∞ and dim Vy ≥ `0 − 1. Proof. We will examine the equation (3.72). The right-hand side of (3.72) contains the exponent `∞. Therefore there exists 1 ≤ i ≤ n such that

`xi = `∞. On the left-hand side of (3.72) there is the exponent `0, hence the polynomial (1 + t)Q1 contains two nonzero terms with exponents `0 and `0 + 1 or `0 − 1 and `0. Therefore, there exists 1 ≤ j ≤ n, such that `xj = `0 − 1 or `xj = `0 + 1. Consequently x := xi and y := xj are `∞ + `0−1 + critical points of f. The inclusions Rt ⊂ Vx and Rt ⊂ Vy give us the + + estimations on dimension of Vx and Vy .

3.5 Continuation of equivariant maps to a gradient.

In this section we shall give an equivariant counterpart of the Reineck continuation theorem (cf. [38]) which says that an isolated invariant set of the flow given by a vector field can be continued, in a sense of Conley index theory, to the isolated invariant set of a gradient flow. Using this theorem Reineck was able to compute the Z2-homology index of isolated invariant set in question by counting the number of connecting orbits between critical points of adjacent indices. Beyond the intrinsic interest of the continuation theorem we will prove it in order to give a simple corollary that G-equivariant Conley index is a homotopy type of a G-complex. In fact, besides some technical details, the proof verifies that the idea given by Reineck works in a G-equivariant setting. (3.74) Theorem. Let f : V → V be a G-equivariant vector field and φt be a G-flow generated by f. Let N be an isolating G-neighborhood for φ and t S = inv(N). There is a continuous family of flows {φλ}λ∈[0,1] such that N t t t t stays an isolating G-neighborhood for all φλ, λ ∈ [0, 1], φ0 = φ and φ1 is a gradient flow on a G-neighborhood of S. 3.5 Continuation of equivariant maps to a gradient 52

t Notice that according to the Proposition (3.32) we have hG(inv(N), φ ) = t hG(inv(N), φ1). The proof is based on the following crucial result due to Robbin and Salamon [39]. (3.75) Lemma. Let N be an isolating neighborhood of a ﬂow φt with S = inv(N). There exists a neighborhood U of N and a smooth function ϕ: U → R such that (3.75.1) ϕ(x) = 0 for all x ∈ S;

d t (3.75.2) dt ϕ(φ (x))|t=0 < 0 for all x ∈ U \ S. Let µ be a normed Haar measure on G. (3.76) Lemma (cf. [3], Theorem 0.3.3.). Let G be a compact group and f : G × R → R, (g, t) 7→ f(g, t) a continuous function, differentiable with respect to variable t with a derivative ∂f/∂t(g, t) which is continuous on R G × R. Then F (t) := G f(g, t) dµ(g) is also differentiable and dF Z ∂f (3.77) (t) = (g, t) dµ(g). dt G ∂t Proof of Theorem (3.74). Let ϕ: U → R be a Lyaponov function from the Lemma (3.75). Since we cannot apriori assume that U and ϕ are G-invariant we can average both U and ϕ over G. Namely, we define ϕˆ: GU → R by Z ϕˆ(x) := ϕ(gx) dµ(g). G One can easily check that ϕˆ has properties (3.75.1) and (3.75.2). Indeed, G-invariance of S implies that ϕˆ vanishes on S. Also for x ∈ GU \ S by the Lemma (3.76) we have Z d t d t ϕˆ(φ (x))|t=0 = ϕ(gφ (x)) dµ(g) |t=0 dt dt G Z d t = ϕ(φ (gx))|t=0 dµ(g) < 0, G dt because gx ∈ GU \ S. In what follows, we shall omit the hat over ϕ and assume that ϕ: U → R is G-invariant. Let N 0 be a compact G-set such that S = inv(N 0) and N 0 ⊂ int(N). Choose a G-invariant function ρ: V → [0, 1] such that: ( 1, dla x ∈ N 0; ρ(x) = 0, dla x ∈ V \ N. Define homotopy h: V × [0, 1] → V setting h(x, λ) := ρ(x)[(1 − λ)f(x) − λ ∇ϕ(x)] + (1 − ρ(x))f(x). One can check at once that h( · , 0) = f, h( · , 1) = −ρ ∇ϕ + (1 − ρ)f h( · , 1) = − ∇ϕ on N 0, h( · , λ) = f on V \ N 3.5 Continuation of equivariant maps to a gradient 53

t t t The homotopy h generates a continuous family of ﬂows φλ such that φ0 = φ . Notice that if x 6∈ S, then ∇ϕ(x) 6= 0 and for x ∈ U \ S

d (3.78) ϕ(φt(x))| = h∇ϕ(φ0(x)), f(φ0(x))i = h∇ϕ(x), f(x)i < 0. dt t=0

t By (3.78) the function ϕ is a Lyaponov function for the family of ﬂows φλ for all λ ∈ [0, 1]

d ϕ(φt (x))| = h∇ϕ(x), ρ(x)[(1 − λ)f(x) − λ ∇ϕ(x)] + (1 − ρ(x))f(x)i dt λ t=0 = (1 − λ)ρ(x)h∇ϕ(x), f(x)i − λρ(x)h∇ϕ(x), ∇ϕ(x)i + (1 − ρ(x))h∇ϕ(x), f(x)i = (1 − λρ(x))h∇ϕ(x), f(x)i − λρ(x) k∇ϕ(x)k2 < 0.

In order to see that N (as well as N 0) is an isolating neighborhood for each t φλ, λ ∈ [0, 1], notice that all equilibria of ∇ϕ are contained in the zero level ϕ−1(0). Then, if x ∈ N \ S and ϕ(x) ≤ 0 there exists T > 0 such that t φλ(x) 6∈ N for some t ∈ (0,T ] by the compactness of N and the fact that ϕ t is strictly decreasing along the flow lines of φλ. Similarly, if ϕ(x) ≥ 0 then t φλ(x) 6∈ N for some t ∈ [−T, 0), T sufficiently large. Notice that, in general, one can expect that the isolated invariant set changes it structure, even disappears via the continuation. Of course, it cannot vanish if its index is nontrivial. It is also clear that setting Ω := int(N) and g := ∇ϕ we obtain a ∇G-admissible pair (g, Ω). Hence by the continuation property of the index and Proposition (3.33) we obtain the promised statement. (3.79) Corollary. Let φt be a flow generated by a G-equivariant vector t t field. If S is an isolated invariant set of φ then the Conley index hG(S, φ ) is a homotopy type of a finite G-complex. Chapter 4

On the invertibility in U(G)

Let V be a finite dimensional real orthogonal representation of a Lie group G. Denote by D(V ) and S(V ) the unit disc and the unit sphere respectively. Since G acts via isometries both D(V ) and S(V ) are G-sets, hence SV defined to be D(V )/S(V ) is a G-set. The following theorem due to Gołębiewska and Rybicki [18] asserts that the G-complex SV represents an invertible element of the Euler ring U(G). (4.1) Theorem. The element u(SV ) is invertible in U(G). The aim of this chapter is to give a simple proof of the above theorem if the group G is finite and abelian. Our proof avoids all technicalities used in the proof of the general case. The nature of this theorem seems to lie far from the subject taken up in this thesis. However, there is a link between Theorem (4.1) and the theory of topological methods in nonlinear analysis. Namely, by equality ∇ V degG(−id) = u(S ), the gradient degree of the minus identity map is invertible in U(G). This was a basis for Gołębiewska and Rybicki to define the degree of gradient strongly indefinite and G-invariant functionals defined on a Hilbert G-representation, see [18].

4.1 Technicalities.

n 2 o Let Σ = (x, t) ∈ V ⊕ R; |x| + (t − 1)2 = 1 . We will regard Σ as a pointed space with distinguished point p = (0,..., 0, 2). (4.2) Lemma. SV is G-homeomorphic to Σ (as a pointed G-spaces). Proof. Deﬁne ξ : SV → V ∪ {∞} by ( |x| x, for x ∈ D(V ) \ S(V ); ξ(x) := 1−|x| ∞, for x ∈ S(V ). Clearly ξ is a G-homeomorphism. The inverse is given by ( |y| y, for y 6= ∞; ξ−1(y) := 1+|y| [S(V )] , for y = ∞ Consider the stereographic projection π : V ∪ {∞} → Σ given by the formula: ∞ 7→ p and 2 ! 4x1 4xn 2 |x| π(x1, . . . , xn) := ,..., , . |x|2 + 4 |x|2 + 4 |x|2 + 4 4.2 Self-invertibility of SV 55

The natural G-action on Σ inherited from the action on V ⊕R coincide with the action deﬁned by

g ∗ z := π(gπ−1(z)), z 6= p (4.3) g ∗ p := p.

Now it is obvious that π is G-equivariant. The composition π ◦ ξ gives us desired G-homeomorphism. From now on we will identify SV with Σ via the above G-homeomorphism. Recall that a smash product of X and Y with base points x0 and y0 respectively is a quotient (X × Y )/(X ∨ Y ), where X ∨ Y = X × {y0} ∪ {x0} × Y is a wedge of X and Y . (4.4) Lemma. SV ⊕W is G-homeomorphic to SV ∧ SW . Proof. Let π : V ⊕ W → SV ⊕W \{p} be the stereographic projection. Set ϕ: SV ∧ SW → SV ⊕W as h i ( π(x, y), for (x, y) 6∈ SV ∨ SW ; ϕ([x, y]) := p, else. and deﬁne ψ : SV ⊕W → SV ∧ SW to be

( [π−1(z)], for z 6= p; ψ(z) := h i SV ∨ SW , for z = p.

It is easy to see that both ϕ and ψ are G-equivariant maps. Moreover

(ψ ◦ ϕ)([x, y]) = ψ(π(x, y)) = [π−1(π(x, y))] = [x, y] h i if [x, y] 6= SV ∨ SW and

(ϕ ◦ ψ)(z) = ϕ([π−1(z)]) = π(π−1(z)) = z, h i provided z 6= p. The same equalities holds true for [x, y] = SV ∨ SW and z = p. This completes the proof.

4.2 Self-invertibility of SV .

Notice that each real ﬁnite dimensional orthogonal G-representation (G ﬁnite and abelian) is of the form V = V1 ⊕...⊕Vk, where Vi’s are irreducible either one or two dimensional. If dim V = 1 then the action is determined by a homomorphism G → Z2. In the case dim V = 2 the action is given by a homomorphism G → SO(2). (4.5) Lemma. Let V = (R2, ρ), where ρ: G → O(2) be an irreducible representation of a compact Lie group G. Then either V = VG or there exists a closed subgroup H ⊂ G, H 6= G such that V \{0} = (V \{0})(H). 4.2 Self-invertibility of SV 56

Proof. Let H := Gx. It is clear, that for all x ∈ V one has Ker ρ ⊂ H. Let x ∈ V \{0} and suppose that g ∈ H and ρ(g) 6= id. Each element of the group O(2) has the form "cos θ − sin θ# "− cos θ − sin θ# R = or T = θ sin θ cos θ θ − sin θ cos θ for θ ∈ [0, 2π). Hence, it has to be θ = 0. It contradicts the fact that ρ(g) 6= id. This implies Ker ρ = H, and each point beside the origin has an orbit type (H), except the case when G acts trivially. (4.6) Lemma. Suppose that V is a finite dimensional orthogonal representation of a finite abelian group G. Then u(SV ⊕V ) is the identity in U(G). Proof. We will distinguish two cases. Firstly, suppose that dim V = 1. Let 0 6= x ∈ V and H := Gx. Notice that the index of H in G equals 2. Then V ⊕ V has the following cell decomposition: ν(G, 0) = 1 and V ⊕V G ν(H, 1) = ν(H, 2) = 2. Therefore, by Proposition (3.18) u(S ) = u(G). If dim V = 2, then according to Lemma (4.4) u(SV ⊕V ) = u(SV )2. If 0 6= x ∈ V and H := Gx, then due to Lemma (4.5) the G-cell decomposition of V satisfies: ν(G, 0) = 1 and ν(H, 1) = ν(H, 2). It implies that u(SV ) = G u(G). (4.7) Theorem. Suppose that V is a real finite dimensional orthogonal representation of finite abelian group G. Then u(SV ) is self-invertible in U(G), i.e., u(SV )−1 = u(SV ). Proof. It is an immediate consequence of Lemma (4.6).

(4.8) Example. Let G = Z2 ⊕ Z2 = {e, g1, g2, g3}. There are three subgroups of G isomorphic to Z2: (1) (2) (3) Z2 = {e, g1}, Z2 = {e, g2}, Z2 = {e, g3}. Since each element e 6= g ∈ G has rank 2, therefore all irreducible representations of G are one-dimensional. Denote them by (Vi, ρi) (see table of n (1) (2) (3) o characters). Let Φ(G) = G, Z2 , Z2 , Z2 ,E . It is easily seen that

V0 G Vi G G u(S ) = −uG, u(S ) = uG − u (i) for i = 1, 2, 3. Z2

V1⊕V1 V1 2 G G G 2 Observe that u(S ) = u(S ) = uG − 2u (1) + (u (1) ) . On the other Z2 Z2 V1⊕V1 G G 2 hand, by a direct computation one has u(S ) = uG. Thus (u (1) ) = Z2 G V1 2 G V1 2u (1) , and u(S ) = uG, i.e., u(S ) is a self-invertible element of U(G). Z2 4 L mi The same arguments holds for the remaining spheres. Let V = Vi be i=1 V Q4 Vi mi an arbitrary representation of a group G. Then u(S ) = i=1 u(S ) is Vi mi G G G invertible, since u(S ) is uG or uG − u (i) depending on mi is even or odd Z2 respectively. 4.2 Self-invertibility of SV 57

e g1 g2 g3 χ1 1 1 1 1 χ2 1 -1 1 1 χ3 1 1 -1 1 χ4 1 1 1 -1

Table 4.1: Characters of irreducible representations of G = Z2 ⊕ Z2

The following example attest to Theorem (4.7) does not hold for infinite groups. (4.9) Example. Let G = SO(2) and let V = R[1, 1] be an orthogonal two-dimensional representation of G (cf. Example (3.62)). Here Φ(G) = V {Zk; k = 0, 1,...}, where Z0 := G, and Z1 := E. The sphere S , a one-point compactification of V has the following G-cell decomposition: ν(0, Z0) = 1 and ν(1, ) = 1. Therefore u(SV ) = uG −uG . Using the following relations Z1 Z0 Z1 uG uG = uG and uG uG = 0 for i 6= j (cf. (3.63)) one can obtain that Z0 Zi Zi Zi Zj u(SV )−1 = uG + uG . Z0 Z1 We will show explicitly, that an element uG + uG cannot be repre- Z0 Z1 sented by the G-homotopy type of the one-point compactification of any G- representation. Notice that if R is a one-dimensional trivial G-representation, then u(SR) = u(S1) = −uG and Z0 u(SV ⊕R) = u(SV ∧ S1) = u(SV )u(S1) = −uG + uG . Z0 Z1 We easily find out that uG = u(S2), where S2 is a compactification of a two Z0 dimensional trivial G-representation. According to the formula u(X ∨ Y ) = u(X) + u(Y ) one has

uG + uG = u(SV ⊕R ∨ S2 ∨ S2), Z0 Z1 i.e., u(SV )−1 = u(SV ⊕R ∨ S2 ∨ S2). Chapter 5

Appendix

Let V be an orthogonal finite dimensional representation of a compact Lie group G. Assume that Φ: V → R is a smooth G-invariant function and the origin is a nondegenerate critical point of Φ. It is rather standard fact, that near the origin the G-flow given by an equation x˙ = − ∇Φ(x) is equivalent to the G-flow given by

x˙ + = A1x+ + g1(x) + − (5.1) , x = (x+, x−) ∈ V ⊕ V x˙ − = A2x− + g2(x) where |g1,2(x)| = o(|x|), the norms |g1,2(x)| < τ and kDg1,2(x)k < τ, where τ is arbitrary small. The linear maps A1,2 : V → V are such that

2 hA1x+, x+i ≤ − λ |x+| (5.2) 2 hA2x−, x−i ≥λ |x−| . for some λ > 0. Here V + (resp. V −) denotes the eigenspace of the Hessian ∇2Φ(0) corresponding to the positive (resp. negative) eigenvalues. For the sake of completeness we include the proof and next we will show how to find the G-index pair for an isolated zero. The equivalence above means that there is a G-neighborhood U 3 0 and a G-homeomorphism h: U → h(U) such that h(0) = 0 and h maps orbits in U of the first system onto orbits of the second one preserving the direction in time. In particular, such an equivalence takes place when the second system is obtained by the smooth (diffeomorphic) G-equivariant change of coordinates y = h(x), i.e., the flows defined by x˙ = f(x) and y˙ = g(y) are equivalent provided that f(x) = (Dh(x))−1g(h(x)). Let A := ∇2Φ(0). Then ∇Φ(x) = Ax + φ(x) where |φ(x)| = o(|x|) as |x| → 0. Choose a Jordan basis {vi} of V such that A with respect to {vi} has a matrix representation

A 0 ! A = 1 , 0 A2 where A1 := A|V + and A2 := A|V − are a diagonal matrices. The inequalities (5.2) are clear since A1 (resp. A2) has only negative (resp. positive) entries + − on the main diagonal. For an element x = (x+, x−) ∈ V ⊕ V deﬁne its norm |x| = max {|x+| , |x−|}. The linear change of coordinates x 7→ εx, 1 gives us an equivariant map Fε(x) = Ax + φε(x), where φε(x) := ε φ(εx). 59

(5.3) Lemma. For any τ > 0 there exists an ε > 0 such that |φε(x)| < τ and kDφε(x)k < τ uniformly for x ∈ B2(0). Proof. Fix τ > 0. Since |φ(x)| = o(kxk) as x tends to 0 there is a δ > 0 such |φ(εx)| τ that |εx| ≤ 2 provided that |εx| < δ. Let ε be chosen such that |εx| < δ. Then 1 |φ(εx)| τ |φ (x)| = |φ(εx)| = |x| ≤ 2 = τ ε ε |εx| 2

The derivative of φε(x) is Dφε(x) = Dφ(εx). Since Dφ(x) is continuous and Dφ(0) = 0 for any τ > 0 one can take δ1 > 0 such that kDφ(εx)k ≤ τ if only |εx| < δ1. Taking ε small enough we are done. In order to find the index pair for the isolated invariant set {0} for the flow given by (5.1) we proceed as follows. Let N be the square {|x| ≤ 1}. It is easily seen that N is G-set since the action is orthogonal. If |x+| ≥ 2 2 |x−| then d/dt |x+| = 2hx˙ +, x+i = 2hA1x+, x+i + 2hg1(x), x+i ≤ −λ |x+| provided that 2τ ≤ λ. The same argument shows that if |x+| ≤ |x−| 2 2 then d/dt |x−| ≥ λ |x−| . Therefore, the flow of (5.1) leaves the square − + N via the set N = {x ∈ N; |x−| = 1} while the entrance set is N = − {x ∈ N; |x+| = 1}. That is the pair (N,N ) is a G-index pair for {0}. To see that N − is a G-set suppose, to the contrary, that x ∈ N − and gx ∈ {|x| = 1}\ N − for some g ∈ G (the sphere {|x| = 1} is obviously a G-set). If so, there exists sufficiently small t > 0 such that φ(0,t)(gx) ⊂ N while φ(0,t)(x) 6⊂ N and by the G-invariance of N one has gφ(0,t)(x) 6⊂ N. But this contradicts that φt is a G-map. Bibliography

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