Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups

Journal of Homotopy and R elate d Structu r es, vol. 1(1), 2006, pp.1–1 2 EXA CT SEQUENCES OF FIBRA TIONS O F CROSSED COMPLEXES, HOMOTOPY CLASSIFICA TION OF MAPS, AND NO NABELIAN EXTENSI ONS OF GR OUPS R ONAL D BRO WN Abstr a ct The class ifying space of a crossed complex generalis es the construction of Eile n b erg-Ma c Lane spaces. W e show ho w the theory o f fibra tions of cr ossed co mplexes allows the analysis of homotopy class es of maps from a fr ee cros s ed complex to such a cla s sifying space. This gives r esults on the homotopy classification of maps from a C W -complex to the classifying space of a crossed module and also , m or e generally , of a crossed complex whose homotop y groups v anis h in dimensions betw een 1 and n . T he r e sults a re ana logous to those for the obstr uctio n to an abstract k ernel in group extension theory . In tro duction A ma jor result in the theory of cros s ed complexes is the homotopy classification theorem of [ BH91 ]: [Π X ∗ , C ] ∼ = [ X , B C ] (1) where: • on the left, Π X ∗ is the fundamental crossed complex of the skeletal filtration of the CW-complex X , and square brack ets denote homotopy classe s in the category Crs of crossed complexes, and • on the right, B C is the class ifying space o f the cross ed complex C , and squa re brack ets denote homotopy classes of maps of topolo gical spaces. Results of Eilenberg –Mac Lane, including the case when C is just a gr oup, and also the case of a loca l co efficient system, are o btained by giving sp ecial cases for the crossed co mplex C . A new a pplication is to the ca se C is a cro ssed mo dule, and this is exploited in, for example, [ FK08, F ar08, PT07 ]. The purp ose of this pap er is to s how how this ho motopy classification can b e analysed using the no tion of exact sequence of a fibratio n of crossed complexes. In this w ay we show how cros sed complexes give a conv enien t co mmon setting for the homotopy cla ssification of maps to an n -as pherical space Y ( i.e. one in which Y is . 2000 M athematics Sub ject Classification: 13D02,18G50,20J05 ,55S37,55S45 Key w ords and phrases: homotopy classification, crossed complexes, fibrations, exact sequences, nonabelian extensions c  ????, Ronald Brown. Pe rmi ssion to copy for priv ate use grante d. Journal of Homotopy and R elate d Structu r es, vol. 1(1), 2006 2 path connected and π i Y = 0 fo r 1 < i < n ) as origina lly found by Olum in [ Olu5 3 ], and for the classical theo r y of nonab elian extensions of gr o ups and abstract kernels, [ ML63 ]. A topolo gical account of this extension theory is given in [ Ber85 ]. An example of the convenience of the se tting of crossed complexes in these a reas is that the traditional Sc hreier notion o f fa c to r set for an extension 1 → K → E → G → 1 of gr oups is conv eniently formulated a s a morphism of crossed complexes F st ( G ) → AUT( K ) from the sta ndard free crossed resolution F st ( G ) o f G to the crossed mo dule AUT( K ) := ( K → Aut( K )), and equiv alence of factor sets is just homotopy of mo rphisms, [ BH82 ]. By replacing F st ( G ) b y an equiv alent free crossed resolution, we can g et calculatio ns, as exploited in [ BP96 ]. That exp osition also shows the use of extensions of the type of a crossed module, due orig inally to Dedeck er, [ Ded58 ]. This relates to w ork of [ BM94 ]. Our tec hniques in volve the clos ed monoida l structur e on the categor y of crossed complexes from [ BH87 ], the ex a ct s e quences of a fibration of cro ssed co mplex es established by Howie in [ How79 ], and the mo del ca tegory prop erties of cros sed complexes from [ BG89, BH91 ]. F or more on model ca tegories, s ee [ Ho v99 ]. This pap er should a ls o b e seen as a mo dern development of pioneering work of J.H.C. Whitehead in [ Whi 49 ], which in so me ways go es further than the work of Olum publis he d later . Whitehead uses the term ‘homo to p y system’ for wha t we call ‘reduced, free cross ed co mplex’. These ideas are ex plo ited by Ellis in [ El l88 ]. Although Whitehead do es not hav e the notion of classifying s pace, he gives his view that, in our terminology , crossed complexes hav e be tter realisation proper ties than chain complexes with a group of op e rators. W e are thus co nfirming tha t crossed co mplexes, though only a linear mo del of homotopy types , are conv enient in that b order b etw een homolo gy and homotopy which includes the Relative Hurew ic z Theor em, [ BH81 ], the Ho mo topy Addition Lemma [ BS07 ], co mputation with cross e d mo dules [ BW9 5 ], and this level of the homotopy classifica tion of maps . The referee is thank ed for some helpful commen ts. 1. Crossed complexes A crossed complex C is in part a seq uence of the form · · · / / C n δ n / / C n − 1 δ n − 1 / / · · · · · · δ 3 / / C 2 δ 2 / / C 1 where a ll the C n , n > 1 are gro upo ids ov er C 0 . The str ucture and axioms fo r a crossed co mplex are thos e universally sa tisfied by the main top ological example, the fun damental cr osse d c omplex Π X ∗ of a filtered space X ∗ , wher e (Π X ∗ ) 1 is the fundamen tal gro up oid π 1 ( X 1 , X 0 ) and (Π X ∗ ) n is the family o f rela tive homotopy groups π n ( X n , X n − 1 , x 0 ) for all x 0 ∈ X 0 . The fundamen tal g r oup oid of the crossed complex C is π 1 C = Co k er δ 2 and the homology groups of C are for n > 2 the families of ab elian groups H n ( C, x ) = (K er δ n ( C n ( x ) → C n − 1 ( x )) / ( δ n +1 C n +1 ( x )) x ∈ C 0 . Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 3 The cr ossed complex C is aspheric al if it is connected (i.e. the gr oup oid C 1 is connected (also called transitive)) and H n ( C, x ) = 0 for all n > 2 , x ∈ C 0 . W e assume the bas ic facts on cros sed complexes a s surveyed in fo r ex ample [ Bro99, Bro0 4 ]. In par ticular, we will use the monoidal clo sed structure o n the category Crs which gives an exp onential la w of the form Crs ( A ⊗ B , C ) ∼ = Crs ( A, CR S ( B , C )) , for crossed complexes A, B , C , a nd also in the p ointed form Crs ∗ ( A ⊗ ∗ B , C ) ∼ = Crs ∗ ( A, CR S ∗ ( B , C )) for p ointed cro ssed complexe s A, B , C ; b oth of these structures are develope d in [ BH87 ]. Here the elements of CRS ( B , C ) 0 are morphis ms B → C ; the elements of the g roup oid CRS ( B , C ) 1 are (left) homotopies be tween these morphisms; and the further elements of CRS ( B , C ) ar e forms of higher homotopies. A full expos ition of the theory of cr ossed complexes will be given in [ BHS08 ]. 2. Fibrations of crossed complexes A model categor y structure o n the catego ry Crs has been studied b y Brown and Golasi ´ nski in [ B G8 9 ], ex plo iting the notion o f fibration o f cr o ssed complexes as defined in [ How79 ]. W e recall some of this ma terial, but with a slightly different emphasis. Recall that a morphism p : E → B of group oids is a fibr ation (c overing morphism) if it is star surjective (bijective) [ Bro06 ]. The extension to cros sed complexes is quite simple (cov ering mo rphisms of crossed complexes are applied in [ BRS99 ]). Definition 2.1. A morphism p : E → B of crossed complexes is a fibr ation if (i) the morphism p 1 : E 1 → B 1 is a fibr ation of group oids; (ii) for each n > 2 a nd x ∈ E 0 , the mor phism o f groups p n : E n ( x ) → B n ( px ) is surjective. The morphism p is a trivial fibr ation if it is a fibration, and also a weak eq uiv alence, by which is mean t that p induces a bijection o n π 0 and isomo r phisms π 1 ( E , x ) → π 1 ( B , px ), H n ( E , x ) → H n ( B , px ) for all x ∈ E 0 and n > 2 . ✷ W e now follow model catego ry ideas a s in [ BG89 ]. Definition 2.2. Conside r the following dia gram. A / / i   E p   C / / > > B . If given i the dotted completion exists for all mor phisms p in a class F , then we say that i has the left lifting pr op erty (LLP) with respec t to F . W e say a morphis m i : A → C is a c ofibr ation if it has the LLP with resp ect to all trivial fibrations. W e say a cro s sed co mplex C is c ofibr ant if the inclusion ∅ → C is a cofibration. ✷ Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 4 T o give our most imp orta n t example imp ortant example of a cofibr ation, we need some definitions. Definition 2.3 . W e write C ( n ) for the free cros s ed co mplex on o ne genera tor c n of dimension n , and S n − 1 for the sub complex o f C ( n ) gener ated b y the elements of dimension ( n − 1). Th us C (1) is essentially a group oid, often wr itten I , which, with the inclusio ns 0 , 1 → C (1), is a unit interv al ob ject in the catego r y o f crosse d complexes, [ KP97 ]; C ( n ) is a mo del of the n - disc, a nd S n − 1 is a mo del of the ( n − 1)- s phere. Definition 2.4. Let A b e a crossed c o mplex. A morphism i : A → F o f crossed complexes is said to b e r elatively fr e e if i is the canonical morphism when F is obtained by attaching in orde r of increa sing dimension co pies of C ( n ) by means of morphisms S ( n − 1 ), analog ously to the corre spo nding notion for relative C W - complexes. The images of the elements c n are called ba sis elements of F . In the case A is empt y , then F is called a fr e e cr osse d c omplex . The following is [ BG89 , Cor olllary 2.4]. W e give the pro of for the conv enience of the reader. Prop osition 2 .5. Le t i : A → F b e a r elatively fr e e morphism of cr osse d c omplexes. Then i is a c ofibr ation. Pr o of. W e co nsider the following diagram A i   α / / E p   F f / / g ? ? B in which p is supp o sed a trivial fibra tion, and the morphisms f , α satisfy f i = pα . W e construct the regula r completio n g on a relatively fr ee basis X of F by induction. F or n = 0, we just lift a p oint in B to a p oint in the cor resp onding comp onent in E . This defines g 0 on A ∪ X 0 . F or the case n = 1 , consider a bas is element x ∈ X 1 ( a, b ), so that f ( x ) ∈ B ( f a, f b ), and g 0 a, g 0 b b elong to the sa me co mponent of E , by the conditio n o n π 0 . So there is an element e ∈ E ( g 0 a, g 0 b ). Hence f x − pe is a lo op in B 1 ( g 0 a ). By the co ndition fo r p on π 1 , ther e is a lo op e 1 ∈ E 1 ( g 0 a ) s uc h that pe 1 is equiv alent to f x − pe , i.e. pe 1 = f x − pe + δ 2 b 2 for some b 2 ∈ B 2 ( f a ). By the fibration condition, b 2 = pe 2 for so me e 2 ∈ E 2 ( g 0 a ). Then p ( e 1 + δ 2 e 2 + e ) = f x . So we can choose g 1 x = e 1 + δ 2 e 2 + e to obtain an extension on x . Suppo se n > 2 and g is defined on X n − 1 . Consider an element x of the free basis in dimension n . Then g δ x is defined and pg δ x = f δ x . By the fibration condition, we ca n choo se e n ∈ E n such that p e n = f x . Let w = g δ x − δ e n ∈ E n − 1 . Then pw = 0 , δ w = 0. B y the triviality co nditio n, w is a bo undary , i.e. w = δ z for some z ∈ E n . Then δ ( z + e n ) = g δ x . So we can extend g by defining it on x to be z + e n . Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 5 The follo wing is [ BG89 , Prop os itio n 2.1.] with a different pro of. Prop osition 2. 6. The fol lowing ar e e quivalent for a morphism p : E → B in Crs : (i) p is a trivial fibr ation: (ii) p 0 is su rje ctive; if e, e ′ ∈ E 0 and b 1 ∈ B 1 ( p 0 e, p 0 e ′ ) , then t her e is e 1 ∈ E 1 ( e, e ′ ) such t hat p 1 e 1 = b 1 ; if n > 1 and e ∈ E n satisfies δ 0 e = δ 1 e for n = 1 , δ e = 0 for n > 2 , and b ∈ B n + l satisfies δ b = p n e , t hen ther e is e ′ ∈ E n +1 such t hat p n +1 e ′ = b and δ e ′ = e ; (iii) p has the RL P with r esp e ct to S ( n − 1) → C ( n ) for al l n > 0 ; (iv) if F is a fr e e cr osse d c omplex then p has the RLP with r esp e ct t o S ( n − 1) ⊗ F → C ( n ) ⊗ F for al l n > 0 ; (v) if F is a fr e e cr osse d c omplex then the induc e d morphism p ∗ : CR S ( F, E ) → CRS ( F, B ) is a trivial fi br ation. Pr o of. The pro of that (i), (ii), (iii) ar e equiv alent follows easily from the definition and Prop osition 2.5. F urther (iii) implies (iv) since we know that under the co ndition of (iv), S ( n − 1) ⊗ F → C ( n ) ⊗ F is relatively free [ BH91 , Pro pos ition 5.1 ]. Finally (iv) triv ially implies (iii). W e also need a pointed version of (iv) of the previous prop osition. Prop osition 2.7. If F , E , B ar e p ointe d r e duc e d cr osse d c omplexes with F fr e e, and p : E → B is a trivial fi br ation, then so also is p ∗ : CRS ∗ ( F, E ) → CRS ∗ ( F, B ) . Pr o of. This relies on the p ointed exp onential la w, [ BH87 ], and the clear fact that S n − 1 ⊗ ∗ F → C ( n ) ⊗ ∗ F is relativ ely free, which follows from metho ds analogous to those of [ BH91 ]. Definition 2.8. If G is a gro upo id, w e write K ( G, 1) for the crosse d complex which is G in dimension 1 and triv ia l elsewhere. Th us K ( G, 1) is certainly aspherical. ✷ Prop osition 2.9. L et F , C b e r e duc e d cr osse d c omplexes with F fr e e and C aspher- ic al. L et G = π 1 ( C ) . Then ther e ar e bije ctions π 0 CRS ∗ ( F, C ) = [ F , C ] ∗ ∼ = [ F, K ( G, 1)] ∗ ∼ = Hom( π 1 F, π 1 C ) , and for al l f : F → C and n > 1 we have π n ( CRS ∗ ( F, C ) , f ) = 0 . Pr o of. Let G = π 1 C . Since C is a spherical, the natural morphism p : C → K ( G, 1) is not only a fibration but a lso a weak equiv alence of cro ssed complexes. It follows that p ∗ : CRS ∗ ( F, C ) → CRS ∗ ( F, K ( G, 1)) is a trivial fibra tio n and so a weak equiv- alence. In particular, p ∗ induces a bijection of π 0 . This gives the firs t r e s ult, since π 0 CRS ∗ ( F, K ( G, 1)) is clea rly bijectiv e with Hom( π 1 F, G ). The second r esult follows, s ince all homotopies a nd higher homotopies F → K ( G, 1) are trivial. Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 6 Remark 2.10. The class ical homologic al algebra t yp e o f inductiv e pro of is so me- how hidden in this pro of. ✷ Howie states in [ How79 ] that a fibration p : E → B of crossed co mplexes y ields a family of exact seq uences involving the H n , π 1 and π 0 , akin to the well known family of ho motopy exact sequences of a fibration of spaces, or of group oids; for the latter see [ Bro06 , 7.2.9]. The standard proper ties of these sequences are: • depe ndency on base p o in ts; • non ab elian features in dimension 1; • sets with base p oint in dimension 0 but with some useful information obtain- able from op erations. Let x ∈ E 0 and let F x = p − 1 ( px ) b e the sub cr ossed complex of E consis ting of all elements of E 0 which map by p to x and o f all element s of some E n , n > 1 , which map by p to the identit y at px . Here is Ho wie’s exact sequence. Theorem 2.1 1. Ther e is an exact s e quenc e · · · → H n ( F x , x ) i n − → H n ( E , x ) p n − → H n ( B , px ) ∂ n − → · · · · · · → π 1 ( F x , x ) i 1 − → π 1 ( E , x ) p 1 − → π 1 ( B , px ) ∂ 1 − → π 0 ( F x ) i ∗ − → π 0 ( E ) p ∗ − → π 0 ( B ) . Her e t he terms of the se qu enc e ar e al l gr oups, exc ept t he last thr e e which ar e sets with b ase p oints the classes x F , x E , x B of x, x, px r esp e ctively. (i) Ther e is an op era tion of the gr oup π 1 ( E , x ) on the gr oup π 1 ( F x , x ) making the morphism i 1 : π 1 ( F x , x ) → π 1 ( E , x ) into a cr osse d mo dule. (ii) Ther e is an op er ation of the gr oup π 1 ( B , px ) on the set π 0 ( F x ) such that t he b oundary π 1 ( B , px ) ∂ 1 − → π 0 ( F x ) is given by ∂ 1 ( α ) = α · x F . F u rther we have additional exactness at the b ottom end as fol lows: (a) ∂ 1 α = ∂ 1 β if and only if ther e is a γ ∈ E ( x ) such that p 1 γ = − β + α ; (b) if ¯ u denotes the c omp onent in F x of an obje ct u of F x , then i ∗ ¯ u = i ∗ ¯ v if and only if ther e is an α ∈ B ( y ) such t hat α # ¯ u = ¯ v ; (c) if ˆ y denotes the c omp onent of y in B then i ∗ [ π 0 F x ] = p − 1 ∗ [ ˆ y ] . Pr o of. The pro of of this theorem is a developmen t of the pa r t of the theor em which deals with fibrations of g roup oids and w hich is given for example by Brown in [ Bro06 ]. W e leav e the details as an ex ercise. Remark 2.12. It may be useful to p oint out tha t the exact sequence of a fibr a tion of gr o upo ids is gener alised to a May er-Vietoris type sequenc e for a pullback of a fibration in [ BHK83 ]. See also [ Bro06 , 10.7.6]. Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 7 3. Homotop y classification of maps t o an n -aspherical space W e hav e alr eady analysed a simple cas e of [ F , C ] ∗ in Prop osition 2.9. W e now pro ceed to some slightly more co mplicated examples, us ing the exact sequences o f a fibration. W e also concentrate on the p ointed homotopy cla ssification, since this is nearest to clas s ical results on gr oup extensions, but the unp ointed case is handled similarly . Definition 3.1. A s pace X is called n -aspheric al if it is path-connected and π i ( X ) = 0 for 1 < i < n . Thus any path-c o nnected spa ce is 2-aspher ic al. Analog ous terms are applied also to crossed complexes. Definition 3.2. F or a group or group oid Q , Q -mo dule A , and n > 2 , let K ( Q, 1; A, n ) denote the crossed complex whic h is Q in dimension 1 , A in dimension n , with the given actio n o f Q , and a ll bounda ries are trivial. ✷ Remark 3.3. Let F b e a cross ed complex . W e will usua lly wr ite φ : F 1 → π 1 F for the quotient morphism. Then for a n y morphism θ : π 1 F → Q the co mpo s ite θφ is co mpletely determined b y θ . How ever so metimes w e start with a gro up (or group oid) Φ and c ho ose what we hav e called a fr e e cr osse d r esolution F of Φ. T his is an as pherical free cro ssed complex together with a choice of isomo r phism π 1 F → Φ, or, equiv alently , with a quotient morphism φ : F 1 → Φ with kernel the image of δ 2 : F 2 → F 1 . In suc h case φ is not determined b y F . Prop osition 3. 4. L et p : E → B b e a morphism of cr osse d c omplexes. Then p is a fi br ation if and only if for any fr e e cr osse d c omplex F , the induc e d m orphism p ∗ : CR S ( F, E ) → CRS ( F, B ) is a fibr ation. If further p and F ar e p ointe d, then the induc e d morphism of p ointe d internal homs p ∗ : CRS ∗ ( F, E ) → CRS ∗ ( F, B ) is a fibr ation of cr osse d c omplexes. Pr o of. The forward implication follows e a sily from the exp onential law, as in the pro of of Prop osition 2.6. T o pro ve it in the other direction, one takes again F to b e C ( n ), the free crossed complex on one generator of dimension n . Definition 3.5. L et F , C b e cr osse d c omplexes, let A b e a su b c omplex of F , with inclusion i : A → F and let f : A → C b e a morphism. We write [ F , C ; f ] for the set of homotopy classes r el A of morphisms F → C which extend f . We write similarly [ F, C ; f ] ∗ for the p ointe d homotopy classes in the c ase A, F , C , i ar e p ointe d. Theorem 3.6. L et F b e a r e duc e d fr e e cr osse d c omplex and let Φ = π 1 F . Then [ F, K ( Q, 1; A, n )] ∗ is the disjoint union of sets [ F , K ( Q, 1; A, n ) : θ φ ] ∗ one for e ach morphism θ : Φ → Q , n amely those homotopy classes inducing θ . F urther, t he morphisms F → K ( Q, 1; A, n ) inducing θ : Φ → Q may b e given the struct ur e of ab elian gr oup which is inherite d by homotopy classes. Pr o of. The morphism q : K ( Q, 1; A, n ) → K ( Q, 1) which is the identit y in dimension 1 and 0 elsewhere is a fibration inducing a fibration q ∗ : CRS ∗ ( F, K ( Q , 1 ; A, n )) → CRS ∗ ( F, K ( Q , 1 )) . Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 8 The induced map on π 0 is surjective since every mo rphism f : F → K ( Q, 1) may b e lifted by 0 to a morphism F → K ( Q , 1 ; A, n ). By Prop osition 2.9, π 0 CRS ∗ ( F, K ( Q , 1 )) ∼ = Hom(Φ , Q ). So w e can write, using the exact sequence of Theorem 2.11, π 0 CRS ∗ ( F, K ( Q , 1 ; A, n )) ∼ = G θ :Φ → Q [ F, K ( Q, 1; A, n ); θφ ] ∗ . The ab elian group structur e o n each set [ F , K ( Q, 1; A, n ); θ φ ] ∗ by addition of v a lues in dimension n is clear from the dia gram · · · F n +1 / /   F n / /   F n − 1 / /   · · · / / F 2 / /   F 1 φ / /   Φ   θ          · · · 0 / / A / / 0 / / · · · / / 0 / / Q Q Definition 3. 7 . W e write H n θ φ ( F, A ) for [ F, K ( Q, 1; A, n ); θ φ ], and call this a belia n group the n th c ohomolo gy over θ φ of F with co efficients in A . Thus [ F , K ( Q, 1; A, n )] ∗ is the disjoint union o f the ab elian groups H n θ φ ( F, A ) for all morphisms θ : Φ → Q . When conv enient and clear, we abbrevia te θ φ to θ . A generalisa tion of the previous example is as follows. Example 3.8. L et C b e a r e duc e d cr osse d c omplex such that C 1 = Q , and δ 2 = 0 : C 2 → C 1 . L et F b e a fr e e cr osse d c omplex. Then Crs ∗ ( F, C ) and [ F , C ] θ φ may b e given the struct ur e of ab elian gr oup by addition of values. ✷ W e now o btain fo r homotopy class ification of maps a result which is a nalogous to and in fact directly generalis e s the classica l theory of abstra c t kernels and ob- structions, [ ML63 , Ch.IV, Thm.8.7]. First w e give a definition. Definition 3.9. Le t F b e a fr ee r educed cros s ed complex, and let φ : F 1 → G = π 1 ( F ) b e the ca nonical morphism. If θ : G → Q is a morphism o f groups, and A is a Q -mo dule, and n > 2, we define the n th c ohomolo gy of F with c o efficients in A with r esp e ct t o θ φ to be the ab e lian group H n θ φ ( F, A ) = [ F , K ( Q, 1 ; A, n ) : θ φ ] ∗ . Theorem 3.10. L et n > 2 and let F, C b e r e duc e d cr osse d c omplexes such that F is fr e e, C is n -aspheric al, and C i = 0 for i > n . L et Φ = π 1 ( F ) , Q = π 1 C, A = Ker δ n : C n → C n − 1 . L et θ : Φ → Q b e a morphism of gr oups. Then ther e is define d an element k θ ∈ H n +1 θ φ ( F, A ) , c al le d t he obstr uc tio n cla ss of θ , su ch that the vanishing of k θ is ne c essary and sufficient for θ to b e r e alise d by a morphism F → C . If k θ = 0 , then the set [ F , C ; θ φ ] of homotopy classes of morphisms F → C r e alising θ φ is bije ct ive with H n θ φ ( F, A ) . Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 9 Pr o of. Consider the morphisms of cros sed c o mplexes C j − → ξ C p − → ζ C as shown in the following diagra m: · · · / / 0 / /   C n δ n / /   C n − 1 / /   · · · / / C 2   / / C 1   C j   · · · / / A / /   C n δ n / /   C n − 1   / / · · · / / C 2 / /   C 1   ξ C p   · · · / / A / / 0 / / 0 / / · · · / / 0 / / Q ζ C Then ξ C is a spherical, ζ C = K ( Q, 1; A, n + 1), and p : ξ C → ζ C is a fibratio n of crossed complexes. Since F is a free reduced cross e d co mplex, we hav e a n induced fibration of cr ossed complexes p ∗ : CRS ∗ ( F, ξ C ) → CRS ∗ ( F, ζ C ) . (2) On applying π 0 to this w e get, considering previous iden tifications, a map of sets p ∗ : Hom(Φ , Q ) − → G θ ∈ Hom(Φ ,Q ) H n +1 θ φ ( F, A ) . (3) Lemma 3.11. A gr oup morphism θ : Φ → Q maps to 0 in H n +1 θ φ ( F, A ) if and only if θ is induc e d by a morphism F → C . Pr o of. Supp ose θ is induced by a morphism f : F → C . Then f factor s through p j and is therefore 0 in H n +1 θ φ ( F, A ). Suppo se conv ersely that θ determines 0 in H n +1 θ φ ( F, A ). W e know that θ is induced by a morphism f ′ : F → ξ C . Then pf is homotopic to 0 and so b y the fibra tio n condition f ′ is ho motopic to f ′′ such that pf ′′ = 0. Hence f ′′ determines f : F → C such that j f = f ′′ . Then f also induces θ . This prov es the lemma. Let F ( f ) denote the fibr e of p ∗ ov er pf . Then we hav e an exact sequence → π 1 ( CRS ∗ ( F, ξ C ) , f ) → π 1 ( CRS ∗ ( F, ζ C ) , pf ) → π 0 F ( f ) → π 0 CRS ∗ ( F, ξ C ) → π 0 CRS ∗ ( F, ζ C ) . By Prop ositio n 2.9, π 1 ( CRS ∗ ( F, ξ C ) , f ) = 0 , a nd so the ab ov e sequence tra nslates to 0 → H n θ φ ( F, A ) → [ F, C ; θφ ] → Hom(Φ , Q ) . F urther we have a free action of the ab elian gro up H n θ φ ( F, A ) o n the set [ F , C ] θ φ . This completes the pro of of the theor em. This result ge ne r alises the cla ssical theory of extensions of g roups and Q -kernels. T o a pply the theo ry to that case, the cros sed co mplex F is taken to be a free cross ed resolution o f the group G . If F is the standard free cr ossed reso lution of G , then the relation w ith facto r systems is s hown in [ BP96 ]. The adv antage of this a pproach Journal of Homotopy and R elate d Struct ur es, vol. 1(1), 2006 10 is that it is clear that the standard free cros s ed res olution ma y b e replaced by an y free crossed r esolution of G , a nd in many cases it is p oss ible to co nstruct small such r esolutions; in thes e c a ses, for example if G is F P 3 , we may obtain a finite description of the classes of extensions. 4. Applications to spaces F or the applications to s paces we need to kno w when a space is o f the homoto p y t yp e o r homo top y n - t yp e o f B C for some crossed complex C . It is shown in [ BH8 1 ] that for any cro s sed complex C there is a filtered space Y ∗ such that Π Y ∗ ∼ = C . Whitehead in [ Whi49 ] gives an example of a 5-dimensional free cr ossed complex which is not isomor phic to Π Y ∗ for any C W - filtr ation Y ∗ . A key result is the following: Theorem 4.1. L et Y ∗ b e the skeletal filt r ation of a C W - c omplex Y . Then ther e is 1-e quivalenc e q : Y → B Π Y ∗ such that if y ∈ Y 0 then the exact se qu en c e of the homotopy fibr e over y is e quivalent to Whitehe ad’s exact se quenc e [ Whi50 ]: · · · → Γ n ( Y , y ) → π n ( Y , y ) ω − → H n ( e Y y ) → · · · → Γ ( π 2 ( Y , y )) → π 3 ( Y , y ) ω − → H 3 ( e Y y ) → 0 wher e ω is the Hur ewicz morphism. Pr o of. The o riginal form of this co mes fr o m [ BH81 , section 8], which in essence used a cubical version of the classifying space. The simplicial version is dealt with in [ Ash88 ]. Corollary 4 .2. If further Y is n -aspheric al, then the homotopy fibr e of q : Y → B Π Y ∗ is n -c onne cte d. Remark 4.3. Whitehead in [ Whi49 ] obtains his clas sification results o n maps X → Y where X is n -dimensiona l a nd when Y is J n -complex, w hich is essentially the conditio n that q ab ove is an n -equiv alence, and which gener a lises the co ndition of n -aspheric ity . ✷ References [Ash88] Ashley , N. ‘Simplicial T -complexes and cro s sed complexes: a nona b elian version of a theorem of Dold and Kan’. D issertationes Math. (Ro zpr awy Mat.) 265 (1988) 1–61 . With a preface by R. Brown. [Ber85] Berr ick, A. J., ‘Gr oup ex tens ions and their trivialisatio n’, Enseign. Math. (2) , 31 (1985) 151–1 72. [Bro99] Brown, R. ‘Group oids and crossed ob jects in algebra ic top ology’. Ho- molo gy Homotopy Appl. 1 (1999) 1–78 (electronic). [Bro04] Brown, R. ‘Cro ssed complexes and homotopy gr oupo ids as non commu- tative to ols for higher dimensional lo cal-to-g lobal problems’. In ‘Galois theory , Hopf algebras, a nd semia b elian categorie s ’, Fields Inst . Commun. , V olume 43 . Amer. Math. 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