{'fraction_non_alphanumeric': 0.06359910581222057, 'fraction_numerical': 0.013822652757078986, 'mean_word_length': 3.6974098704935248, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Localization of grouplike function and section spaces with compact domain

We extend the standard localization theory for function and section spaces due to Hilton-Mislin-Roitberg and Moller outside the CW category to the case of compact metric domain in the presence of a grouplike structure. We study applications in two cases directly generalizing the gauge group of a principal bundle. We prove an identity for the monoid of fibre-homotopy self-equivalences of a Hurewicz fibration -- due to Gottlieb and Booth-Heath-Morgan-Piccinini in the CW category -- in the compact case. This leads to an extended localization result for this monoid. We also obtain an extended localization theory for groups of sections of a fibrewise group. We give two applications in rational homotopy theory.


Introduction
We study two generalizations of the homotopy classification problem for gauge groups. Let X be a space, G a connected CW topological group and h : X → BG a map. The gauge group G(P ) corresponding to this data is the topological group of G-equivariant self-maps of P : E → X where P is the principal G-bundle induced by h. Fixing G and X, the problem is that of determining the number of distinct homotopy types or, alternately, the number of H-homotopy types corresponding to maps in [X, BG]. Complete results in special cases are given by Kono [20], Crabb-Sutherland [6] and Kono-Tsukuda [21] among many others.
After rationalization, the gauge group classification problem admits a complete solution with considerable generality. By [19,Theorem D], when X is a compact metric space and G is a homotopy finite, connected, CW group, the rationalization of the connected component of the identity of G(P ) is H-commutative and independent of the classifying map h : X → BG. When X is actually a finite CW complex, this result may be deduced from an identity for the gauge group due to Gottlieb, (1), below, combined with the standard localization theory for function spaces due to Hilton-Mislin-Roitberg [17] (see [19,Theorem 5.7]). Alternately, the result for X finite CW follows from corresponding localization results for section spaces due to Möller [24] and a result of Crabb-Sutherland [6, Proposition 2.2]. The proof for X compact metric in [19] entails an extension of the localization results of Hilton-Mislin-Roitberg and Möller to the case of non-CW domain. In this paper, we expand on this analysis to develop corresponding extended localization theories for two natural generalizations of the gauge group.
The two generalizations we consider are related to two basic identities for the gauge group. First, let map(X, Y ) denote the space of all continuous maps with the compact-open topology. Let map(X, Y ; f ) denote the path-component of a given map f : X → Y. Then, for X a finite CW complex, there is an H-equivalence (1) G(P ) ≃ Ωmap(X, BG; h) (see [12] and [1,Proposition 2.4]). Second, let Ad(P ) : E × G G ad → X denote the adjoint bundle. Here the total space is the quotient of the product by the diagonal action where G ad = G is a left-G space via the adjoint action and the projection is induced by the projection E → X. This is an non-principal G-bundle. We have an isomorphism where the latter space is the group of sections with fibrewise multiplication (see [1, p.539]). Now suppose given a Hurewicz fibration ξ : E → X. We assume here and throughout that X has a distinguished, nondegenerate basepoint. By the fibre of ξ, we will mean the fibre over this basepoint. We consider the monoid aut(ξ) consisting of all fibre-homotopy self-equivalences of ξ covering the identity of X topologized as a subspace of map(E, E). The monoid aut(ξ) is a natural generalization of the gauge group to the fibre-homotopy setting. Unlike the gauge group, the rational H-homotopy type of this monoid is generally a nontrivial invariant of the fibrehomotopy theory of ξ (see [11]).
By [2,Theorem 3.3], the identity (1) extends to a corresponding identity for aut(ξ) when both X and the fibre F are finite CW complexes. Our first main result extends this identity, in turn, to the case X is compact metric. Recall the universal F -fibration, for F finite CW, may be identified, up to homotopy type, as a sequence F → Baut * (F ) → Baut(F ) where aut(F ) and aut * (F ) are the monoids of free and based homotopy self-equivalences of F . The spaces Baut(F ) and Baut * (F ) are the Dold-Lashof classifying spaces for these monoids. (See [32,25] for the classification theory and [13] for the identification with Dold-Lashof [8].) Theorem 1. Let X be a compact metric space, F a finite CW complex and h : X → Baut(F ) a map. Let ξ : E → X be the corresponding F -fibration. Then there is an H-equivalence As a consequence, we obtain an extended localization theory for aut(ξ). Let P be a collection of primes. We say a space Y is nilpotent if Y is connected, has the homotopy type of a CW complex and has a nilpotent homotopy system (see [17,Definition II.2.1]). In this case, Y admits a P-localization ℓ Y : Given a monoid G we write G • for the path component of the identity. Given a space Z with distinguished basepoint we write Ω 0 Z for the space of loops based at the basepoint. For the function space map(X, Y ; f ) we assume f is the basepoint.
Theorem 2. Let X be a simply connected compact metric space, F a finite CW complex and h : X → Baut(F ) a map. Let ξ : E → X be the corresponding F -fibration. Then aut(ξ) • is a nilpotent space and we have an H-equivalence The second generalization we consider is based on (2) in which the gauge group corresponds to a group of sections. Recall ζ : E → X is a fibrewise group if there is a fibrewise map m : E × X E → E over X, a section e : X → E and a map i : E → E over X satisfying: (i) m is associative, (ii) e is a two-sided unit and (iii) i is an inverse with respect to the maps m and e. The space of sections Γ(ζ) is then a topological group with the multiplication of sections induced by m. More generally, relaxing the group axioms to require identities only up to homotopy, ζ is a fibrewise grouplike space ([5, p.62]) and Γ(ζ) is a grouplike space. Our motivating example is the adjoint bundle Ad(P ) of a principal G-bundle P, as above. Observe that, if P : E → X is classified by a map h : X → BG, then Ad(P ) is the pullback by h of the universal G-adjoint bundle EG × G G ad → BG which is, in particular, a CW fibration.
Suppose generally that ζ : E → X is a fibrewise grouplike space with connected grouplike fibre G and, further, that ζ is the pullback of a CW fibration. In this case, we may still identify a fibrewise P-localization ζ → ζ (P) of ζ (see the remarks preceding Theorem 2.4, below). Our third main result extends [24,Theorem 5.3] from the case the base space is finite CW to the case of compact metric base in this context. Theorem 3. Let ζ : E → X be a fibrewise grouplike space with connected, CW grouplike fibre G and base X a compact metric space. Suppose ζ is the pullback of a CW fibration. Then Γ(ζ) • is a nilpotent space and the map induced by a fibrewise P-localization ζ → ζ (P) is a P-localization map.
Remark 1.1. In many circumstances, the fibrewise P-localization ζ (P) of a fibrewise grouplike space ζ : E → X will itself be a fibrewise grouplike space over X and the fibre map ζ → ζ (P) will be equivariant. In this case, the equivalence of Theorem 3 is actually an H-equivalence. For example, this is the case for the adjoint bundle and, generally, when ζ is a CW fibration. For general compact metric X, the fibrewise grouplike structure for ζ (P) is not assured due to the lack of uniqueness of fibrewise P-localization outside the CW category.
The paper is organized as follows. We prove Theorems 1-3 in Section 2. In Section 3, we apply our results to obtain two consequences. We prove that the rationalization of aut(ξ) • is H-commutative and independent of the classifying map for fibrations ξ with fibre satisfying a famous conjecture in rational homotopy theory. (Theorem 3.3). This result extends [10, Theorem 4]. As an application of Theorem 3, we extend [19, Theorem F] on the classification of projective gauge groups (Theorem 3.6). In Section 4, we deduce based versions of our main results.

Localization of Function and Section Spaces
The standard results on P-localization of function spaces are as follows. By Milnor [27], the path components of map(X, Y ) are of CW homotopy type when X is a compact metric space and Y is a CW complex. By Hilton-Mislin-Roitberg, when X is finite CW and Y is a nilpotent space then map(X, Y ; f ) is itself a nilpotent space and These results all hold for spaces of basepoint-preserving functions, as well. In what immediately follows, we will focus on the basepoint free case. We discuss the based case in Section 4. The preceding results also hold with alternate hypotheses: namely, X may be any CW complex when Y is a finite Postnikov piece (see [18]). We will not consider these alternate hypotheses here as they are not amenable to our methods.
The results of Hilton-Mislin-Roitberg were extended to section spaces by Möller [24] (see also, Scheerer [28]). Let ξ : E → X be a fibration of connected CW complexes with X finite. Let F be the fibre over a basepoint of X. If F is a nilpotent space then ξ then admits a fibrewise P-localization which is a fibre map ξ → ξ (P) over X inducing P-localization on the fibres [26]. By [24,Theorem 5.3], under these hypotheses, the component of Γ(ξ; s) corresponding to a given section s : X → E is a nilpotent space and where s 0 is the section of ξ (P) induced by s.
Our extensions of these localization results depend on a classical result of Eilenberg-Steenrod in [9]. Let X be a compact metric space. By [9, Theorem X.10.1], there is an inverse system of finite complexes X j with structure maps g ij : X j → X i for i ≤ j and compatible maps g j : X → X j such that the induced map g : [9, Theorem X.11.9]. We apply this result to study the function space map(X, Y ; f ), as follows.
both induced by precomposition. We have: Theorem 6.4]) Let X be a compact metric space and Y a CW complex. Then, for all n ≥ 1, the maps (g j ) * above induce an isomorphism This result leads to an extension of the Hilton-Mislin-Roitberg result, mentioned above, provided the space map(X, Y ; f ) is known a priori to be a nilpotent space [19,Theorem 7.1]. Essentially the same proof yields the following: Proof. Consider the commutative square with vertical maps induced by ℓ Y . That the left vertical map is a P-localization map follows from [17,Theorem II.3.11]. Thus the right vertical map is a Plocalization map, as well. Looping, we see that ℓ Y induces a weak H-equivalence and so an honest H-equivalence since the spaces are CW, again by [27] .
Next let ξ : E → X be a Hurewicz fibration over a compact metric space with a fixed section s : X → E. Suppose ξ is the pullback of a CW fibration ξ 0 : E 0 → B for some map h : X → B. Then Γ(ξ; s) is of CW homotopy type [19, Proposition 3.2] Given an inverse system of finite complexes X j for X with compatible maps g j : X → X j as above, choose an index m so that h : X → B factors as h = h m • g m for some map h m : X m → B. Given j ≥ m, write ξ j : E j → X j for the pullback of the fibration ξ 0 via the map h j = h m • g mj : X j → B. Restricting again to indices j ≥ m gives a direct system of spaces of sections Γ(ξ j ) with structure maps γ ij : Γ(ξ i ) → Γ(ξ j ) and compatible maps γ j : Γ(ξ j ) → Γ(ξ) induced by g ij and g j for j ≥ i ≥ m.
By [19,Theorem 6.5], with this set-up the induced map γ : of pointed sets. Thus we may choose an index m ′ ≥ m and a section s m ′ of ξ m ′ such that s m ′ induces a section homotopic to s via γ m ′ . Restricting now to indices j ≥ m ′ , let s j be the section induced on ξ j by s m ′ . We obtain a direct system of connected spaces Γ(ξ j ; s j ) with compatible maps γ j : Quoting the theorem for n ≥ 1, we have: Let ξ : E → X be a Hurewicz fibration over X a compact metric space with a fixed section s : X → E. Suppose ξ is the pullback of a CW fibration. Then, for all n ≥ 1, the maps γ j above induce an isomorphism lim − → j π n (Γ(ξ j ; s j )) ∼ = π n (Γ(ξ; s)).
By the work of May [26], a fibration ξ : E → X of CW complexes with nilpotent fibre F admits a fibrewise P-localization which is a fibration ξ (P) : E 0 → X and a map g : E → E 0 over X such that g induces P-localization F → F P on fibres. This fibrewise P-localization of ξ is unique up to fibre-homotopy equivalence by Llerena [22,Theorem 6.1].
We may directly extend this construction to non-CW fibrations which are nevertheless the pullback of an appropriate CW fibration. Specifically, let ξ : E → X be the pullback of a fibration ξ 0 : E 0 → B of CW complexes with nilpotent fibre via a map h : X → B. We take ξ (P) = h −1 ((ξ 0 ) (P) ), the pullback of the fibrewise P-localization of ξ 0 . We note that uniqueness is no longer assured. Proof. Choose an inverse system X j of finite complexes for X and let ξ j be the corresponding fibrations over X j with fibre F and compatible sections s j . Consider the commutative diagram, as in the proof of Theorem 2.2: The horizontal maps are isomorphisms by Theorem 2.3. The left vertical map is P-localization by [24,Theorem 5.3]. Thus the right vertical map is a P-localization, as well.

Theorem 2.4 implies:
Proof of Theorem 3. Recall we are assuming ζ : E → X is fibrewise grouplike over a compact metric space X with fibre G a connected CW grouplike space. Further, we assume ζ is the pullback of CW fibration. By [19,Proposition 3.2], Γ(ζ) • is of CW type and so a nilpotent space since grouplike. The result is thus a direct consequence of Theorem 2.4.
We now turn to the proof of Theorem 2. Fix a compact metric space X and a finite CW complex F. We write ξ ∞ : E ∞ → Baut(F ) for the universal F -fibration.
(Here E ∞ ≃ Baut * (F ).) Given a map h : X → Baut(F ), let ξ : E → X be the pullback. In [2], the authors define a fibration in which the total space is the set of all homotopy equivalences f a,b : E a → (E ∞ ) b between fibres of ξ and fibres of ξ ∞ suitably topologized. The projection is given by f a,b → a ∈ X and so the fibre over a basepoint of X may be identified with aut(F ). Next, the authors define a fibration Φ : Γ(ξ 1 ξ ∞ ) → map(X, Baut(F )) with fibre Φ −1 (h) ≈ aut(ξ). We observe that the construction of Φ and the identification of the fibre both hold in our case. For the former, the key is [2, Proposition 2.1] whose proof depends on the use of exponential laws for functional fibrations which hold for X compact by [3]. The identification of the fibre over h with aut(ξ) requires that ξ be an F -fibration in the sense of [25,Definition 2.1] where F is the category of "fibres" homotopic to F. This fact is a consequence of our assumption that ξ is the pullback of a CW fibration (use [25, Lemma 3.3 and Proposition 3.4]).
Using Theorem 1 we obtain: Proof of Theorem 2. Recall we are assuming X is a simply connected, compact metric space and ξ : E → X is the pullback of the universal F -fibration via h : X → Baut(X). Since X is simply connected, h lifts to a maph : X → Baut(X) • to the universal cover. Further, we have an H-equivalence as these spaces are evidently weakly H-equivalent and both are CW complexes [27]. The result now follows from Theorem 2.2 and Theorem 1

Consequences in Rational Homotopy Theory
There are good algebraic models for the rational homotopy theory of the monoid aut(ξ) and the space of sections Γ(ζ). (See [11] for the former and [16,4] for the latter.) However, these models require finiteness and/or nilpotence conditions on the fibration and so are not directly applicable to the case of compact domain. In our main results above, we require that ξ and ζ occur as the pullback of a CW fibration. In this section, we observe that, in certain special circumstances, the structure of the rationalization of these CW fibrations allow for a description of the rational homotopy theory of these grouplike spaces of maps.
For the monoid aut(ξ), the CW fibration in question is the universal F -fibration. The following result is the P-local version of [2, Proposition 6.1] in our setting. where the latter is the space of null maps into the grouplike space (aut(F ) • ) P which is a grouplike space with pointwise multiplication.
Proof. The proof is the same as that given in [19,Example 4.7]. We give the details for completeness. Using the homotopy inverse for (Baut(F ) • ) P , we obtain a homotopy equivalence between map(X, (Baut(F ) • ) P ;h P ) and map(X, (Baut(F ) • ) P ; 0). Looping this equivalence and applying Theorem 2 yields H-equivalences: The condition on (Baut(F ) • ) P is, of course, very strong and will be rarely satisfied. However, as we explain now, there is a famous class of spaces in rational homotopy theory, identified by Halperin in [15], that conjecturally (and in many known cases) do have this property after rationalization. In the rational case, observe that it is sufficient to check (Baut(F ) • ) Q is an H-space to apply Theorem 3.1 since, rationally, all H-spaces are grouplike (see, e.g., [29]).
We say a space F is an F 0 -space if F is a simply connected elliptic space (i.e., a finite complex with π * (F ) ⊗ Q finite-dimensional) and satisfying H odd (F ; Q) = 0. Halperin's conjecture (see [15]) translated to our setting is: Examples of F 0 -spaces include products of even-dimensional spheres and complex projective spaces for which Conjecture 3.2 is easily confirmed. Homogeneous spaces G/H of equal rank, compact pairs H ⊆ G are also F 0 -spaces and satisfy Conjecture 3.2 by [31].
We note that, if F is an F 0 -space satisfying Conjecture 3.2, then (Baut(F ) • ) Q is a rational H-space since the Sullivan model of a nilpotent space with no odd rational homotopy has trivial differential. Further, the (odd) rational homotopy groups of aut(F ) • can be, in practice, directly computed from the minimal model (∧V ; d) of F using Sullivan's identity (∧V ; d)).
(See, e.g., [14].) Here the latter space is the homology of the DG vector space of negative degree derivations of (∧V ; d).
(2) The monoid aut(ξ) • is rationally H-commutative and equivalent to a product of Eilenberg-Mac Lane spaces with homotopy groups given by for n ≥ 1.
The computation of the rational homotopy groups of map(X, aut(F ); 0) reduces to the problem of computing homotopy groups of the space of maps into an Eilenberg-Mac Lane space. Here we have the identity of Thom [33]: π q (map(X, K(π, n)) ∼ = H n−q (X; π) which holds, with ordinary cohomology when X is CW. For compact spaces, the same identity holds with rationalČech cohomology by writing X = lim ← − X j and using the continuity ofČech cohomology (see the proof of [23,Theorem 5.6].) We prove a related result for grouplike spaces of sections. We say a fibration ξ : E → X of connected CW complexes is nilpotent if the spaces E and X are nilpotent. In this case, F is also nilpotent by [17,Theorem II.3.12] and term-byterm P-localization gives a fibration ξ P : E P → X P with fibre F P [17, Proposition II.2.13]. Note that the term-by-term P-localization ξ P : E P → X P is not the same as the fibrewise P-localization ξ (P) : E 0 → X. In fact, we have: Lemma 3.4. Let ξ : E → X be a nilpotent fibration with term-by-term Plocalization ξ P and fibrewise P-localization ξ (P) . Then there is a fibre-homotopy equivalence ξ (P) ≃ ℓ −1 X (ξ P ) where ℓ X : X → X P is a P-localization map.
As a direct consequence, we have: Theorem 3.5. Let ζ : E → X be a fibrewise grouplike space over a compact metric space X with fibre G connected CW. Suppose ζ is the pullback of nilpotent fibration ξ with P-localization (ξ) P fibre-homotopically trivial. Then Proof. By the remarks preceding the proof of Theorem 3, the fibrewise localization of ζ may be taken as the pullback of the fibrewise localization ξ (P) of ξ. Since the latter is the pullback of ξ P which is, by hypothesis, fibre-homotopically trivial, we see ζ (P) is fibre-homotopically trivial, as well. Now use Theorem 3.
We apply this last result to a natural generalization of the gauge group. Given a group G let P G = G/Z(G) denote the projectification. Given a map h : X → BP G write P h : E → X for the corresponding principal P G-bundle and Pad(P h ) : E × P G G ad → X the associated G-bundle where P G acts on G ad = G by the adjoint action. This is a fibrewise group. We call the group of sections Γ(Pad(P h )) the projective gauge group. When G = U (n), the projective gauge group corresponds to the group of unitaries of a complex matrix bundle (see [19,Example 3.7]).
The classification of rational H-types of projective gauge groups for G a compact, connected Lie group and X a compact metric space is given by [19,Theorem F]. The following result extends this to all connected Lie groups but at the expense of the H-structure.
Theorem 3.6. Let X be a compact metric space and G a connected Lie group. Then: (1) The rational homotopy type of Γ(Pad(P h )) • is independent of h ∈ [X, BP G].
Proof. We note that Pad(P h ) is the pullback by h of the universal projective adjoint bundle Pad(η P G ) : EP G× P G G ad → BP G. Here η P G : EP G → BP G is the universal principal P G-bundle. By [19,Lemma 5.11], the total space EP G× P G G ad is a nilpotent space. (This result assumes G is compact Lie. However, all that is used there is that Z(G) have vanishing higher rational homotopy groups, which is true for G connected Lie.) Thus Pad(η P G ) is nilpotent since BP G is simply connected. Below we show the rationalization of Pad(η P G) ) is fibre-homotopically trivial. The result (1) then follows from Theorem 3.5. The result (2) is proved using the arguments given in the proof of Theorem 3.3 (2).
To show Pad(η P G ) Q : (EP G × P G G ad ) Q → (BP G) Q is fibre homotopically trivial, we first show the total space is an H-space. Let EG × G G ad denote the total space of the universal G-adjoint bundle. Then EG × G G ad is also a nilpotent space. In fact, we have EG × G G ad ≃ map(S 1 , BG; 0) the free loop space of the classifying space (see, e.g., [19,Lemma 9.1]). Since BG is simply connected, map(S 1 , BG; 0) is nilpotent by [17,Theorem II.3.11]. Further, by this last result again, Now since G is a connected Lie group, it has oddly graded rational homotopy groups. It follows, as above, that (BG) Q is an H-space. Thus map(X, (BG) Q ; 0) is an H-space, as well. Next, by [19,Lemma 5.9], the natural map induces a surjection on rational homotopy groups. (Again, the lemma is stated for G compact Lie but the proof uses only that Z(G) is rationally aspherical.) Thus, by [23,Lemma 5.8], since EG× G G ad is a rational H-space so is EP G× P G G ad . We have shown that Pad(η P G ) is a sectioned, nilpotent fibration of rational H-spaces with simply connected base. By Lemma 3.7 below, (Pad(η P G )) Q is fibre-homotopically trivial, as needed.
Lemma 3.7. Let ξ : E → B be a fibration of nilpotent spaces with E, B and the fibre F each a rational H-space of finite type. Suppose B is simply connected and the linking homomorphism ∂ : π k (B) −→ π k−1 (E) is trivial after rationalization for all k ≥ 2. Then ξ Q is fibre homotopically trivial.
Proof. By [15,Theorem 4.6], ξ is a rational fibration as in [15,Definition 4.5]. In general, this means that ξ Q admits a Koszul-Sullivan model which is a sequence Here (∧V j ; d j ) is a free DG algebra over Q for j = 1, 2, 3 and a Sullivan model for F, E, B, respectively. The maps P and J are Sullivan models for the projection and fibre inclusion. The models (∧V 1 ; d 1 ) and (∧V 3 ; d 3 ) are minimal, meaning d j (V j ) is contained in the decomposables of ∧V j for j = 1, 3. Since B and F are assumed to be rational H-spaces, this forces d 3 = d 1 = 0. The model (∧V 2 ; d 2 ) is not, in general, minimal. However, by [15,Theorem 4.12], the vanishing of the rational linking homomorphism implies (∧V 2 ; d 2 ) is a minimal model for E, as well. Since H * (E; Q) ∼ = H * (∧V 2 ; d 2 ) is free we conclude d 2 = 0. Now we may directly obtain a DG algebra map J : ∧ (V 2 ; 0) → ∧(V 3 ; 0) with J • P = 1 ∧V3 . The maps J, J induce an isomorphism A : ∧ (V 2 ; 0) → (∧V 1 ; 0) ⊗ (∧V 3 ; 0) of DG algebras satisfying π 2 • A = P where π 2 : (∧V 1 ; 0) ⊗ (∧V 3 ; 0) → ∧(V 3 ; 0) is the projection. Using the correspondence between (homotopy classes of) maps between minimal DGAs and maps between rational spaces, we obtain α : F Q × B Q −→ E Q , the needed fibre homotopy equivalence.

The Based Case
We deduce versions of our main results in the basepoint preserving cases. First, let ξ : E → X be a fibration over a based space X and let F be the fibre over a fixed basepoint. We denote by aut F (ξ) the submonoid of aut(ξ) consisting of equivalences inducing the identity on F. This is the natural based version of aut(ξ). Note that aut F (ξ) is the fibre over the identity map 1 F of the restriction map res: aut(ξ) → aut(F ). We have the following based version of the identity (1) in our setting. Write map * (X, Y ) for the space of basepoint perserving maps from X to Y . Proof. First, note that, by [30], aut F (ξ) is of CW type since aut(ξ) and aut(F ). Now compare the long exact homotopy sequence of the restriction fibration above to that of loops on the evaluation fibration ω : map(X, Baut(F ); h) → Baut(F ) with fibre map * (X, Baut(F ); h). The result follows from Theorem 1 and the 5-lemma.
As before, this gives a corresponding localization result for aut F (ξ) • : Theorem 4.2. Let X be a compact metric space, F a finite CW complex and h : B → Baut(F ) a map. Let ξ : E → X be the corresponding F -fibration. Then aut F (ξ) • is a nilpotent space and we have an H-equivalence (aut F (ξ) • ) P ≃ Ω • map * (X, (Baut(F ) • ) P ; (h) P ).
Next, given a fibration ζ : E → X write Γ * (ζ) for the space of basepoint preserving sections of ζ. We have: Theorem 4.3. Let ζ : E → X be a fibrewise grouplike space with connected, CW grouplike fibre G and base X a connected compact metric space. Suppose ζ is the pullback of a CW fibration. Then Γ * (ζ) • is a nilpotent space and the map Γ * (ζ) • → Γ * (ζ (P) ) • induced by a fibrewise P-localization ζ → ζ (P) is a P-localization map.
Proof. In this case, the relevant evaluation fibration ω : Γ(ζ) • → E with fibre Γ * (ζ) • is not of use since E is not well-behaved under fibrewise localization. However, we may repeat the entire argument in the based case to achieve the needed result. We note that Möller's theorem [24,Theorem 5.3], for X a finite complex, is proved in the based setting. The rest of the argument thus proceeds as before.
Finally, we remark that our applications Theorems 3.3 and 3.6 also hold in the respective based settings but with ordinaryČech cohomology replaced by reduceď Cech cohomology in the rational homotopy calculations.