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First Steps in Twisted Rabinowitz-Floer Homology

Rabinowitz-Floer homology is the Morse-Bott homology in the sense of Floer associated with the Rabinowitz action functional introduced by Kai Cieliebak and Urs Frauenfelder in 2009. In our work, we consider a generalisation of this theory to a Rabinowitz-Floer homology of a Liouville automorphism. As an application, we show the existence of noncontractible periodic Reeb orbits on quotients of symmetric star-shaped hypersurfaces. In particular, our theory applies to lens spaces.


Introduction
In this paper, we introduce an analogue of the twisted Floer homology [Ulj17] in the Rabinowitz-Floer setting. See the excellent survey article [AF12b] for a brief introduction to Rabinowitz-Floer homology and [AS19] for an overview of common different Floer theories. Following [CF09] and [AF10], we construct a Morse-Bott homology for a suitable twisted version of the standard Rabinowitz action functional, generalising the standard Rabinowitz action functional.
The proof is straightforward, once we have computed the Z m -equivariant twisted Rabinowitz-Floer homology of the spheres S 2n−1 ⊆ C n . Indeed, by invariance we may assume that Σ = S 2n−1 , as Σ is star-shaped. Then we use the following elementary topological fact (see Lemma 1.1 below). Let Σ be a simply connected topological manifold and let ϕ : Σ → Σ be a homeomorphism of finite order m that is not equal to the identity. If the induced discrete action Lemma 1.1. If γ ∈ L ϕ (Σ, x) for some x ∈ Σ, then π • γ ∈ L (Σ/Z m , π(x)) is not contractible. Conversely, if γ ∈ L (Σ/Z m , π(x)) is not contractible, then there exists 1 ≤ k < m such thatγ x ∈ L ϕ k (Σ, x) for the unique liftγ x of γ satisfyingγ x (0) = x.
For a more detailed study of twisted loop spaces of universal covering manifolds as well as a proof of Lemma 1.1 see Appendix A. We put Theorem 1.2 into context. If Σ 2n−1 /Z m satisfies the index condition  Figure 1: The projection π • γ ∈ L (Σ/Z m , π(x)) of γ ∈ L ϕ (Σ, x) is not contractible for the deck transformation ϕ = id Σ .
where M denotes the completion of W . Note that even in the case of lens spaces this need not be the case, as for example RP 2n−1 is not Liouville fillable for any odd n ≥ 2 by [GN20, Theorem 1.1]. As the index condition (1) is only required for contractible Reeb orbits and they come from the universal covering manifold Σ, we can say something in the case where Σ is strictly convex. Indeed, the Hofer-Wysocki-Zehnder Theorem [FK18, Theorem 12.2.1] then implies that Σ is dynamically convex, that is, holds for all periodic Reeb orbits γ. Thus for n ≥ 2, the index condition is satisfied and we actually computeŠH * (S 2n−1 /Z m ) via the Z m -equivariant version ofŠH * (S 2n−1 ). In the case of hypertight contact manifolds, there is a similar construction without the index condition (1). See for example [  Let (M, ω) be a symplectic manifold and ϕ ∈ Symp(M, ω). Given a twisted loop γ ∈ L ϕ M and ε 0 > 0, we say that a curve is smooth. Since γ ε (t + 1) = ϕ(γ ε (t)) holds for all ε ∈ (−ε 0 , ε 0 ) and t ∈ R, we call such a variation a twisted variation. Then the infinitesimal variation Lemma 2.1. Let (M, ω) be a symplectic manifold and let ϕ ∈ Symp(M, ω) be of finite order. Let γ ∈ L ϕ M and let X ∈ X(γ) be such that Then there exists a twisted variation of γ such that δγ = X.
Proof. As ϕ is assumed to be of finite order, there exists a ϕ-invariant ωcompatible almost complex structure J on M by [MS17, Lemma 5.5.6]. With respect to the induced Riemannian metric the symplectomorphism ϕ is an isometry. Define the exponential variation , for ε 0 > 0 sufficiently small and ∇ J denoting the Levi-Civita connection associated with m J . Such an ε 0 > 0 does exist by naturality of geodesics [Lee18, Corollary 5.14]. Then we compute by naturality of the exponential map [Lee18,Proposition 5.20].
Remark 2.1. The statement of Lemma 2.1 remains true if ord ϕ = ∞.
This discussion together with Lemma A.1 motivates the following definition of the tangent space to the free twisted loop space.
Definition 2.2 (Tangent Space to the Free Twisted Loop Space). Let (M, ω) be a symplectic manifold and ϕ ∈ Symp(M, ω). For γ ∈ L ϕ M define the tangent space to the free twisted loop space at γ by Definition 2.3 (Twisted Hamiltonian Function). Let (M, ω) be a symplectic manifold and ϕ ∈ Symp(M, ω). A function H ∈ C ∞ (M × R) is said to be a twisted Hamiltonian function, iff We denote the space of all twisted Hamiltonian functions by C ∞ ϕ (M × R) and the subspace of all autonomous twisted Hamiltonian functions by C ∞ ϕ (M ). Recall, that an exact symplectic manifold is by definition a pair (M, λ) such that (M, dλ) is a symplectic manifold. An exact symplectomorphism of an exact symplectic manifold (M, λ) is a diffeomorphism ϕ ∈ Diff(M ) such that ϕ * λ − λ is exact.
Definition 2.4 (Perturbed Twisted Rabinowitz Action Functional). Let (M, λ) be an exact symplectic manifold and ϕ ∈ Diff(M ) an exact symplectomorphism with ϕ * λ − λ = df . For H, F ∈ C ∞ ϕ (M × R) define the perturbed twisted Rabinowitz action functional and call A H ϕ the twisted Rabinowitz action functional.
Definition 2.5 (Differential of the Perturbed Twisted Rabinowitz Action Functional). Let ϕ ∈ Diff(M ) be an exact symplectomorphism of an exact symplectic manifold (M, λ). For H, F ∈ C ∞ ϕ (M × R), define the differential of the perturbed twisted Rabinowitz action functional where γ ε is a twisted variation of γ such that δγ = X.
Fix a Riemannian metric on M and let X v denote the unique parallel vector field along γ| I such that X v (t 0 ) = v. As Int I is open, there exists δ > 0 such thatB δ (t 0 ) ⊆ Int I. Fix a smooth bump function β ∈ C ∞ (I) for t 0 supported in B δ (t 0 ). By shrinking δ if necessary, we may assume that we have that βX v ∈ T γ L ϕ M and thus we compute = τ ϕ k * X Ht (γ(t + k)) + ϕ k * X Ft (γ(t + k)) = τ X ϕ k * Ht (γ(t + k)) + X ϕ k * Ft (γ(t + k)) = τ X H t+k (γ(t + k)) + X F t+k (γ(t + k)) for all t ∈ I and k ∈ Z. The other direction is immediate.
Corollary 2.1. The differential of the perturbed twisted Rabinowitz action functional is well-defined, that is, independent of the choice of twisted variation, and linear.
Preservation of energy of an autonomous Hamiltonian system yields the following corollary.
There is a natural R-action on the twisted loop space L ϕ M given by If (M, λ) is an exact symplectic manifold and H ∈ C ∞ ϕ (M ) for an exact symplectomorphism ϕ ∈ Diff(M ) of finite order such that supp f ∩ H −1 (0) = ∅, then the twisted Rabinowitz action functional A H ϕ is invariant under the induced S 1action on Crit A H ϕ . In particular, the unperturbed twisted Rabinowitz action functional is never a Morse function.
Definition 2.7 (Symplectic Connection). Let (M, ω) be a symplectic manifold. A symplectic connection on (M, ω) is defined to be a torsion-free connection ∇ in the tangent bundle T M such that ∇ω = 0.
Remark 2.3. Every symplectic manifold admits a symplectic connection by [Gut06,p. 308], but in sharp contrast to the Riemannian case, a symplectic connection on a given symplectic manifold is in general not unique.
Corollary 2.3. The Hessian of the twisted Rabinowitz action functional is a well-defined, that is, independent of the choice of twisted two-parameter family, symmetric bilinear form.
In fact, the Hessian of the twisted Rabinowitz action functional is also independent of the choice of symplectic connection.
Lemma 2.3. Let ϕ ∈ Diff(M ) be an exact symplectomorphism of an exact symplectic manifold (M, λ) and H ∈ C ∞ ϕ (M ). If (γ, τ ) ∈ Crit A H ϕ , then with φ H denoting the smooth flow of the Hamiltonian vector field X H .
Proof. One computes Inserting this into (4) yields But as ∇ has no torsion by assumption, we compute for all t ∈ I. where Proof. We follow [FK18,. Let (X, η) ∈ ker Hess A H ϕ | (γ,τ ) and define v : We claim that is an isomorphism. First, we show that the above homomorphism is indeed well-defined. The assumption that (X, η) lies in the kernel of the Hessian of the twisted Rabinowitz action functional at the critical point (γ, τ ) is by Corollary 2.4 equivalent tov = ηX H (γ(0)) and Integrating the first equation yields That (8) is an isomorphism follows by considering the inverse This establishes the canonical isomorphism (8).
In what follows, we assume that the energy hypersurface H −1 (0) is a contact manifold. A contact manifold is a pair (Σ, α), where Σ is an odd-dimensional manifold and α ∈ Ω 1 (Σ) is a global contact form. Every contact manifold (Σ, α) admits a unique vector field R ∈ X(Σ), called the Reeb vector field, defined implicitly by Recall, that a strict contactomorphism of a contact manifold (Σ, α) is defined to be a diffeomorphism ϕ ∈ Diff(Σ) such that ϕ * α = α. Note that the Reeb flow always commutes with a strict contactomorphism.

Compactness of the Moduli Space of Twisted
Negative Gradient Flow Lines Definition 3.1 (Liouville Domain). A Liouville domain is defined to be a compact connected exact symplectic manifold (W, λ) with connected boundary such that the Liouville vector field X defined implicitly by i X dλ = λ is outwardpointing along the boundary.
is said to be a Liouville automorphism, iff ϕ(Σ) = Σ, ϕ * λ−λ is exact and compactly supported in Int W , and ord ϕ < ∞ near the boundary. The set of all Liouville automorphisms on the Liouville domain (W, λ) is denoted by Aut(W, λ).
Remark 3.2. The set Aut(W, λ) of Liouville automorphisms of a Liouville domain (W, λ) is in general not a group. Indeed, for ϕ, ψ ∈ Aut(W, λ) it is not necessarily true that ϕ • ψ is of finite order unless ϕ and ψ commute. Definition 3.4 (Twisted Defining Hamiltonian Function). Let (W, λ) be a Liouville domain with boundary Σ and ϕ ∈ Aut(W, λ). A twisted defining Hamiltonian function for Σ is a Hamiltonian function H ∈ C ∞ (M ) on the completion (M, λ) of (W, λ), satisfying the following conditions: Denote by F ϕ (Σ) the set of twisted defining Hamiltonian functions for Σ.
This is not true in general if ϕ does not induce a strict contactomorphism on Σ.
Definition 3.5 (Adapted Almost Complex Structure). Let (W, λ) be a Liouville domain with boundary Σ. An adapted almost complex structure is defined to be a dλ-compatible almost complex structure J on (W, λ) such that J restricts to define a dλ| Σ -compatible almost complex structure on the contact distribution ker λ| Σ and JR = ∂ r holds near the boundary.
Definition 3.6 (Rabinowitz-Floer Data). Let (M, λ) be the completion of a Liouville domain (W, λ) with boundary Σ and ϕ ∈ Aut(W, λ). Rabinowitz-Floer data for ϕ is defined to be a pair (H, J) consisting of a twisted defining Hamiltonian function H ∈ F ϕ (Σ) for Σ and of a smooth family J = (J t ) t∈R of adapted almost complex structures on W such that Remark 3.4. For simplicity we ignore the fact, that in order to achieve transversality of the moduli spaces in general one actually needs a dependence of the smooth family of almost complex structures on the Lagrange multiplier (see [AM18]). But this technicality does not significantly alter our arguments as explained in [FS16].
Indeed, that ψ is an exact symplectic embedding follows from the computation that is, ϕ and ψ commute. Note that (12) makes sense because ϕ(Σ) = Σ by assumption. Indeed, (12) follows from the uniqueness of integral curves and the where we used the ϕ-invariance of the Liouville vector field on U δ , that is, and assumption (11). Next we construct the defining Hamiltonian H ∈ C ∞ (M ). Let h ∈ C ∞ (R) be a sufficiently small mollification of the piecewise linear function Figure 2. Define Then H is a defining Hamiltonian for Σ and dH is compactly supported by construction. Moreover, H is ϕ-invariant by (12). Finally, Next we construct the family J = (J t ) t∈R of dλ-compatible almost complex structures on W . Fix a dλ| Σ -compatible almost complex structure J on the contact distribution ker λ| Σ and choose a path (J Σ t ) t∈I ⊆ J (ker λ| Σ , dλ| Σ ) from J to ϕ * J. Extend this smooth family to (J Σ t ) t∈R satisfying ϕ * J Σ t+1 = J Σ t for all t ∈ R. Finally, extend this family to ((−δ, +∞) × Σ, d(e r λ| Σ )) by where π : ker λ| Σ ⊕ R → ker λ| Σ denotes the projection. Choose a smooth family J W \Σ t t∈R on W \ Σ twisted by ϕ, and let β Σ , β W \Σ be a partition of unity subordinate to {U δ , W \ Σ}. Define a smooth family (m t ) t∈R of Riemannian metrics on W by and let (J t ) t∈R be the corresponding family of dλ-compatible almost complex structures on W .
With respect to the L 2 -metric (15), the gradient of the twisted Rabinowitz action functional grad J A H ϕ ∈ X(L ϕ M × R) is given by Lemma 3.2 (Fundamental Lemma). Let (H, J) be Rabinowitz-Floer data for a Liouville automorphism ϕ ∈ Aut(W, λ). Then there exists a constant C = C(λ, H, J) > 0 such that Proof. The proof [CF09, Proposition 3.2] goes through with minor modifications as f ϕ ∞ < +∞ by assumption.
Definition 3.9 (Energy). Let (H, J) be Rabinowitz-Floer data for a Liouville automorphism ϕ ∈ Aut(W, λ). The energy of a twisted negative gradient flow line (u, τ ) is defined by Theorem 3.1 (Compactness). Let (H, J) be Rabinowitz-Floer data for a Liouville automorphism ϕ ∈ Aut(W, λ). Suppose (u µ , τ µ ) is a sequence of negative gradient flow lines of the twisted Rabinowitz action functional A H ϕ such that there exist a, b ∈ R with Then for every reparametrisation sequence (s µ ) ⊆ R there exists a subsequence µ ν of µ and a negative gradient flow line (u ∞ , τ ∞ ) of A H ϕ such that [CF09,p. 268] goes through without any changes as we have a twisted version of the Fundamental Lemma. However, for convenience, we reproduce the main arguments here. We need to establish  Lastly, the uniform L ∞ -bound on the derivatives of u µ follows from standard bubbling-off analysis. Indeed, if the derivatives of u µ are unbounded, then there exists a nonconstant pseudoholomorphic sphere as in [MS12,Section 4.2]. This is impossible as M is an exact symplectic manifold and thus in particular symplectically aspherical.

Definition of Twisted Rabinowitz-Floer Homology
In this section we define twisted Rabinowitz-Floer homology for the setting introduced in the last section. Note that here we make implicit use of the requirement that a Liouville automorphism has finite order near the boundary. This is crucial because then the arguments go through as in the case of loops by Remark 2.2.
Let (H, J) be Rabinowitz-Floer data for ϕ ∈ Aut(W, λ). ζ (γ,τ ) (γ, τ ) satisfying the Novikov finiteness condition Define a boundary operator Proof. If there do not exist any nonconstant twisted periodic Reeb orbits, for any H ∈ F ϕ (Σ). Since Fix(ϕ| Σ ) is a properly embedded submanifold of Σ by [Lee18, Problem 8-32] or [MS17, Lemma 5.5.7], A H ϕ is Morse-Bott. Let x, y ∈ Fix(ϕ| Σ ). As Σ is simply connected by assumption, there exists a path γ from x to y in Σ and a homotopy from γ to ϕ • γ with fixed endpoints. Extend this homotopy to a path in L ϕ Σ from c x to c y . Choose a Morse function h on Fix(ϕ| Σ ) and any critical point c x ∈ Fix(ϕ| Σ ). Then a Z-grading of RFC ϕ (Σ, M ) is given by the index µ((c y , 0), (c x , 0)) + ind h (c y ) = ind h (c y ) ∀c y ∈ Crit(h), and consequently, as there are only twisted negative gradient flow lines with zero cascades, that is, ordinary Morse gradient flow lines of h. Indeed, suppose that there is a nonconstant twisted negative gradient flow line (u, τ ) of A H ϕ such that lim s→±∞ (u(s), τ (s)) = (γ ± , τ ± ) ∈ Crit A H ϕ .

Invariance of Twisted Rabinowitz-Floer Homology Under Twisted Homotopies of Liouville Domains
Twisted Rabinowitz-Floer homology is stable under twisted homotopies of Liouville domains. This property is crucial for proving Theorem 1.2. Remark 5.1. Invariance of twisted Rabinowitz-Floer homology allows us to define twisted Rabinowitz-Floer homology also in the case where A H ϕ is not necessarily Morse-Bott. Indeed, as the proof of Theorem 5.1 shows, we can perturb the hypersurface Σ slightly to make it Morse-Bott. Thus we can define the twisted Rabinowitz-Floer homology of such a hypersurface to be the twisted Rabinowitz-Floer homology of any Morse-Bott perturbation. This is well-defined by Theorem 5.1. Proof. Note that F ϕ (Σ) is a convex space. Indeed, set Then ϕ * H σ = H σ , dH σ has compact support and X Hσ | Σ = R for all σ ∈ I. Moreover, for the Liouville vector field X ∈ X(M ) we compute for any H ∈ F ϕ (Σ), and thus H < 0 on Int W and H > 0 on M \ W . Consequently, H −1 σ (0) = Σ and so H σ ∈ F ϕ (Σ) for all σ ∈ I. Hence (H σ ) σ∈I is a twisted homotopy of Liouville domains in M and Theorem 5.1 implies the claim.

Twisted Leaf-Wise Intersection Points
Definition 6.1 (Twisted Leaf-Wise Intersection Point). Let (M, λ) be the completion of a Liouville domain (W, λ) and let ϕ ∈ Aut(W, λ) be a Liouville automorphism. A point x ∈ Σ is a twisted leaf-wise intersection point for a Hamiltonian symplectomorphism ϕ F ∈ Ham(M, dλ), iff and χ ∈ C ∞ (S 1 , I) such that 1 0 χ = 1. Any time-dependent Hamiltonian function χH is said to be weakly timedependent.
(ii) supp χ ⊆ 0, 1 2 and F t = 0 for all t ∈ 0, 1 2 . Lemma 6.1. Let ϕ ∈ Aut(W, λ). For all H ∈ F ϕ (Σ) and ϕ F ∈ Ham(M, dλ) there exists a corresponding twisted Moser pair M such that the flow of χX H is a time-reparametrisation of the flow of X H .
where the last equation is the observation that twisted Rabinowitz-Floer homology in the autonomous case extends to the weakly time-dependent case without any issues. Crucial is, that the period-action equality (see Remark 2.4) is still valid. Indeed, we compute

Existence of Noncontractible Periodic Reeb Orbits
We define an equivariant version of twisted Rabinowitz-Floer homology following [AF12a,p. 487]. Denote by (C n , ω) the standard symplectic vector space with symplectic form and coordinates z j := x j + iy j . Then ω = dλ for Consider the free smooth discrete action on the odd-dimensional sphere S 2n−1 := (z 1 , . . . , z n ) ∈ C n : n j=1 |z j | 2 = 1 generated by the rotation ϕ : C n → C n , ϕ(z 1 , . . . , z n ) := e 2πik1/m z 1 , . . . , e 2πikn/m z n for m ≥ 1 and k 1 , . . . , k n ∈ Z coprime to m. Define a twisted defining Hamiltonian function H ∈ F ϕ (S 2n−1 ) by for some sufficiently small mollification of the piecewise linear function Fix a ϕ-invariant ω-compatible almost complex structure on (C n , ω). Then the rotation ϕ induces a free Z m -action on Crit A H ϕ and on the moduli space of twisted negative gradient flow lines with cascades of A H ϕ . Therefore, we can define Z m -equivariant twisted Rabinowitz-Floer homology as the homology of the Z-graded chain complex (see Remark 4.1) for some ϕ-invariant Morse function h on Crit A H ϕ . Theorem 7.1. Let n ≥ 2. For m ≥ 1 consider the rotation ϕ : C n → C n , ϕ(z 1 , . . . , z n ) := e 2πik1/m z 1 , . . . , e 2πikn/m z n for k 1 , . . . , k n ∈ Z coprime to m. Then If m is even, then RFH ϕ k (S 2n−1 /Z m ) is generated by a noncontractible periodic Reeb orbit in the lens space S 2n−1 /Z m for all k ∈ Z.
Proof. First we consider the special case ϕ : C n → C n , ϕ(z) = e 2πi/m z.
Arguing as in [FK18, p. 235-236] we conclude Fix z 0 ∈ S 2n−1 and define η := φ τ0R (z 0 ). Note that φ τ k R (z) belongs to the same equivalence class in π 0 L ϕ S 2n−1 as η for all z ∈ S 2n−1 and k ∈ Z because S 2n−1 is simply connected for n ≥ 2. Let h ∈ C ∞ (S 2n−1 ) be the standard height function. By Remark 4.1, RFH ϕ (S 2n−1 , C n ) carries the Z-grading We claim that the number of twisted negative gradient flow lines between the minimum of S 2n−1 × {k + 1} and the maximum of S 2n−1 × {k} must be odd, so that the critical manifold Crit A H ϕ looks like a string of pearls, see Figure  4. Indeed, if there is an even number of such negative gradient flow lines, then RFH ϕ (S 2n−1 , C n ) = 0, contradicting Theorem 6.1 as S 2n−1 is displaceable in the completion C n . The resulting chain complex is given by where 1 ∈ M m×m (Z 2 ) has every entry equal to 1 and A ∈ M m×m (Z 2 ) is defined by where e ij ∈ M m×m (Z 2 ) satisfies (e ij ) kl = δ ik δ jl . Thus the resulting chain complex looks like a rope ladder. Compare Figure 4. Passing to the quotient via the free Z m -action, we get the acyclic chain complex if m is even and the alternating chain complex if m is odd. From this the statement follows.
For the general case, we note that Crit A H ϕ is a disjoint union of spheres and different copies of Z. It is therefore rather hard to compute the equivariant twisted Rabinowitz-Floer homology directly via analysing the critical manifold. By Theorem 6.1, there is a canonical isomorphism where RFH Zm * (S 2n−1 , C n ) denotes the Z m -equivariant Rabinowitz-Floer homology constructed in [AF12a,p. 487]. A computation similar as before shows The crucial observation is, that Crit A H ∼ = S 2n−1 × Z. In particular, the string of pearl looks like in Figure 5. Finally, RFH ϕ k (S 2n−1 /Z m ) is generated by a noncontractible periodic Reeb orbit in S 2n−1 /Z m for all k ∈ Z by Lemma 1.1. Corollary 7.1 (Tate Homology). Let C m denote the cyclic group of order m ≥ 1. Then for the trivial left C m -module Z 2 we have that whereĤ * (C m ; Z 2 ) denotes the Tate homology group of C m with coefficients in the trivial left C m -module Z 2 .
Using Theorem 7.1 we can prove Theorem 1.2.
Proof. By assumption, Σ bounds a star-shaped domain D with respect to the origin. Thus (D ∪Σ, dλ) is a Liouville domain with λ given by (16). By rescaling we may assume that S 2n−1 ⊆ D. Define a smooth function denotes the Liouville vector field. We claim that δ • ϕ = δ. Indeed, δ(ϕ(x)) is the unique number such that φ X δ(ϕ(x)) (ϕ(x)) ∈ S 2n−1 . As the flow of X and ϕ commute by the proof of Lemma 3.1, we conclude that φ X δ(ϕ(x)) (x) ∈ S 2n−1 . Define a smooth family of star-shaped hypersurfaces (Σ σ ) σ∈I Then we compute σδ(y) (y) : y ∈ ϕ(Σ) = φ X σδ(y) (y) : y ∈ Σ = Σ σ for all σ ∈ I and therefore we can find a twisted homotopy (H σ ) σ∈I of Liouville domains in C n . By Theorem 5.1 we have that giving rise to a canonical isomorphism of the associated Z m -equivariant twisted Rabinowitz-Floer homology But by Theorem 7.1 the latter does not vanish as m ≥ 2 is even.

A Twisted Loops in Universal Covering Manifolds
In this Appendix, we will consider the category of topological manifolds rather than the category of smooth manifolds, because smoothness does not add much to the discussion. Free and based loop spaces are fundamental objects in Algebraic Topology, for a vast treatment of the geometry and topology of based as well as free loop spaces see for example [LO15]. But so-called twisted loop spaces are not considered that much.
where ψ ∈ Aut π (M ) is such that ψ(e) = e , (c) The projectionπ x : defined byπ x (γ) := π • γ is a covering map with number of sheets coinciding with the cardinality of π 1 (M, x). Moreover,π x restricts to define a covering mapπ andπ x gives rise to a principal Aut π (M )-bundle. If M admits a smooth structure, then this bundle is additionally a bundle of smooth Banach manifolds.
For proving part (b), observe that Φ e and Φ e are isomorphisms follows from [Lee11, Corollary 12.9]. Moreover, it is also clear that C ψ is an isomorphism with inverse C ψ −1 . Let [γ] ∈ π 1 (M, x). Then using part (a) we compute Thus by uniqueness [Lee11, Proposition 12.1 (a)], we conclude Finally for proving (c), define a metricd ∞ on Then the induced topology coincides with the disjoint union topology and with respect to this topology,π x is continuous. So left to show is thatπ x is a covering map. Surjectivity is clear. So let γ ∈ L (M, x). Then γ ∈ U η for some [η] ∈ π 1 (M, x). Now note that U η is open in L (M, x) and by part (a) we concludeπ −1 for some fixed e ∈ π −1 (x) and ϕ ∈ Aut π (M ) such that ϕ(e) =η e (1). As the cardinality of the fibre π −1 (x) and of Aut π (M ) coincides with the cardinality of the fundamental group π 1 (M, x) by [Lee11, Corollary 11.31] and part (b), we conclude that the number of sheets is equal to the cardinality of the fundamental group π 1 (M, x).
Then α β is Aut π (M )-equivariant with respect to this action for all β ∈ B. Indeed, using again the commutative diagram (17) we compute for all ξ ∈ Aut π (M ) and γ ∈ L ψ•ϕ•ψ −1 (M , ψ(e)). Note, that here we use again the fact that Aut π (M ) acts transitively on the fibre π −1 (x). Suppose that M admits a smooth structure. Then for every compact smooth manifold N we have that the mapping space C(N, M ) admits the structure of a smooth Banach manifold by [Wit19]. Therefore, the twisted loop space L ϕ (M , e) = ev −1 1 (ϕ(e)) is also a smooth Banach manifold for all ϕ ∈ Aut π (M ) and e ∈ π −1 (x) by the implicit function theorem [MS12, Theorem A.3.3]. As the fundamental group π 1 (M, x) is countable, the topological space E has only countably many connected components being smooth Banach manifolds and thus the total space itself is a smooth Banach manifold. Finally, Aut π (M ) is trivially a Banach manifold with dim Aut π (M ) = 0 as a discrete Lie group.  where ψ ∈ Aut π (M ) is such that ψ(e) = e .
Remark A.1. Using Lemma A.1 one should be able to prove similar results as in Theorem A.1 in the case of free twisted loop spaces. However, in the non-abelian case the situation gets much more complicated as in general it is not true, that lifts of conjugated elements of the fundamental group lie in the same free twisted loop space by [LO15, Theorem 1.6 (i)].