Classification of fiber sequences with a prescribed holonomy action
aa r X i v : . [ m a t h . A T ] F e b Classification of fibrations with a prescribed structural group
Mario Fuentes ∗ February 3, 2021
Abstract
We classify fibrations with a prescribed action of the fundamental group of the baseon the fibre.
Introduction
From the foundational paper of Stasheff [7] to the recent reference [1] of homotopy theoret-ical flavour, the literature is splashed of results which let us classify fibrations, sometimesof a certain type (see for instance [4, 8]), by means of a suitable classifying space. Themost general result in the homotopy category is probably the classical work of May [5]which classifies fibrations with a given category of fibers . However, this treatment doesnot readily apply to classify fibration with a prescribed action of the fundamental groupof the base on the fiber. Although this may be part of the folklore we are not aware ofany detailed reference in the literature and the purpose of this note is to fill this gap in arigourous way. The resulting classifying space is a well known and interesting object, seefor instance [2]. Particularly interesting examples and applications of this classification, inthe rational homotopy category, are being considered in [3].Throughout this text, any considered topological space is compactly generated andweakly Hausdorff. Also, F shall always denote a space of the homotopy type of a CW-complex. By a fibration with fiber F we mean a sequence F ω −→ E p −→ X where X is path connected and pointed by x , p is a Hurewicz fibration, and the map ω : F ≃ w −→ F = p − ( x ) is a weak homotopy equivalence. A map between two such fibra-tions, F ω −→ E p −→ X and F ω −→ E p −→ X , is a homotopy commutative diagram of the ∗ The author has been partially supported by the MICINN grant MTM2016-78647-P and the grantFPU18/04140.2020
Mathematics Subject Classification . 55R15, 55R05.
Key words and phrases . Fiber spaces, classifying spaces, action of the fundamental group. E p ❆❆❆❆❆❆❆ f (cid:15) (cid:15) F ω > > ⑥⑥⑥⑥⑥⑥⑥⑥ ω ❆❆❆❆❆❆❆❆ XE p > > ⑥⑥⑥⑥⑥⑥⑥⑥ Observe that the map f is necessarily a weak homotopy equivalence and note also that welose no generality by imposing the right triangle to be strictly commutative. The maps offibrations over X with fibre F generate a equivalence relation, and we denote by F ib( X, F )the corresponding equivalence classes. Remark that this equivalence relation is, in general,less restrictive than the usual one in which f is required to be a homotopy equivalence, andthus a fiber homotopy equivalence. However, if X has the homotopy type of a CW-complex,the set of equivalence classes of both relations coincide (see [6, Lemma 1.1]).Given a fibration F ω −→ E p −→ X and α ∈ π ( X, x ), the induced homotopy equiva-lence ˆ α : F ≃ → F together with the bijection ω ∗ : [ F, F ] ∼ = → [ F, F ] produces the homotopyequivalence α = ω ∗− [ ˆ αf ] ∈ [ F, F ]. That is, ωα ≃ e αω .Let aut F be the topological monoid of self homotopy equivalences of F and write E ( F ) = π (aut F ). Fix a normal subgroup H E E ( F ) and consider aut H F ⊂ aut F thesubmonoid of homotopy equivalences ϕ such that [ ϕ ] ∈ H . Definition 0.1.
A fibration with fiber F is an H -fibration if α ∈ H for any α ∈ π ( X, x ).Denote by F ib H ( X, F ) the set of equivalence classes of H -fibrations over X with fiber F .A trivial inspection shows that this definition is indeed independent of the equivalenceclass of the given fibration and that pullbacks of H -fibrations remain H -fibrations. Also,this definition is independent of the considered weak equivalence ω : F ≃ w → F as two suchmaps produce conjugate equivalences α and H is assumed to be normal. Then, we prove(see Theorem 1.1 for an extended statement) : Theorem 0.2.
For any space X of the homotopy type of a CW-complex there is a naturalbijection F ib H ( X, F ) ∼ = [ X, B aut H F ] . A pointed version of this result is given in Theorem 2.2.
We strongly rely in the classical reference [5] from which we recall some facts. Let G be atopological monoid and let X and Y be right and left G -spaces respectively. The geometric ar construction B ( X, G, Y ) is the geometric realization of the simplicial set whose spaceof j simplices is X × G j × Y and the face and degeneracy operators are given by, d i ( x, g , . . . , g j , y ) = ( x · g , g , . . . , g j , y ) if i = 0 , ( x, g , . . . , g i − , g i · g i +1 , g i +2 , . . . , g j , y ) if 1 ≤ i < j, ( x, g , . . . , g j − , g j · y ) if i = j,s i ( x, g , . . . , g j , y ) =( x, g , . . . , g i , e, g i +1 , . . . , g j , y ) . This is a functor from the category of triples (
X, G, Y ) as above, whose morphisms aretriples ( g, f, h ) where f : G → G ′ is a map of monoids and g : X → X ′ and h : Y → Y ′ are f -equivariant maps. Of particular interest is the map γ : B ( ∗ , G, Y ) → B ( ∗ , G, ∗ ) = BG induced by the trivial G -map Y → ∗ . ConsiderΓ γ : Γ B ( ∗ , G, Y ) −→ BG its associated fibration in which, by definition,Γ B ( ∗ , G, Y ) = { ( z, α ) ∈ B ( ∗ , G, Y ) × BG I , α (0) = γ ( z ) } , Γ γ ( z, α ) = α (1) , where BG I are the paths in BG , and whose fibre F = (Γ γ ) − ( ∗ ) is the homotopy fiber of γ . In [5, §
7] there is an explicit weak homotopy equivalence Y ≃ w → F which exhibits Γ γ asa fibration with fiber Y .In particular, whenever F is of the homotopy type of a CW-complex, the universalfibration Γ γ : Γ B ( ∗ , aut F, F ) −→ B aut F classifies fibrations with fiber F . Explicitly, see [5, Theorem 9.2], for any space X of thehomotopy type of a CW-complex, the map,Λ : [ X, B aut F ] ∼ = −→ F ib( X, F ) , Λ[ f ] = f ∗ Γ γ is a natural bijection. Here f ∗ denotes the pullback over f . Given H E E ( F ) a normalsubgroup, consider the map γ H : B ( ∗ , aut H F, F ) → B ( ∗ , aut H F, ∗ ) = B aut H F. Then, a precise reformulation of Theorem 0.2 reads:
Theorem 1.1.
The map, Λ H : [ X, B aut H F ] ∼ = −→ F ib H ( X, F ) , Λ H [ f ] = f ∗ Γ γ H , s a natural bijection which fits in the following commutative diagram [ X, B aut F ] F ib( X, F )[ X, B aut H F ] F ib H ( X, F ) . ∼ =Λ ∼ =Λ H (1)Here the left vertical map is induced by the inclusion i : aut H F ֒ → aut F . Proof. (i)
For any [ f ] ∈ [ X, B aut H F ], Λ H [ f ] is indeed an H -fibration:This is a consequence of the following general fact. Let Y be a G -space with G agrouplike topological monoid and consider the explicit weak homotopy equivalences Y ≃ w → F and η : G ≃ w → Ω BG of § g ∈ G , as before, let e η g : F ≃ → F denote the induced equivalence on the homotopy fibre. Then, an inspection shows that thefollowing diagram commutes up to homotopy, Y Y
F F . g ≃ w ≃ w e η g That is, with the nomenclature in the Introduction, and whenever Y has the homotopy typeof a CW-complex, Γ γ is a fibration with fibre Y and, for each η g ∈ π ( BG ), η g = g ∈ E ( Y ).In particular, Γ γ H is an H -fibration. Finally, a straightforward argument shows that thepullback of an H -fibration is also an H -fibration. Thus, Λ H [ f ] = f ∗ Γ γ H is an H -fibration. (ii) The diagram (1) commutes:It is enough to show that the fibrations Γ γ H : Γ B ( ∗ , aut H , F )) → B aut H and( Bi ) ∗ Γ γ : ( Bi ) ∗ Γ B ( ∗ , aut F, F ) → B aut H F are equivalent. This is also a consequenceof the following general fact. Let H and G be grouplike topological monoids, let Y beboth a left H -space and a left G -space, and let f : H → G a map of topological monoidmorphism such that g · x = f ( g ) · x for any g ∈ H and any y ∈ Y . Then,Γ B ( ∗ , H, Y ) → Γ B ( ∗ , G, Y ) , ( z, α ) (cid:0) B ( ∗ , f, id Y )( z ) , Bf ◦ α (cid:1) is a fiber map which makes commutative the diagramΓ B ( ∗ , H, Y ) Γ B ( ∗ , G, Y ) BH BG. Γ γ Γ γBf
4n our particular case, this diagram is the outer square of the following diagram where thedashed arrow is the unique map induced by the pullback property and it is necessarily amap of fibrations:Γ B ( ∗ , aut H F, F ) ( Bi ) ∗ Γ B ( ∗ , aut F, F ) Γ B ( ∗ , aut F, F ) B aut H F B aut F. Γ γ H ( Bi ) ∗ Γ γ Γ γBi (iii) The map Λ H is injective:Equivalently, we show that [ X, BH ] → [ A, BG ] is injective. By [5, § Bi : B aut H F → B aut F is equivalent to a quasifibration with fiber B (aut F, aut H F, ∗ ).A long exact sequence argument, using that π n ( Bi ) : π n ( B aut H F ) → π n ( B aut F ) is anisomorphism for n ≥
2, shows that B (aut F, aut H F, ∗ ) is weakly equivalent to the discretespace K = E ( F ) / H .Using the CW approximation theorem, we can find CW-complexes Y, Z and weakhomotopy equivalences κ : Y → B aut H F and λ : X → B aut F such that Y B aut H FZ B aut F κ ≃ w ξ Biλ ≃ w commutes up to homotopy and ξ is a covering map. The associated subgroup of the coveringmap is given by the image of H E E ( F ) under the isomorphisms E ( F ) ∼ = π ( B aut F ) and π ( λ ) : π ( Z ) ∼ = → π ( B aut F ). Therefore we have a commutative diagram[ X, Z ] [
X, B aut F ][ X, Y ] [
X, B aut H F ] λ ∗ ∼ = κ ∗ ∼ = ξ ∗ (2)in which, by the lifting property of covering maps, ξ ∗ : [ A, X ] → [ A, Y ] is injective, so is[
X, B aut H F ] → [ X, B aut F ]. (iv) The map Λ H is surjective: 5iven an H -fibration p : E → X , via Λ − we get a map f : X → B aut F which, withthe notation in (2), produces another map f ′ : X → Z with λf ′ ≃ f . It is then enoughto show that Im π ( f ) ⊂ H . In this case, by the lifting property of covering maps, thereexists ˜ f : X → Y such that ξ ˜ f = f ′ . Therefore, ( Bi ) κ ˜ f ≃ f and Λ H [ κ ˜ f ] = p .To finish, we show that in fact, Im π ( f ) ⊂ H . To give an explicit description of f ,recall from [5, p.49] the existence of a commutative diagram of fibrations with fiber aut FP E B ( P E, aut F, aut F ) E aut FX B ( P E, aut F, ∗ ) B aut F h ϕ q (3)where: • The space
P E is the space of weak homotopy equivalences ψ : F ≃ w → E such that ψ ( F ) ⊂ p − ( x ) for some x ∈ X , and h is defined by h ( ψ ) = x , see [5, Definition 4.3]for more details. • E aut F is the contractible space B ( ∗ , aut F, aut F ) and all the maps in the rightsquare are induced by the functor B . • All the arrows pointing left are weak homotopy equivalences, so, since X is of thehomotopy type of a CW-complex, there exists ϕ , a right homotopy inverse.Then, Λ − [ p ] = [ f ] with f = qϕ . To check that π sends this map into H , we firstchoose a base point B ( P E, aut F, aut F ) which, in turn, determines basepoints in any ofthe spaces in (3). Among the space P E × aut F of 0-simplices of B ( P E, aut F, aut F ) wefix ( ω, id F ) with ω : F ≃ w → F = p − ( x ) a weak homotopy equivalence. With this choice,the fibre of h is the space b F of all weak homotopy equivalences F ≃ w → F .With this choice of base points the long homotopy exact sequences associated to the6ut F -fibrations in (3) yield a commutative diagram,... ... ... π ( P E ) π B ( P E, aut F, aut F ) 0 π ( X ) π B ( P E, aut F, ∗ ) π ( B aut F ) π ( b F ) E ( F ) E ( F )... ... ... π ( h ) δ ∼ = π ( q ) ∼ = ω ∗ id Hence, Im π ( f ) = Im π ( qϕ ) ∈ H if and only if Im w − ∗ δ ∈ H . However, an easy inspectionshows that δ ( α ) = ˆ αω = ωα . That is, ω − ∗ δ ( α ) = α which is in H for any α if, by definition, p is an H -fibration. Remark . As in the ordinary case, we can give a more convenient expression of theuniversal H -fibration. Fix a base point x ∈ F and consider the evaluation fibration aut ∗ H F ֒ → aut H F ev −→ F where aut ∗ H F is the submonoid of aut H F of self homotopy equivalences which fix the basepoint and ev( g ) = g ( x ). This yields a weak homotopy equivalence F ≃ w aut H F/ aut ∗ H F = B (aut H , aut ∗ H , ∗ ) which, in turn, produces equivalent fibrations of fibre F ,Γ B ( ∗ , aut H F, F ) → B aut H F and Γ B ( ∗ , aut H F, aut H F/ aut ∗ H F ) → B aut H F. On the other hand, by [5, Remark 8.9], the quasifibrations B ( ∗ , aut H F, aut H F/ aut ∗ H F ) → B aut H F and B aut ∗ H F → B aut H F are equivalent. That is, the fibration sequences, F → B aut ∗ H F → B aut H F and F → B ( ∗ , aut H F, F ) → B aut H F are equivalent. 7 Based fibrations
A fibration F ω −→ E p −→ X is based if p has a section σ : X → E and the weak equivalence ω sends the basepoint to σ ( x ). Consider the ‘whisker construction’ (see [5, Addenda]), e p : e E −→ X where e E = E ⊔ ( X × [0 , σ ( x ) ∼ ( x, , ∀ x ∈ X , e p ( y ) = p ( y ) , e p ( x, t ) = x. which is a fibration with fiber e π − ( x ) = e F = F ∨ [0 , α ∈ π ( X, x ) the inducedhomotopy equivalence e α : e F ≃ → e F can be taken to send 1 to 1. Since the inclusion of thebasepoint in F is a cofibration, there is a pointed weak homotopy equivalence ˜ ω : F → ˜ F which induces a bijection e ω ∗ : [ F, F ] ∗ ∼ = → [ F, ˜ F ] ∗ between pointed homotopy classes. Thisyields the pointed homotopy equivalence ˙ α = e ω − ∗ [ e αf ] ∗ ∈ [ F, F ] ∗ .Let aut ∗ F be the topological monoid of pointed self homotopy equivalences of F anddenote by E ∗ ( F ) = π (aut ∗ F ) the group of pointed homotopy classes. Fix a normalsubgroup H E E ∗ ( F ) and consider aut ∗ H F ⊂ aut ∗ F the submonoid of pointed homotopyequivalences ϕ such that its (pointed) homotopy class [ ϕ ] ∗ ∈ H . Definition 2.1.
A based fibration with fiber F is an based H -fibration if ˙ α ∈ H for any α ∈ π ( X, x ). Denote by F ib ∗ H ( X, F ) the set of equivalence classes of based H -fibrationsover X with fiber F .Again, pullbacks preserve based H -fibrations and this definition is independent of theequivalence class of the given based fibration. Also, it is independent of the weak equiv-alence ω : F ≃ w → F as two such maps produce conjugate pointed equivalences ˙ α and H isassumed to be normal.Consider the maps γ : B ( ∗ , aut ∗ F, F ) → B ( ∗ , aut ∗ F, ∗ ) = B aut ∗ F and γ H : B ( ∗ , aut ∗ H F, F ) → B ( ∗ , aut ∗ H F, ∗ ) = B aut ∗ H F, both endowed with the section induced by the inclusion of the basepoint in F , and letΛ : [ X, B aut ∗ F ] ∼ = → F ib ∗ ( X, F ) be the natural bijection given in [5, Theorem 9.2 (b)].Then, by requiring all the maps and fibrations involved in the proof of Theorem 1 . Theorem 2.2.
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Departamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga,Campus de Teatinos, s/n, 29071, M´alaga, Spain.
E-mail address : [email protected]@uma.es