H-SPACES OF SELF-EQUIVALENCES OF FIBRATIONS

Let G(p) be the space oí all equlvariant automorphisms oí a principal Gbundle p : E 8 . topologlzed as a subspace oí M(E.E). the space oí maps from E to Itself.Composltion oí automorphisms gives G(p) a group structure and indeed . G(p) is a topological group . The topological group G(p) has been used quite (requently In connectIon wlth certain problems oí Theoretical Physlcs : for example . It appears In the Feynman approach lo Quantum Mechanics as the group of all gauge trans(ormations oí a smooth principal G-bundle p . wlth G a Lie group . In these problems . It is necessary on severa¡ occasions lo know more about the space G(p) or about certain oi Its homotopy groups (sea 181) . Clearly, If p Is a trivial G-bundle over a space B . then G(p) is homeomorphic to the space M(B .G ) . In general . II I EG(p) and x E E. because G acts effectively and transitively on libres there Is a unique gEG such that f (x)=gx . This gives risa lo a homeomorphism 9 from G(p) lo the space MC (E .G) o1 all maps w from E lo G . such that w(gx)=g,p(x)g-1 for all gEG and all x EE . In practica. a diflicult if G Is abellan . 9 : G(p)Q, M(B.G) 161 . Ttiis Is a Is too Iimited . A more general result was obtained BG Is the classifying space for G . k :8 BG Is space lo deal with . Note that better result but . oí course . It

(1111 principal G-bundles .G a topological group : [IV) smooth principal G-bundles .G a Lle group : [VI vector bundles with fibres IsomorphIc to a fixed vector space V : [VI] fibre bundles with fibre F .corresponding to a given effectlve action of a compact topological group G : (VII) principal H-fibrations wlth fibres of the homotopy type of a strlctly associative H-space with strict identity (sea [11 .Ex .3) .
All these categories have In common the fact that each has a Universal Object Theorem 1-If (X,q,A), (Y,r,B) EA then, qlr :XXY -Ax8 Is a Dold fibration .
As  Flnally.we roca¡$ from the Mllnor constructlon of universal bundies that each countable .connected CW-complex X can be viewed as the base space of a universal G-bundle (G Is constructed trom X) : let us take X to be S 4 and let k :S4 " S 4 be a degree k function and let (Ek .pk.S4 ) bé the corresponding for all gEG and all x EE .In practica.a diflicult if G Is abellan .9 : G(p)Q, M(B .G) 161 .Ttiis Is a Is too Iimited .A more general result was obtained BG Is the classifying space for G .k :8 -BG Is space lo deal with .Note that better result but .oí course .It by D .H .Gottlieb In 1972 151 : it the classlfylng map for the principal G-bundle p : E -8 and M(B .BG :k) Is the path-component oí M (B .BG) containing k .then Proposition 1-G(p) d w 11M(8,8G :k) ("w = wesk homotopy equivalence) .As for other types oí fibrations .probably the first result along the lines oí Proposltion I was also obtalned by Gottlleb (4) .To describe It .we must recall the following classification theorem .due to A .Dold :°let EF (B) be the set oí al¡ fibrehomotopy equivalence classes oí Hurewlcz fibrations over a path-connected CWcomplex 8 and with fibres oí the homotopy type oí a fixed space F : then .there Is a CW-complex 8m such that the functors EF and [ ,8.) oí CW Into Sel are naturally equivalenY (here 1X,Y) represents the set oí al¡ homotopy classes oí maps from X Into Y : see 13), Coroilary 16 .9) .Proposition 2-If p : E -8 is a Hurewlcz fibration with libre F, 8 Is a path-connected CW-complex and k : 8 -B® is the classlfying map, the space G(p) of all self-flbre homotopy equivalences of p Is sugh that The purpose oí lhls note Is to report results oí -a joint work -with P .Booth .P .Heath and C .Morgan, concerning the study -In a unified fashion -oí the' homotopy type and certain homotopy groups oí the space G(p) .where p is an object oí en arbltrary category oí fibrations over CW-complexes .Proofs will be given elsewhere.The maln examples oí categories oí fibrations we have In mind are the following (note that all fibrations considered have a path-connected CW-complex as a base space) .[1) Dold fibrations with fibres oí the homotopy type oí a fixed space F (we define a Dold fibration as a fibration satistylng the Wesk Covering Homotopy Property [2l) : 1113 Hurewlcz fibrations with fibres oí the homotopy type oí a fixed space F : n g (G(p)) L , n 0 (nm(B .B m :k)) .

(
E. .pm.B .)from whlch one deduces a Classification Theorem ' of Dold's type : furthermore .In each one of these examples .(E p .pm .B.) also satisfles another type of universaiity which we shall describe ¡atar on and whlch plays a crucial role In our consideratlons .In order to unify these Ideas we begin by taking a category F wlth a distingulshed object F and a falthful underlylng space functor F -K. where K Is the convenient category of k-spaces, that Is to say .K is the Image of Top under the functor k : Top -Top -callad the k-lfication functorobtalned as a left Kan-extenslon of the Imbeddlng C -Top over Itself, where C Is the category of all compact Hausdorfi spaces .lt Is also assumed that for any two objects X .Y E F .F(X;Y) Is non empty .We then define an F-space as a triple (E .p.8)such that 8 Is a CW-complex .E E K .p : E -B Is a map In K and finally .for every b E 8 .Eb =p-1 (b) E F .An t 1 :E .-E' .f 0 :8 -B' fibre Eb Is a morphlsm to be en F-map over pH =h q x l .If A = 8 and h is homotopy over 8 .An F-map g :X there exists en F-map g' : E -X F-map (I 1 ,t 0 ) :(E,p,B) -(E' .p',8') is given by two maps such that p' f 1 = f 0 p and the restriction of t projectlon map .we have the notlon of F-E over B Is en F -homotopy equlvalence If over B such that gg' and g'g are F-homotopic over B to the respective Identity maps .We now once more restrlct the category F by requirlng that every morphlsm of F Is an F-homotopy equivalence over a point .We are now " prepared lo define formally what we Intend for a category of fibrations relatively lo a category F. Definitlon -A category of 1lbrations Is a non-empty .fui¡ subcategory A of the category of F-spaces and F-maps such that : 111 (F,c,)K)EA .where X is a singleton space and c Is the constant map : [21 If (E,p,f3)EA,AECW and f :A -B Is a map .the pullback (t )K (E), p f .A)EA [31 A Is closed under F-Isomorphisms over a fixed base space : 141 if (E,p,B) EA, there Is a numerable open covering (U) of B such that.for every U E (U) p : p -t (U) -U is F-homotopy equivalent to pr : U xF " U .As examples of categories of tibrations we quote the categorles (I) to [VII] described earlier ; we Iimlt ourselves lo define the category F In each case .For the examples numbered f_I1_ and 1111 .F is the category of all spaces Of the same homotopy type as F and al¡ homotopy equivalences between these spaces .For [1111 and [IV] .F is the category whose objects are right G-spaces Y such that .for all y E Y .the functlon y : G " Y defined by y (g)=yg Is a homeomorphism : Its morphlsms are G-maps .For [VI .F consists of all vector spaces Isomorphic lo V and al¡ Isomorphisms between such vector spaces .For [VI] .we first assume that G acts effectively on the left of F .then, we define F by taklng for Its objects al¡ pairs (X,1V) such that X Is a left G-space and tt :F -X is a homeomorphism of left G-spaces: the set of morphlsms from (X .V) lo CX' .Y') Is given by F((X .y).(X'.~')=(>r't71r 1 1 0 EG) wlth the obvlous operation of compositlon .Finally, for [VIII .F Is similar to that of [1111 (DI .Example 3) .We continua as In 111 by defining .for two arbitrary F-spaces (X .qA)and (Y .r,B), che tunctional space and the function XXY = u F(Xa .Yb ) & EA b EB qXr :X31(Y -A xB .g3Kr(1:Xa -Yb ) = (a .b).The topology of XXY Is given as follows .Let Y+ = YU(m) be che k-Ificatlon of the topology deflned by requiring that C Is closed in Y+ lf elther C=V;* or lf C is closed In Y .Now define che function 1 :XXY -M (X .Y+ ) by J«)(,)= f (x) lf xE Xa , t :Xa -Yb and 1(1)(x)=°° otherwise .(M (X,Y+ ) Is endowed with the compact open topology) .Then we give XXY the k-Ification of che Inicial topology wlth respect to / and q*r .In general .(X)KY .gXr.AXB) Is not en F-space : however .che following holds .

For a given
object (E .p.8)ot the category of fibrations A let G(p) be the space of all F-homotopy equivalences of p Into itselt over B, topologlzed as a subspace of M (E,E) : notica that the composition gives to t3(p) a continuous product under which space with a strlct two-sided unit defined by the of F-maps of p Into p over B O(p) becomes an associative H-Identlty morphlsm of p Into Itself ~over 8 .Theorem 2-Let A be a category of fibratlons wlth a weakly contractlble oblect (m .pm.Em ) and Jet (E,p,B) be en arbitrary element of A: suppose that k :B -B m is a clessifying map tor (E,p,8) .Then, there exists en H-map 6 :f1M (8 .8m :k) -G(p) which is a weak homotopy equlvalence .Observe that the Dold fibration FIEm -Bm has flbre FAF and so, if F0F has the homotopy type of a CW-complex, FUE,, Is contractlble : this .In turn .wlll Imply that the H-map S of Theorem 2 Is a homotopy equivalence .This Is precisely the situation of Example UVI .since GAG is homeomorphlc to G .From now on, we shall assume for technlcal reasons that (E .p.B) Is an oblect of the category of fibratlons A which satisfies a strenghtened .verslon of axiom [4I In the definition of a category of fibratlons .implylng that if (X .q.A) and (Y .r.8)are -objects of A then .(XXEY .q#rAXB)Is a Hurewlcz fibratlon : furthermore.we suppose that A has a weakly contractlble universal oblect (E. ..pm .B.) .Thls is the case of examples [II) and [VII .Let F be the flbre of (E .p.8)over a point Z(E B and define G 1 (p) to be the subspace of G(p) ot al¡ F-homotopy equivalences of p over itself over 8 which extend the Identity map 1 F : F --F .The space (G 1 (p) has proved Itself very useful In certain problems of Mathematlcal Physlcs . 1 F where (E .p.B) Is en oblect of the category [IV) (see [81) .* We wlsh to observe that the relation between G 1 (p) and G(p) Is deeper than just the relation 'subspace-space °: In fact .Theorem 9-There is a Hurewlcz fibratlon G(p) " FAF wlth Libre 0 1 (p) over A result similar to Theorem 2 holds for ® 1 (p) : In what tollows M # (8 .8® :k) denotes the space ot al¡ based maps trom 8 to 8, . .Theorem 4-There is an H-map whlch Is a weak homotopy equlvalence (or a strong homotopy equivalence ft FXE m is contrectlble) .Next. consider the Hurewlcz tibration FX m -B m (with fibre FXF ovar b =k (X) c B -) and Its long homotopy sequence because FXE. i s weakly contractable .S : f)MX (B .B. :k) -O1 (P) ----MFXEW )-fTBw -F)KF -FXE -B.: S :()B m -FXF is a weak homotopy equlvalence (strong homotopy equlvalence If F*E .Is contractible) .This tact Is used to prove the tollowing .Theorem 5-Suppose that all path-components of M(B .B .)(resp .MX (B .B,)) have the same homotopy (ype .Then G(p)w M(B .FXF) (resp .d 1 (P) d w M X (B .FXF)) (strong homotopy equlvalence If F31EEM is contractlble) ; turthermore, these weak (strong) equivalences preserve the H-space structures .In connection to the previous theorem the reader should recall that If 8 Is an H-cogroup (e .g ..8 is a suspension space) then the hypothesis of Theorem 5 hold for M1 (B .B.) and if Bm Is an H-group (e .g ..B .= BU .BO .BSp) then these hypothesis hold for both M(B .B.) and Mi, < 2n .then for 0 < i < 2n -m tr i (O(p)) a n i (M(B .FXF :c)) ni ((3 1 (p)) a tr i (M X (B .F)IEF :c)) whero c : B -FXF Is the constent map to 1 f .We complete these notes wlth a few computatlons .Ii (E .p.8)Is a smooth principal Sp (1)-bundle and B Is a manifold ot dimension m < 5 .slnce Sp (1) g Ss and Sp (1)XSp (1) ºF Sp (1) .theorem 6 shows that If 0 < / < 6-m .n l (G(p)) L, n¡(M(B .Sp(1)) and particular .B=S n and n >O .then principal G-bundle .Then .n ¡ (G 1 (A )) Ri n~(M X (B .Sp (1)) .If p Is a smooth principal G-bundle over a sphére S n , n >0 .then G 1 (p) e MX (Sn .G) and thus the homotopy groups of G i (p) are totally determlned by the homotopy groups of G .sInce .for every ¡>O .n¡(G1(p))q! n/+n(G) .If p is a smooth principal U-bundle over a manifold B . then G(p)-M(B,U) and G 1 (P) -m X (8 .U) : ¡f.In 0 .11¡=oven .n=oven Z .lt / =oven .n=odd n ¡ ( G 1 (p» L' tt¡+n(U)L, Z .11y =odd .n=oven or 0 .11¡=odd .n=odd On the other hand .Theorem 2 .2 of (7) shows that M (S n .U) E U XM X (Sn .U) and so .0 .I(/=oven .n=ovenZ .If/=oven .n=oddn / (G(p)) a IZ 8Z , If /=odd .n=oven or Z, If / =odd .n=odd