On the cohomology algebra of a fiber Luc Menichi Mathematics Subject Classification. 55R20, 55P62, 18G55, 57T30. Key words and phrases. homotopy fiber, bar construction, homotopical cate- gory, free model, Hopf algebra up to homotopy, loop space homology, divided powers algebra. Research supported by the University of Lille (URA CNRS 751) and by the University of Toronto (NSERC grants RGPIN 8047-98 and OGP000 7885). 1 Introduction Let f : E ! B be a continuous map between pointed spaces. The inverse image of the base point f-1 (*) is not in general a homotopy invariant of f. But after replacing E up to homotopy by a space X such that the new map p : X i B is a fibration, the space p-1(*), called the homotopy fiber of f, becomes a unique homotopy invariant of f. In particular, if f : E i B was initially a fibration, the fiber of f, f-1 (*), has the homotopy type of the homotopy fiber. We work over a field _. The normalized singular cochain functor induces a morphism of differential graded algebras (DGA's) C*(f) : C*(B) ! C*(E). If f is a weak homotopy equivalence then C*(f) is a DGA morphism such that the map induced in homology H*(f) is an isomorphism (We say that C*(f) is a quasi-isomorphism.). So if two topological spaces X and Y are weakly homotopy equivalent then C*(X) and C*(Y ) are linked by a chain of DGA quasi-isomorphisms, and we say that they are weakly DGA-equivalent. Note that the weak homotopy type of the DGA C*(X) is a much stronger homotopic invariant of X than the cohomology algebra H*(X). Considering a DGA morphism A ! M, the homology of the complex M A _ is not invariant by DGA quasi-isomorphisms. But replacing M by 1 an A-module P "free" in the category of A-modules such that there is a quasi-isomorphism of A-modules P !' M (We say that P is an A-semifree resolution of M.), the homology H(P A _) becomes an invariant called the differential torsion product denoted TorA (M; _). This differential torsion product generalizes the standard definition of torsion product in the non- graded non-differential case. Let F denote the homotopy fiber of f : E ! B. The link between topology and algebra is provided by the Eilenberg-Moore formula which gives the isomorphism of graded vector spaces *(B) * H*(F ) ~=Tor C (C (E); _): Generally this formula is used implicitly by applying the well-known Eilenberg- Moore spectral sequence. This formula allows the computation of the coho- mology of F , H*(F ), as a vector space. On the contrary, we don't know how to compute in general H*(F ) as an algebra. Given a particular map f : E ! B of homotopy fiber F , your best chance for computing the algebra H*(F ) is to apply the formidable machinery of the Eilenberg-Moore or Serre spectral sequences using all their algebraic structure. But it does not always work. In this article, we are interested in this problem: how to compute the cohomology algebra H*(F )? Other works on the subject are [7] and [21]. When A ! M is a morphism of commutative differential graded alge- bras (CDGA's), TorA (M; _) has a natural structure of algebra ([18] Theorem VIII.2.1 in the non-graded non-differential case). When _ = Q, Sullivan [22] proved that for any simply-connected topological space X, C*(X) is natu- rally weakly DGA-equivalent to a CDGA APL (X). Replacing C*(B) and C*(E) by APL (B) and APL (E), Tor APL(B)(APL (E); _) has now an algebra structure and a theorem proved by Grivel [11], Thomas (unpublished) and Halperin [13], called the theorem of the model of the fibre showed that this algebra coincides with that of H*(F ). Over a field _ of characteristic 0, this theorem solves completely the problem of computing the algebra H*(F ). Over a field _ of positive characteristic p, extending Sullivan's result, Anick ([2] dualize Proposition 8.7(a)) proved that if X is a finite r-connected CW-complex of dimension rp (We say that X is in the Anick range.), C*(X) is weakly DGA-equivalent to an CDGA A(X) that we will call an Anick model of X. A natural question was to generalize the Grivel-Thomas- Halperin theorem in this new context and that is the main result of this paper: 2 Theorem 9.2 Assume the characteristic of the field _ is an odd prime p. Let f : E ,! B be an inclusion of CW-complexes with trivial r-skeleton and of dimension rp. Let A(E) and A(B) denote their respective Anick models. If F is the homotopy fiber of f then H*(F ; _) ~=Tor A(B)(A(E); _)as graded algebras. The case of the inclusion * ,! B has been proved by Halperin in [14]. In fact, he proved that there is an isomorphism of Hopf algebras H*(B; _) ~= TorA(B)(_; _). In rational homotopy, the Grivel-Thomas-Halperin theorem, by staying at the level of semifree resolutions without taking their homology, not only gives the cohomology algebra of F but also its weak rational homotopy type: it gives a CDGA weakly CDGA-equivalent to APL (F ), so in particular weakly DGA-equivalent to C*(F ). Our Theorem does not give in general a CDGA weakly DGA-equivalent to C*(F ) (Remark 9.10). However, in this article we will adopt this idea that it is better to work at the level of semi-free resolu* *tion and we will not speak about Tor after this introduction. In particular, we will give a formulation of our main theorem as close as possible to the usual formulation for the Grivel-Thomas-Halperin theorem ([10], 15.5). To prove our theorem, surprisingly, we will not use the previous Eilenberg- Moore formula but another Eilenberg-Moore formula. Consider a G-fibration ss : E i X: it means in particular that ss is a fibration whose fiber G is a topological monoid acting on E. Then there is an isomorphism of graded vector spaces H*(X) ~=Tor C*(G)(C*(E); _): Let A be a DGA, M an A-module. A general way to compute TorA (M; _) is to consider the bar construction B(M; A; A) which is an A-semifree res- olution of M and to take the homology of B(M; A) := B(M; A; A) A _. In this second Eilenberg-Moore formula, following the general idea that to manipulate semi-free resolution is better than working with Tor , Felix, Halperin and Thomas remarked that it is more fruitful to consider the bar construction B(C*(E); C*(G)) instead of its homology Tor C*(G)(C*(E); _): They constructed a natural coalgebra structure on the bar construction B(C*(E); C*(G)) and proved that the differential graded coalgebra (DGC) B(C*(E); C*(G)) is weakly DGC-equivalent to the DGC C*(X) [9]. Let f : E ! B be a continuous map between path connected pointed spaces of homotopy fiber F . Starting Barratt-Puppe sequence, they showed 3 that B(C*(F ); C*(B)), where the Moore loop space B acts on F by the holonomy action, is weakly DGC-equivalent to C*(E). Pursuing Barratt- Puppe sequence, we easily see that, when F is path connected, B(C*(B); C*(E)) is weakly DGC equivalent to C*(F ) (Proposition 3.10). That is the starting observation of our paper. We now give the plan. Section 2. We set up the notations, introduce some definitions and give some elementary properties. Section 3. We review the work of Felix, Halperin and Thomas in [9]. In particular, we give a simple form of the Felix-Halperin-Thomas diagonal on the bar construction ([9] 4.1), analogous to the definition of the diagonal on C*(X) ([18] p. 245). Section 4. We carefully review the notion of homotopy in the category of DGA's and of chain complexes using cylinders: the notion of homotopy does not depend of the cylinder considered, homotopies can be composed with maps, added, DGA homotopies are closely linked with derivations. We conclude by giving two lifting lemmas. Section 5. We prove that the bar construction transforms homotopies of pairs of DGA's into chain complexes homotopies. Section 6. Let f : E ,! B be an inclusion of simply connected CW- complexes of homotopy fiber F . The natural coalgebra structure on the bar construction B(C*(B); C*(E)) is determined by the Hopf algebras morphism C*(f) : C*(E) ! C*(B). Theorem 6.2 allows us to replace in the bar construction, this strict Hopf algebras morphism by the inclusion (T X; @) ,! (T Y; @) between the Adams-Hilton models [1] of E and B, after having first equipped both DGA's (T X; @) and (T Y; @) with a structure of Hopf algebra up to homotopy (HAH) such that the HAH structure on (T Y; @) extends the one on (T X; @). Therefore, Theorem 6.2 gives the isomorphisms of algebras H*(F ) ~=H*(B(T Y ; T X)_) ~=H*((T Y TX _)_): Section 7. As an application of Theorem 6.2, we compute the coalgebra H*(Ff ) where Ff is the homotopy fiber of a suspended map injective in homology (Theorem 7.3). Section 8. We define the homotopy cofiber of a CDGA morphism A ! M: it is a CDGA defined up to weak CDGA-homotopy type, whose homology co- incides with the the algebra TorA(M; _) ([18] Corollary VIII.2.3). At the level of CDGA's, we rediscovered that the algebra TorA (M; _) can be computed 4 either with an A-semifree resolution of M or with an A-semifree resolution of _ and is invariant by CDGA quasi-isomorphisms. Section 9. We prove Theorem 9.2 and show how to apply it to compute the cohomology algebra of some fiber. Section 10. Let F be the homotopy fiber of an inclusion of CW-complexes in the Anick range. Then the cohomology algebra H*(F ) is a divided powers algebra (Theorem 10.8). Acknowledgments: I wish to thank my supervisor Nicolas Dupont. He introduced me to the problem of computing the cohomology algebra of a fiber with algebraic models. I also wish to thank Steve Halperin. In particular, he gave me Theorem 9.2 to prove, with the counterexample 9.10. 2 Algebraic preliminaries and notation We work over an arbitrary field _. References for these algebraic preliminaries are [9], [14], [15], [10], [5] and [2]. We just give our notations and recall t* *he less-known definitions. The symbol ~= denotes an isomorphism. The homology functor from differential graded objects to graded objects is denoted H. The denomination "chain" will be restricted to objects with a non-negative lower degree and "cochain" to those with a non-negative upper degree. The degree of an element x is denoted |x|. The suspension of a graded vector space M is the graded vector space sM such that (sM)i+1 = Mi. Let C_be_an augmented complex. The kernel of the augmentation is denoted C . A differential graded algebra, or DGA, is a complex A equipped with two morphisms of complexes : A A ! A and j : _ ! A called the multiplication and the unit such that O ( 1) = O (1 ) (associativity) and O (j 1) = 1 = O (1 j) (unitary). The commutator isomorphism ~= o : A B ! B A is given by o(a b) = (-1)|a||b|b a. A commutative DGA or CDGA is a DGA such that O o = . If 1_22 _, a differential divided powers algebra or -algebra is an augmented CDGA A together with the maps __ __ flk : A 2n! A 2nk (k 2 N; n 2 Z) such that: 5 (i)fl0(a) = 1; fl1(a) = a. (i + j)! i+j (ii)fli(a)flj(a) = _______fl (a). i!j! X (iii)flk(a + b) = fli(a)flj(b). i+j=k ae 0 if |a|, |b| odd and i 2 (iv) fli(ab) = i i __ a fl (b) if |a|, |b| 2 A even (v) flj(fli(a)) = _(ij)!_(i!)jj!flij(a) if i, j > 0. (vi) the differential d satisfies: dflk(a) = d(a) flk-1(a). Let A, B be two -algebras. A -morphism f : A ! B is a morphism of augmented CDGA's such that fflk(a) = flkf(a). A derivation D in a graded algebra A is a linear map of degree |D| such that Dxy = Dx:y + (-1)|D||x|x:Dy, x; y 2 A. The tensor algebra T V on a complex V is the free DGA on V . The free CDGA on V is denoted V . The free divided powers algebra V on V is the free commutative graded algebra generated by fli(v) for v 2 Veven; i 2 N* and v for v 2 Vodddivided by (i + j)! i+j i vi the relations fli(v)flj(v) = _______fl (v). Over Q, V ~= V by fl (v) 7! __. i!j! i! Let A be a CDGA, V and W two graded vector spaces. A -derivation in A W is a derivation D such that Dflk(w) = D(w)flk-1(w), k 1, w 2 W even. Any linear map V W ! V W of degree k extends to a unique -derivation over V W . Let A be a DGA, M a right A-module, N a left A-module. The tensor product of M and N over A, denoted M A N, is the complex quotient of M N by the sub-complex generated by m:a__n - m a:n, m 2 M, n 2 N, a 2 A. If A is augmented, M A _ = M=M . A. Let A ! B, A ! C be two morphisms in a category. If it exists, the push out and the morphism given by the universal property will be denoted 6 as in the commutative diagram: A _________B- | | | |A | | A | | A | | |? |? A ______- A f C B [A Cp A H H pp A H H pp A(f;g) HgH ppAp H pAp HHjR AU D If it exists, the sum of B and C is denoted B q C. The push out exists in the category of DGA's. The push out is B A C in the category of CDGA's. In particular, the tensor product of DGA's is the sum in the category of CDGA's. A quasi differential graded coalgebra, or quasi DGC, is an augmented complex C equipped with a morphism of augmented complexes : C ! C C called the diagonal. Let C and C0 be two quasi DGC's. A morphism of augmented chain complexes f : C ! C0 is a morphism of quasi DGC's if f = (f f). A differential graded coalgebra, or DGC, is a quasi DGC such that (1)O = (1)O (coassociativity) and ("1)O = 1 = (1")O (counitary). A DGC is cocommutative if o O = . The dual Hom(C; _) of a DGC C is a DGA denoted C_ . A quasi differential graded Hopf algebra, or quasi DGH, is an augmented DGA K equipped with a morphism of augmented DGA's : K ! K K. A differential graded Hopf algebra, or DGH, is a quasi DGH such that ( 1) O = (1 ) O and (" 1) O = 1 = (1 ") O . The notion of homotopy we use in the category of augmented chain com- plexes and of augmented DGA's, are recalled in section 4. The symbol t stands for homotopic morphisms. A coalgebra up to homotopy is a chain quasi DGC such that ( 1) O t (1 ) O and (" 1) O t 1 t (1 ") O . Let C and C0 be two coalgebras up to homotopy. A morphism of augmented chain complexes f : C ! C0 is a morphism of coalgebras up to homotopy if f t (f f). A Hopf algebra up to homotopy, or HAH, is a quasi DGH such that ( 1) O t (1 ) O and (" 1) O t 1 t (1 ") O . Let K, K0 be two 7 HAH's. A morphism of augmented DGA's f : K ! K0 is a HAH morphism if f t (f f). Let K be a quasi DGH. A quasi left K-coalgebra D is both a quasi DGC and a left K-module such that the action K D ! D is a quasi DGC mor- phism. A K-coalgebra D is a coassociative and counitary quasi K-coalgebra. Since : K ! K K is a morphism of augmented DGA's then _ is K- coalgebra. Property 2.1 The tensor product of a quasi right K-coalgebra C and a quasi left K-coalgebra D over K, C K D, is a quasi DGC. If C and D are coassociative, counitary, cocommutative then so is C K D. Further, the following example can be useful for computation: Example 2.2 Let (A; d) and (A q T V; D) be two counitary quasi DGH's such that the inclusion (A; d) ,! (A q T V; D) is a quasi DGH morphism. Then __ (A q T V; D) (A;d)(_; 0) ~=(T (A V ); D ) as quasi DGC's. The diagonal on T (A V ) is the morphism of complexes T(AV ) : T (A V ) ! T (A V ) T (A V ) where the image of a typical element a1 v1 . . .an vn is X ([a01a11 v11 . . .a1p v1p a1p+1a02a21 v21 . . .vnp"(anp+1)] [a001b11 w11 . . .b1q w1q b1q+1a002b21 w21 . . .wnq"(bnq+1)] ) X if A ai = a0i a00i; X AqTV vi = ai1 vi1 . . .aip vip aip+1 bi1 wi1 . . .biq wiq biq+1 and is the sign of the permutation a01a001v11...a1pv1pa1p+1b11w11...b1qw1qb1q+1a02a002...an1vn1...anpvnpanp+1bn1* *wn1...bnqwnqbnq+1 a01a11v11...a1pv1pa1p+1a02a21v21...vnpanp+1a001b11w11...b1qw1qb1q+1a002b21* *w21...wnqbnq+1: 8 The differential is given by the formula: __ Xn j -|a | D (a1 v1 . . .an vn) = (-1) i ia1 v1 . . .dai . . .vn i=1 n-1X X + (-1)ji a1 v1 . . .aici1 ui1 . . .cir uir cir+1ai+1 vi+1 i=1 X +(-1)jn a1 v1 . . .ancn1 un1 . . .cnr unr"(cnr+1) X if Dvi = ci1 ui1 . . .cir uir cir+1 and ji = |a1| + |v1| + |a2| + . .+.|vi-1| + |ai|: A differential graded Lie algebra, or DGL, is a complex L equipped with a morphism of complexes: [ ; ] : L L ! L such that for x, y, z 2 L: o [x; y] = -(-1)|x||y|[y; x] o (-1)|x||z|[x; [y; z]] + (-1)|z||y|[z; [x; y]] + (-1)|y||x|[y; [z; x]] = 0 o [x; [x; x]] = 0, x 2 Lodd The universal enveloping algebra of L is denoted UL. A quasi-isomorphism is denoted !'. Two objects A and B in a category C are weakly C-equivalent, denoted A ~ B, if they are connected by a chain of C-quasi-isomorphisms of the form: A ' A(1) '! . ....'. A(n) '! B: If the C-quasi-isomorphisms are natural, we say that A and B are naturally weakly C-equivalent. Let A be an augmented DGA, M a right A-module, N a_left A-module. Denote d1 be the differential of the complex M T (sA ) N obtained by tensorization._We_denote_the tensor product of the elements m 2 M, sa1 2 sA , : :,:sak 2 sA and n 2 N by m[sa1|_. .|.sak]n. Let d2 be the differential on the graded vector space M T (sA ) N defined by: d2m[sa1| . .|.sak]n= (-1)|m|ma1[sa2| . .|.sak]n k-1X + (-1)"im[sa1| . .|.saiai+1| . .|.sak]n i=1 -(-1)"k-1m[sa1| . .|.sak-1]akn; 9 Here "i = |m| + |sa1| + . .+.|sai|. Remark 2.3 We only find the above formula in the non-graded case in the literature ([18] X.(2.5)). We obtain the appropriate signs by Mac Lane's condensation of complexes of complexes ([18] X.9). If we set N = _, we recover the same formula as in [9] x4. The bar construction_of_A with coefficients in M and N, denoted B(M; A; N), is the complex (M T (sA ) N; d1 + d2). We use mainly B(M; A) = B(M; A; _). The reduced bar construction of A, denoted B(A), is B(_; A). Let L be a DGL, M a right UL-module. If 1_22 _, B(M; UL) has a subcomplex C*(M; L) = (M sL; d1 + d2) [14, x1]. Its dual, denoted C*(M; L), is called the Cartan-Chevalley-Eilenberg complex with coefficients in M. Again C*(L) denotes C*(_; L). Let A be a DGA. A semifree extension of an A-module M is an inclusion of A-modules: (M; d) ae (M (A V ); D) such that: o V = k2NV (k) as graded vector space. o D : V (k) ! M (A V (< k)); k 2 N where V (< k) = k-1i=0V (i). An A-semifree module is an A-module (A V; D) such that 0 ae (A V; D) is a semifree extension of 0. A free extension is an inclusion of augmented DGA's: (A; d) ae (A q T V; D) such that V = k2NV (k) and D : V (k) ! A q T V (< k); k 2 N. A free DGA is a DGA (T V; D) such that _ ae (T V; D) is a free extension. A relative Sullivan model is an inclusion of augmented CDGA's: (A; d) ae (A V; D) such that V = k2NV (k) and D : V (k) ! A V (< k); k 2 N. A Sullivan model is a CDGA (V; D) such that _ ae (V; D) is a relative Sullivan model. A Sullivan model of a DGA A is a Sullivan model (V; D) equipped with a DGA quasi-isomorphism (V; D) '! A. A -free extension is an inclusion of augmented CDGA's: (A; d) ae (A V; D) such that V = k2NV (k), D : V (k) ! A V (< k); k 2 N and D is a -derivation. In particular, if A is a -algebra, than the -free extension (A; d) ae (A V; D) is a -morphism. Note that the condition on the graded vector space V in these four similar definitions is always satisfied in chain. In particular, an inclusion of chain DGA's (A; d) ae (A q T V; D) is always a free extension. 10 The_complex_of_indecomposables of the augmented DGA A, denoted Q(A) is A =A . A. The augmented DGA A is minimal if the differential on Q(A) is zero. An inclusion of augmented DGA's A ,! B is minimal if the augmented DGA B [A _ is minimal. In particular, an inclusion of augmented CDGA's A ,! B is minimal if the augmented CDGA B A _ is minimal. Property 2.4 (i)If A ae A q T V is a free extension then A q T V is (left and right) A-semifree. (ii)[10, 14.1] If A ae A V is a relative Sullivan model then A V is A-semifree. (iii)If A ae A V is a -free extension then A V is A-semifree. The normalized singular chain complex of a topological space X with coeffi- cients in _ is denoted C*(X). Let G be a topological monoid. A right G-Serre fibration is a Serre fibration p : E i B such that E is a right G-space, for each b 2 B the fiber p-1(b) is stable by G and for each z 2 E the map g 7! z:g is a weak homotopy equivalence from G to p-1(p(z)). Convention 2.5 Let f : E ! B be a continuous map. When we will speak about the homotopy fiber of f, except if specified, we will choose the homotopy fiber where the holonomy acts on the left and denotes it by Ff. 3 The bar construction with coefficients as a DGC Property 3.1 ([18] X.7.2) Let A (respectively B) be an augmented DGA, M (respectively N) a left A-module (respectively B-module) and P (respectively Q) a right A-module (respectively B-module). Then we have an Alexander- Whitney morphism of complexes AW : B(P Q; A B; M N) ! B(P ; A; M) B(Q; B; N) 11 where the image of a typical element p q[s(a1 b1)| . .|.s(ak bk)]m n is X k (-1)iip[sa1| . .|.sai]ai+1. .a.km qb1 . .b.i[sbi+1| . .|.sbk]n: i=0 Xk j-1X ! Xk ! Here ii = |q| + |bl||aj| + |q| + |bj||m| j=1 l=1 j=1 Xk i-1X + (j - i)|aj| + (k - i)|m| + |i||q| + (i - j)|bj|: j=i+1 j=1 AW is natural and associative (AW O (AW id) = AW O (id AW )). Remark 2.3 holds here too. Corollary 3.2 Let K be a quasi DGH, C a quasi right K-coalgebra, D a quasi left K-coalgebra. Then B(C; K; D) is a quasi DGC with the diagonal B(C;K ;D) AW B(C; K; D) --------! B(C C; K K; D D) - ! B(C; K; D) B(C; K; D) B("C;"K ;"D) and the counit B(C; K; D) -------! B(_; _; _) = _: If K, C and D are coassociative, counitary then so is B(C; K; D). This coalgebra structure on B(C; K; D) is functorial. Proof. It is obvious with commutative diagrams using AW 's associativity, _____ naturality and the functoriality of the bar construction. |QED_| Property 3.3 Moreover, if C is K-semifree then B(C; K; D) '! C K D is a quasi-isomorphism of quasi DGC's. Proof. The morphism of quasi left K-coalgebras B(K; K; D) '! D remains a quasi-isomorphism of quasi DGC's after applying C K - ([9] 2.3 (i) and _____ Property 2.1). |QED_| Remark 3.4 When K is a DGH and C is a K-coalgebra, the coalgebra structure on B(C; K) coincides with the one defined in ([9] 4.1). The proof is a tedious calculation. Anyway, we don't need to give it, since we will verify that the following theorem is valid independently of the functorial coalgebra structure chosen on the bar construction, either the one defined by Felix- Halperin-Thomas, or the one defined in Corollary 3.2. 12 Theorem 3.5 ([9] 5.1) Let p : E i B be a right G-Serre fibration with B path connected. Then there is a natural DGC quasi-isomorphism B(C*(E); C*(G)) '! C*(B). Remark 3.6 This natural quasi-isomorphism is the identity on C*(E) for the *-fibration id : E ! E. Proof. As shown in Theorem 8.3 of [10], if m : M '! C*(E) is a right C*(G)-semifree resolution of C*(E) then we have the commuting diagram of complexes. m M __________________-C*(E)' | | | | i | |C (p) | | * | | |? |? M C*(G)_ ______________-C*(B)'_m In particular, we can take M = B(C*(E); C*(G); C*(G)). Since m, C*(p) and i are DGC morphisms and i is an epimorphism, __mis a DGC morphism _____ too. |QED_| Remark 3.7 Suppose further that G acts from the left on a space Y and that the map q : E xG Y i B defined by q(z; y) = p(z) for z 2 E and y 2 Y is a Serre fibration such that for each z 2 E the map y 7! (z; y) is a weak homotopy equivalence from Y to q-1(p(z)). Then there is a natural DGC quasi-isomorphism B(C*(E); C*(G); C*(Y )) '! C*(E xG Y ). The proof is the same as above interpreting the general Eilenberg-Moore formula ([12] Theorem 3.9) H*(E xG Y ) ~=Tor C*(G)(C*(E); C*(Y )) at the chain level. Proposition 3.8 ([9] 6.7) Let f : E ! B be a continuous map between path connected spaces and Ff its homotopy fiber then there is a natural DGC quasi-isomorphism B(_; C*(B); C*(Ff)) '! C*(E). Remark 3.9 This natural quasi-isomorphism is the identity on C*(E) for the map E ! *. Proof. The Moore path space fibration P B i B with P B being the Moore paths that begin at the basepoint, is a left B-fibration. So, by pull back, 13 we obtain a left B-fibration p0 : Ff i E. We apply Theorem 3.5 to p0. _____ |QED_| Proposition 3.10 Let f : E ! B be a continuous pointed map with E, B and Ff path connected. Then C*(Ff) is naturally weakly DGC equivalent to B(C*(B); C*(E)). Proof. We have the morphism of topological monoids f : E ! B. So B is a right E-space and C*(f) : C*(E) ! C*(B) is a DGH morphism. Consider the previous map p0 : Ff i E. Let F"p0denote its homotopy fiber where the holonomy acts on the right. There is a natural morphism of right E-spaces j : F"p0'!B which is a homotopy equivalence and is the identity if E = * ([10] I.2.(c)). By Proposition 3.8 applied to p0, we have the chain of natural DGC quasi-isomorphisms C*(Ff) -' B(C*(F"p0); C*(E)) -'! B(C*(B); C*(E)): B(j;id) _____ |QED_| Remark 3.11 This chain of natural quasi-isomorphisms is just the identity on C*(B) for the map * ! B. So by naturality, we have the commutative diagram of DGC's C*(B) _________B(C*(B);-C*(E)) | H H | || H H ||6 C*(@)| H H ' |B(j;id) | H H | |? Hj | C*(Ff) __________B(C*(F"p0);oC*(E))e' where @ : B ,! Ff is the inclusion. 4 Homotopy of augmented chain complexes and of augmented DGA's and lifting lem- mas We recall the notion of homotopy of augmented chain complexes and of augmented DGA's using cylinders since our proof will rely heavily on it. 14 To develop homotopy theory using cylinders in a category, a good frame- work is to have a structure of cofibration category where all objects are fibrant. Definition 4.1 (Compare [4] I.1.1) A cofibration category where all objects are fibrant is a category C with two classes of morphisms called cofibrations (denoted by ae) and weak equivalences (denoted by !'), subject to axioms C1, C2, C3 and C4. The axioms in question are: (C1) Composition axiom: The isomorphisms in C are weak equivalences and also cofibrations. For two maps f g A ! B ! C if any two of f, g and g O f are weak equivalences, then so is the third. The composite of cofibrations is a cofibration. (C2) Push out axiom: For a cofibration i : B ae A and map f : B ! Y there exists the push out in C f B _________Y- | | | | | |_ i | |i | | |? |? A ______A-[B_Yf _ and iis a cofibration. Moreover: __ (a) if f is a weak equivalence, so is f, _ (b) if i is a weak equivalence, so is i. (C3) Factorization axiom: For a map f : B ! Y in C there exists a com- mutative diagram f B _____________________Y- @ @ ' i@ g @@R A where i is a cofibration and g is a weak equivalence. 15 (C4) All objects are fibrant: Each cofibration which is also a weak equivalence ' i : R ae Q in C admits a retraction r : R ! Q, r O i = 1. The axiom C4 can be replaced in Definition 4.1 by the Property: Property 4.2 ([4] II.1.11=II.2.11a)) Given a commutative diagram of un- broken arrows B ________-X | p | | pp | | h pp ' |p i| pp | | pp | |?p |? A _________Y-g where i is a cofibration and p is a weak equivalence then there is a map h for which the upper triangle commutes. Property 4.3 (i)([4] II.6.4) The category of augmented chain complexes is a cofibration category where all objects are fibrant and where cofi- brations are injections and weak equivalences, quasi-isomorphisms. (ii)([4] II.7.10) The category of augmented DGA's is a cofibration category where all objects are fibrant and where cofibrations are free extensions and weak equivalences, quasi-isomorphisms. Remark 4.4 In a cofibration category where all objects are fibrant, Prop- erty 4.2 is the key to obtain the basic properties of the notion of homotopy. In the particular case of augmented chain complexes and of augmented DGA's (Property 4.3), we can use the following lifting lemma instead of Property 4.2. Property 4.5 Given a commutative diagram of unbroken arrows B ________-X | p | | pp | | h pp ' |p i| pp | | pp | |?p |? A _________Y-g 16 where i is a cofibration and p is both a surjection and a weak equivalence then the dotted arrow h exists such that both triangles commute. We can now define the notion of homotopy in the category of augmented chain complexes and in the category of augmented DGA's. In this section, the morphism Y ! X is going to be either (i)a cofibration and we define homotopy relative Y or under Y and follow [4] IIx2 in the case of a cofibration category where all objects are fibra* *nt, (ii)or just the unit of X, _ ! X and we define absolute homotopy for augmented DGA's following [9] x3. Remark 4.6 In a cofibration category C with an initial object , homotopy relative is called absolute homotopy and we call an object X cofibrant if ae X is a cofibration. Following case (i), absolute homotopy is defined only when X is cofibrant. In the category of augmented chain complexes, all objects are cofibrant and therefore by case (i), absolute homotopy is defined for every complex X. In the category of augmented DGA's, only free DGA's are cofibrant but case (ii) define absolute homotopy even when X is a not a free DGA. Definition 4.7 ([19] 1.1) An object denoted X" is a left homotopy object on Y ! X if there is a factorization of the folding map (id;id) X [Y X __________________X- @ @ ' i@ p @@R X" Remark 4.8 Let i0 (respectively i1) be the composite of the first (respec- tively second) inclusion X ! X [Y X with i. Then by universal property, i = (i0; i1) and we use this last notation. Definition 4.9 A cylinder on Y ! X, denoted IY X is a left homotopy object on Y ! X such that (i0; i1) is a cofibration. If the category has an initial object , IX will stand for a cylinder on ! X instead of I X. 17 Let u : Y ! U be a fixed morphism. Let x; y : X ! U be two morphisms such that for each of them the following diagram commutes: Y | | @ | @ u | @ | |? @@R X _________U- Definition 4.10 The morphisms x and y are homotopic for the left homo- topy object X" if there is a commutative diagram (x;y) X [Y X __________________U- @ @ (i0;i@1) h @@R X" We call h a homotopy from x to y, and denoted it h : x t y. Property 4.11 If we fix a cylinder IY X, then for any homotopy h : x t y starting from a left homotopy object X", there exists a homotopy h0: x t y starting from IY X. In particular, all cylinders define the same notion of homotopy between morphisms. Proof. By the lifting lemma (Property 4.5), we obtain a morphism m : IY X ! X" such that the following diagram commutes U * (x;y) h (j0;j1) X [Y X _____-X" | p | | pp | (i ;i|) m pp ' |q 0 |1 pp | | pp | |?p p |? IY X _______-X 18 _____ and we set h0= h O m. |QED_| Property 4.12 The homotopy relation defined with a cylinder is an equiv- alence relation. Definition 4.13 (i)The homotopy x O p : x t x is called the trivial homotopy and is denoted 0. (ii)Let h : x t y be a homotopy. By the lifting lemma (Property 4.5), we obtain a morphism n : IY X ! IY X such that the following diagram commutes U * | (y;x) |6 ||h (x;y) | (i0;i1) | X [Y X ___X-[YTX ____-IY X | ppp* | | pppp | (i ;i|) n ppp ' |p 0 |1 pppp | | pppp | |? pp p |? IY X ___________________-X Here T is the interchange map of the two factors. The homotopy h O n : y t x is called a negative of the homotopy h and is denoted -h. (iii)Let h : x t y and g : y t z be two homotopies for the same cylinder IY X. The push out of two cylinders is a left homotopy object. So again as in Property 4.11, we can apply the lifting lemma (Property 4.5) to the diagram U * | (x;z) |6 ||(h;g) | | X [Y X ____________IY-Xi[X[IYiX | ppp* 0 | 1 | pppp | (i ;i|) m ppp ' |(p;p) 0 |1 pppp | | pppp | |? pp p |? IY X ___________________-X 19 The homotopy (h; g) O m : x t z, is called the sum of the homotopies and is denoted h + g. Property 4.14 The notion of homotopy is stable by composition. Proof. o Let g : U ! V be a morphism and h : x t y be a homotopy. Then g O h : g O x t g O y is a homotopy. o Let f0 B _________Y- | | | | | | | | | | |? f |? A _________X- be any commutative diagram with B ae A a cofibration or the unit of A. Then by the lifting lemma (Property 4.5), we obtain a morphism If : IB A ! IY X such that the following diagram commutes U * | (xOf;yOf) |6 ||h (x;y)| f[f (j0;j1)| A [B A ___-X [Y X ____-IY X | ppp* | | pppp | (i ;i|) Ifppp ' |q 0 |1 pppp | | pppp | |? ppp f |? IB A ________A- _________X- _____ So h O If : x O f t y O f is the desired homotopy. |QED_| Definition 4.15 We denote by If : IB A ! IY X any morphism from a cylinder on B ! A to a cylinder on Y ! X such that the preceding diagram commutes. Remark 4.16 In the category of augmented DGA when X is a free DGA, there is a canonical cylinder IX called the Baues-Lemaire cylinder and a 20 canonical map If : IA ! IX ([4] II.7.15). For this cylinder, a given homo- topy h : x t y has a canonical negative -h and the sum of two homotopies is canonically defined ([4] II.17.3). For the Baues-Lemaire cylinder, any homotopy h from x to y corresponds uniquely to an (x; y)-derivation H ([9] 3.5, [4] I.7.12). Of course, the canoni* *cal negative of the homotopy h corresponds to the (y; x)-derivation -H. And the composite of the homotopy h and of the canonical map If, hOIf corresponds to the (x O f; y O f)-derivation H O f. Warning, the sum H + G of an (x; y)- derivation H and an (y; z)-derivation G is not in general an (x; z)-derivation. In section 6, we will use for DGA's two lifting lemmas other than Property 4.5, the first of which refines Property 4.2. Property 4.17 ([4] II.1.11=II.2.11a)) Given a commutative diagram of un- broken arrows B ________-X | p | | pp | | h pp ' |p i| pp | | pp | |?p |? A _________Y-g where i is a cofibration and p is a weak equivalence then (i)there is a map h for which the upper triangle commutes and for which p O h is homotopic to g relative to B, and (ii)this map h is unique up to homotopy relative to B. __ Proof. We recall just the proof of (ii). Let h and h be two maps satisfying (i) and let H and G be homotopies_relatively to B for a cylinder Z from p O h to g and from g to p O h respectively. We apply Property 4.2 to the commutative diagram (h;_h) A [B A ______X- | p | | pp | i [i| F pp ' |p 0 |1 pp | | pp | |?p |? Z [A Z ______-Y(G;H) __ _____ Now F is a homotopy from h to h for the cylinder Z [A Z. |QED_| 21 Property 4.18 ([9] 3.6) Consider the following diagram, that commutes up to a homotopy H: T V ________X- | p | | pp | | h pp ' |p i| pp | | pp | |?p |? T W ________Y-g where T V and T W are free DGA's, i is a cofibration and p is a weak equiv- alence. Then there exists a map h for which the upper triangle commutes and such that p O h is homotopic to g. The homotopy G from p O h to g can be chosen such that G O Ii = H (G extends H). 5 Bar construction and homotopies After reviewing Felix-Halperin-Thomas diagonal on the bar construction and the notion of homotopy defined with cylinders, we prove in this section the key lemma from which derives all our theorems. This lemma is a homotopic version of Corollary 3.2. First, we need a "functoriality up to homotopy" of the bar construction provided by the Property. Property 5.1 Let h:'t'0 A ________-A0 | | | | f | |g | | | | |? |? M ________M0-h0:t0 be a "diagram" of chain augmented DGA's where h : IA ! A0and h0: IM ! M0 are homotopies, and where Of = gO', and 0Of = gO'0. Consider one of the morphisms If : IA ! IM (Definition 4.15). If h0OIf = gOh (naturality of the homotopies) then the morphisms of augmented chain complexes B(; ') and B(0; '0) are homotopic. 22 Proof. Since the bar construction is a functor preserving quasi-isomorphisms from the category of pairs of chain augmented DGA's to the category of augmented chain complexes ([9] 4.3(iii)), B(IM; IA) is a left homotopy ob- ject on 0 ae B(M; A) in the category of augmented chain complexes. So _____ B(h0; h) : B(; ') t B(0; '0) is a homotopy. |QED_| Lemma 5.2 (i)Let K (respectively C) be a strictly counitary chain HAH, coassociative up to a homotopy hassocK(respectively hassocC): ( 1) O t (1 ) O . Let f : K ! C be a morphism of augmented DGA's such that C f = (f f)K and hassocCIf = (f f f)hassocK(f commutes with the diagonals and the homotopies of coassociativity). Then B(C; K) with the diagonal B(C;K ) AW B(C; K) ------! B(C C; K K) -! B(C; K) B(C; K) is a strictly counitary coalgebra up to homotopy. (ii)Consider the following cube of augmented chain DGA's ' K _______________-K0 | pp | @ K pp@ K0 || @ pp @ | @@R ppp @@R | _____ppg__-'0' 0 f| K K pp K K | | p | | | pp | | | pp | | | pp | |? || ? || C ppppffpp|pppppppC0pppppp- |gg | pp | @ | ppC0 | @ | pp | C | pp | @@R |? R |? C C ___________C0-C0 where all the faces commute exactly except the top and the bottom ones. Suppose that the top face commutes up to a homotopy htop : (' ')K t K0' and the bottom face commutes up to a homotopy hbottom: ( )C t C0 such that hbottomIf = (g g)htop. Then the morphism of augmented chain complexes B(; ') : B(C; K) ! B(C0; K0) commutes with the diagonals up to homotopy. 23 Proof. the same as the proof of Corollary 3.2, with Property 5.1 replacing _____ the functoriality of the bar construction. |QED_| Remark 5.3 The results of this section remain true if we replace the bar construction by any functor B preserving quasi-isomorphisms from the cate- gory of pairs of chain augmented DGA's to the category of augmented chain complexes equipped with a natural, associative morphism of augmented com- plexes AW : B(P Q; A B) ! B(P ; A) B(Q; B). In particular, Lemma 5.2 is valid for the functor B(M; A) = M A _ if f : K ! C is a free extension and K is a free DGA (The extra hypothesis is needed to preserve quasi-isomorphisms.). Remark 5.4 There is a generalization of Lemma 5.2(ii) to homotopy com- mutative cubes. In [20], we define diagonals on B(C; K) and B(C0; K0) and a morphism of augmented chain complexes from B(C; K) to B(C0; K0) com- muting with the diagonals up to homotopy provided that the homotopies in each face of the cube satisfies a compatibility condition. 6 HAH structure on free models Let X be a graded vector space. We denote a free DGA (T X; @) simply by T X except when the differential @ can be specified. In particular, a free DGA with zero differential is still denoted by (T X; 0). Definition 6.1 ([19] D.28) An explicit HAH is a free DGA T X equipped with a morphism of augmented DGA's : T X ! T X T X such that (" 1) O = 1 = (1 ") O , a homotopy hassoc: ( 1) O t (1 ) O and a homotopy hcom : t o. Note that if (T X; ; hassoc; hcom) is an explicit HAH then (T X; ) is a strictly counitary HAH, coassociative and cocommutative up to homotopy. Let (T X; TX ; hassocTX; hcomTX ) and (T Y; TY ; hassocTY; hcomTY ) be two explicit HAH's. Let f : T X ! T Y be an augmented DGA morphism. Then f is a morphism of explicit HAH's if f(X) Y , TY f = (f f)TX , hassocTYIf = (f f f)hassocTX and hcomTY If = (f f)hcomTX . Theorem 6.2 Let f : E ! B be a map between path connected pointed topological spaces with a path connected homotopy fiber F . We consider a 24 commutative diagram of augmented chain algebras as follows: ' T X _____C*(E)- | | X | | m(f)| |C (f) | | * | | |? |? ' T Y _____C*(B)-Y where T X, T Y are free DGA's and m(f) : T X ae T Y is a free extension. Then 1. T X (respectively T Y ) can be endowed with an explicit HAH structure such that X (respectively Y ) commutes with the diagonals up to a homotopy hX (respectively hY ) and such that m(f) is a morphism of explicit HAH's and hY extends (C*(f) C*(f))hX . 2. B(Y ; X ) : B(T Y ; T X) !' B(C*(B); C*(E)) is a morphism of coalgebras up to homotopy. 3. The homology of the coalgebra up to homotopy T Y TX _ is isomorphic to H*(F ) as coalgebras. Remark 6.3 o The isomorphism of graded coalgebras between H*(T Y TX _) and H*(F ) fits into the commutative diagram of graded coalgebras: ~= H*(T Y ) _______________H*(B)- | H*(Y) | | | H (q)| |H (@) * | | * | | |? ~ |? H*(T Y TX _) ____________-H*(F=) where @ : B ,! F is the inclusion and q : T Y i T Y TX _ the quotient map. o The quasi DGC T Y TX _ can be made explicit using Example 2.2. Remark 6.4 o The exact commutativity of the diagram in Theorem 6.2 is not important. If the diagram commutes only up to homotopy, since m(f) is a cofibration, by Property 4.18, we can replace Y by another Y which is homotopic to it, so that now the diagram strictly commutes. 25 o But it is important that m(f) is a cofibration. We will show it in Remark 7.5. Indeed, the general idea for the proof of 1 is to control the homotopies using the homotopy extension property of cofibrations. Proof of Theorem 6.2 1. By Property 4.17(i), we put a diagonal on T X, TX , such that X commutes with the diagonals up to a homotopy hX . The diagram of un- broken arrows T X ______T-X2 _____T-Y 2 | ppp* | | pppp | | TY ppp |' | pppp | | pppp | |? pp |? T Y _____C*(B)- __C*(B)2- commutes, with homotopy C*(f)2 hX : By Property 4.18, there exists a diagonal on T Y , TY , satisfying 8 < TY extends the diagonal on T X such that there exists a (*) homotopy hY between (Y Y )TY and C*(B) Y : extending C 2 *(f) hX . We can assume that, both the diagonal of T X and the diagonal of T Y are counitary. Let's give a sketch of proof of that: Since C*(E) has a counitary diagonal, by Property 4.17(ii), TX is counitary up to a homotopy hunitTX. That is, the diagram T X ____T-X T X @ || @ |("1;1") (1;1@) || @@R |? T X x T X commutes up to the homotopy hunitTX. Furthermore, TY is counitary up to a homotopy hunitTY extending hunitTX. We can change the diagonal of T X up to homotopy to get a counitary one [2, Lemma 5.4 i)]. Moreover, since hunitTYextends hunitTX, we can change up to homotopy the diagonal of T Y to get a counitary one such that the condition (*) is still satisfied with the new counitary diagonals. 26 We give now a detailed proof that TX is cocommutative up to a ho- motopy hcomTX and that TY is cocommutative up to a homotopy hcomTY extending hcomTX : Since the diagonal on C*(E) is cocommutative up to a homotopy hcomC*, by Property 4.17(ii), TX is cocommutative up to a ho- motopy hcomTX . More precisely (Proof of Property 4.17(ii)), hcomTX is given by Property 4.2 in the diagram: (oTX ;TX ) T X q T X __________________________-T X2 | ppppppp1 | | h ppppp | i0[i|1 comppppppTX |' | ppppp | | pppppp | |? ppp |? IT X [TX IT X _____________________-C*(E)2(oh X ;hcomC*X -hX ) where IT X is the Baues-Lemaire cylinder (Remark 4.16). Now, since the ho- motopy of cocommutativity of C*(B) is natural ([2] (23)) and the sums and negatives of homotopies are canonically defined (Remark 4.16), the following cube of unbroken arrows is commutative: T X q T X __________-T X2 | p | | @ hcomTX pp |@ || @ pp || @ | @@R ppp | @@R | __|___- 2 | IT X [TX IT X | C*(E) | | | | | | | | | | | | | | | | |? || |? || T Y q T Y ____|_____-T Y 2 | | p | @ h|comTYpp @ | @ | pp @ | | pp | @@R |?p @@R |? IT Y [TY IT Y ______-C*(B)2 The homotopy of cocommutativity of T Y , hcomTY is given by applying Prop- erty 4.2 to the commutative diagram (IT X [TX IT X) [TXqTX (T Y q T Y ) _________-T Y 2 | ppppppp1 | | pppppp | | ppppp |' | pppppp hcomTY | |? ppp |? IT Y [TY IT Y __________________-C*(B)2 27 A similar proof shows that TX is coassociative up to a homotopy hassocTX and that TY is coassociative up to a homotopy hassocTYextending hassocTX. 2. Now, by Lemma 5.2, the augmented chain complexes quasi-isomorphism B(Y ; X ) : B(T Y ; T X) '! B(C*B; C*E) commutes with the diagonals up to homotopy. Since m(f) is a morphism of explicit HAH's, this diagonal on B(T Y ; T X) is counitary exactly and is coassociative up to homotopy. 3. By Property 2.4(i) and Property 3.3, the augmented chain complexes quasi-isomorphism B(T Y ; T X) '! T Y TX _ commutes exactly with the diagonals. By Remark 5.3, T Y TX _ is a strictly counitary coalgebra up to homotopy, coassociative up to homotopy and co- commutative up to homotopy. By Proposition 3.10, C*(F ) is weakly DGC equivalent to B(C*B; C*E). So now by 2, the coalgebra H*(T Y TX _) _____ is isomorphic to H*(F ). |QED_| 7 The fiber of a suspended map Lemma 7.1 Let X be_a_path_connected space. Then there is a natural DGH quasi-isomorphism T C*(X) !' C*(X). Proof. The adjunction map ad induces a morphism of coaugmented DGC's C*(ad) : C*(X) !_C*(X)._ By universal_property_of the tensor algebra on the complex C*(X) , denoted T C*(X) , C*(ad) extends to a natural DGH morphism. By Bott-Samelson Theorem ([16] appendix 2 Theorem 1.4), it is _____ a quasi-isomorphism, since the functors H and T commute. |QED_| Lemma 7.2 Let f : E ! B be a continuous map between path connected_ ______ spaces. Then C*(Ff ) is naturally weakly DGC equivalent to B(T C*(B); T C*(E)). Proof. It is a direct consequence of Lemma 7.1, Proposition 3.10 and Corol- _____ lary 3.2. |QED_| Theorem 7.3 Let f : E ! B be a continuous map between path connected spaces such that H*(f) is injective. Then the graded coalgebra T H+ (B)TH+(E) _ is isomorphic to H*(Ff ). 28 Remark 6.3 holds here too. Proof of Theorem 7.3 Since H*(f) is injective, we can apply Theorem 6.2 and Lemma 7.2 to the homotopy commutative diagram of DGA's: ' ______ (T H+ (E); 0)_____________T-C*(E) | | | | TH (f) | | _____ + | |TC*(f) (7.4) | | |? __|?__ (T H+ (B); 0)_____________T-C*(B)' Since the horizontal arrows induce the identity in homology, the diagonals on T H+ (E) and T H+ (B) must be obtained by tensorization of the diagonals _____ of H+ (E) and H+ (B). |QED_| Remark 7.5 If H*(f) is not injective, Theorem 7.3 is not true in general: the algebra H*(F ) does not depend only on H*(f). Indeed, since T H+ (f) is not a free extension, we cannot apply Theorem 6.2 to the diagram 7.4. For an example over Fp, we can take a map f from S2p-1 to CP p-1. Let y2 be a generator of H2(Ff ). If f is the Hopf map, there is a map : CP p ! Ff such that H2( ) is an isomorphism. So yp26= 0. If f is the constant map then yp2= 0. Remark 7.6 When H*(B) is of finite type and H1(f) is an isomorphism, the isomorphism given by Theorem 7.3 can be proved using a spectral se- quence argument. Recall first that by the Bott-Samelson Theorem, the adjunction maps ad induce an isomorphism of graded coalgebras between T H+ (B) TH+(E) _ and H*(B) H*(E) _. The inclusion @ : B ! Ff is up to a homotopy equivalence a right E-fibration (Proof of Proposition 3.10). So kerH*(@) contains the left ideal generated by Im H+ (f) and by Property 2.1, H*(@) induces a morphism of graded coalgebras ______ H*(@) : H*(B) H*(E) _ ! H*(Ff ): Since H*(f) is injective, the Serre spectral sequence applied to @ collapses at the E2-term, H*(@) is surjective and kerH*(@) is isomorphic to H*(Ff ) H+ (E). Using again the Bott-Samelson Theorem, kerH*(@) is the left H+(f) H+(ad) ideal_generated by the image of H+ (E) -! H+ (B) -! H+ (B). So H*(@) is an isomorphism. 29 8 Homotopy cofibers for CDGA's In this section, we develop the notions of cofibration, cofiber, homotopy cofiber and homotopy push out in the category of augmented CDGA's. We give an example of homotopy cofiber crucial for the proof of Theorem 9.2 and we notice that the weak CDGA equivalence class of homotopy cofibers of a CDGA morphism is preserved if one changes the CDGA morphism up to quasi-isomorphisms. Definition 8.1 Let i : A ! C be a morphism of augmented CDGA's. Con- sider the A-module structure on C induced by i. If C is an A-semifree module, we say that i is a cofibration (in the category of augmented CDGA's) and we denote i : A ae C. The cofiber of a cofibration i : A ae C is the augmented CDGA _ A C. Property 8.2 The category of augmented CDGA's where the cofibrations are the morphisms as defined above and where the weak equivalences are the quasi-isomorphisms, satisfies axioms C1, C2 and C3 of Definition 4.1 (but not C4!). Remark 8.3 If we restrict our definition of cofibrations to morphisms be- ing relative Sullivan models, the category of augmented Q-CDGA's forms a cofibration category where all objects are fibrant ([4] I.x8). However, over a field of characteristic p, the category of augmented CDGA's is still not a cofibration category. The topological notions of homotopy push out and homotopy cofibers can be defined more generally in any category with a final object satisfying axiom C1, C2 and C3 ([6], chapter 4). Using Property 8.2, we develop now the notion of homotopy push out and of homotopy cofiber in the category of augmented CDGA's: Let f : A ! B, g : A ! C be two morphisms of augmented CDGA's. Consider two factorizations f = p O i, g = q O j where i : A ae D, j : A ae E are cofibrations and p, q quasi-isomorphisms. By Property 8.2, we can 30 construct the commutative diagram of augmented CDGA's: i _____________________-' A _____________________-D p B | | | | | | j| _ | | | j | | | | | |? _ |? |? i _______________-' E __________________-D A E B A E (8.4) | | | | '|q |' | | | | |? |? C __________________-D A C _ __ All the rectangles appearing in this diagram are push outs, iand j are cofi- brations. We have a chain of quasi-isomorphisms of augmented CDGA's D A C -' D A E -'! B A E: In particular, the augmented CDGA's DA C, DA E and BA E are weakly CDGA equivalent and their weak CDGA equivalence class is independent of the factorization chosen of f and g. Definition 8.5 The augmented CDGA's D A C, D A E and B A E obtained by considering various factorizations of f and g as above are called homotopy push outs of f and g. All the homotopy push outs of f and g are weakly CDGA equivalent. The homotopy cofibers of f are the homotopy push outs of f and of the augmentation on A, " : A i _. Example 8.6 Let f : E ! B a chain Lie algebras morphism. If B is positively graded and of finite type then C*(UB; E) = ((UB)_ (sE)_; d1+ d2) equipped with the tensor product algebra structure becomes a CDGA which is a homotopy cofiber of C*(f) : C*(B) ! C*(E). Proof. By [9] 6.10, C*(UB; B) is an acyclic CDGA. Since B is of finite type, C*(UB; B) is C*(B)-semifree. By the universal property of push out, there is a CDGA morphism ~= * C*(E) C*(B)C*(UB; B) -! C (UB; E) 31 which is an isomorphism since B is of finite type. So we get the commutative diagram of augmented CDGA's C*(f) C*(B) ________________-C*(E) || || | | || || |? |? ' ____________- _ ____oC*(UB;eB) C*(UB; E) where the square is a push out and where C*(B) ! C*(UB; B) is a cofi- bration. Therefore, C*(UB; E) is a homotopy push out of C*(f) and of the _____ augmentation of C*(B). |QED_| Proposition 8.7 (particular case of [6] 4.13) Suppose given a commutative diagram of augmented CDGA's f A _________B- | | | | |' |' | | | | |? f0 |? A0 ________-B0 where the vertical arrows are quasi-isomorphisms. Consider two factoriza- tions f = O i, f0 = 0O i0 where i : A ae C is a morphism of augmented CDGA's such that C is an A-semifree module, i0: A0ae C0 is a morphism of augmented CDGA's such that C0 is an A0-semifree module and : C !' B, 0 : C0 !' B0 are quasi-isomorphisms of augmented CDGA's. Then the cofibers _ A C and _ A0 C0 are weakly CDGA equivalent. Proof. By Property 8.2, we have the commutative diagram of augmented CDGA's i _________- A _________C- ' B | | | | | | ' | ' | |' | | | | | | |? _ |? __ |? i pppppp- A0 _____-A0A C ' B0 32 _ __ _ where i is a cofibration and f0 = O i. So the CDGA _ A0 A0A C is a homotopy cofiber of f0 as the CDGA _ A0 C0. Therefore they are weakly _____ CDGA equivalent. |QED_| 9 The fiber of the model in the Anick range Let r 1 be a fixed integer. p is going to be the least noninvertible prime (or +1) in _. We suppose now p 6= 2. Definition 9.1 [14] A topological space X is (r; p)-mild or in the Anick range if it is r-connected and its homology is concentrated in degrees rp and of finite type. Theorem 9.2 Let f : E ! B be a continuous map between two topological spaces both (r; p)-mild with Hrp(f) injective. Consider the homotopy fiber F and the fibration p0 : F i E. Then there are two morphisms of augmented CDGA's, denoted A(f) : A(B) ! A(E) and A(p0) : A(E) ! A(F ) such that 1. there is a commutative diagram of cochain complexes C*(f) C*(p0) C*(B) ____-C*(E) ____-C*(F ) | | | | | | |' |' |' | | | | | | |? |? |? D1(B) ____-D1(E) ____-D1(F ) | | | |6 |6 |6 |' |' |' | | | | | | | | | D2(B) ____-D2(E) ____-D2(F ) | | | | | | |' |' |' | | | | | | |? A(f) |? A(p0) |? A(B) _____-A(E) _____-A(F ) where all the vertical maps are quasi-isomorphisms and where all the maps are DGA morphisms except : D2(F ) !' A(F ) who induces a morphism of graded algebras only in homology. 33 2. for any factorization A(f) = O i where i : A(B) ae C is a morphism of augmented CDGA's such that C is an A(B)-semifree module and where : C !' A(E) is a quasi-isomorphism of augmented CDGA's, we have a commutative diagram of augmented CDGA's A(f) A(p0) A(B) _____-A(E) _____-A(F ) @ ||6 ||6 @ |' |' i@ || || @@R | | C ________D3- @ || @ |' @ || @@R |? _ A(B) C In particular, the cohomology algebra of the homotopy fiber of f, H*(F ), is isomorphic to the cohomology of the homotopy cofiber of A(f), H*(_A(B) C). Remark 9.3 Over Q, the functor APL due to Sullivan [22] is such that the two CDGA morphisms APL (f) : APL (B) ! APL (E) and APL (p0) : APL (E) ! APL (F ) verifies 1 and 2: by Corollary 10.10 of [10], for any topological space X, there are natural quasi-isomorphisms of cochain algebras C*(X) '! D(X) ' APL (X) and by the Grivel-Thomas-Halperin theorem "the fiber of a model is a model of the fiber" ([10], 15.5), _ A(B) C is weakly CDGA-equivalent to APL (F ). 34 Proof. By naturality of Proposition 3.10 with respect to continuous maps, we have a commutative diagram of DGC's C*(p0) C*(f) C*(F ) _________________C*(E)- ________________-C*(B) | | | |6 |6 |6 |' |' |' | | | | | | | | | G(F ) _________________-G(E) _________________-G(B) (9.4) | | | | | | |' |' |' | | | | | | |? |? BC*(f) |? B(C*(B); C*(E)) ________-BC*(E) _____________-BC*(B) There is a commutative diagram of augmented DGA's ' ___-' T X _____C*(E)- C*(E) | | | | | | m(f)| |C (f) |C (f) | | * | * | | | |? |? |? ' ___-' T Y _____C*(B)- C*(B) where denotes the cobar construction ([8] Theorem I), T X is a minimal free DGA and m(f) : T X ae T Y is a minimal free extension. Since the indecomposables functor Q preserves quasi-isomorphism between free DGA's ([5] 1.5), X ~=s-1H+ (E) and Y ~= s-1H+ (E) s-1cokerH+ (f) kerH+ (f): So X and Y are graded vector spaces of finite type concentrated in degree r and rp - 1. Denote by X the composite T X !' C*(E) '! C*(E) and by Y the composite T Y '! C*(B) !' C*(B). By Theorem 6.2, m(f) : T X ae T Y is a morphism of explicit HAH's and B(Y ; X ) : B(T Y ; T X) '! B(C*(E); C*(B)) is a morphism of coalgebras up to homotopy. By the naturality of Anick's Theorem ([19] D.29 and D.21, see also the proof of Theorem 8.5(g)[2]), there exists a DGL morphism L(f) : L(E) ! 35 L(B) and a commutative diagram of DGA's ~= UL(E) _____-T'X | | | | UL(f)| |m(f) | | | | |? ~ |? UL(B) _____-T=Y where ' and are two DGA isomorphisms equipped with two DGA homo- topies htop: (' ')UL(E) t TX ' and hbottom: ( )UL(B) t TY such that hbottomI(UL(f)) = (m(f) m(f))htop (the horizontal arrows commute with the diagonals up to natural homotopies). ~= By Lemma 5.2(ii), the isomorphism B(; ') : B(UL(B); UL(E)) ! B(T Y ; T X) commutes up to chain homotopy with the diagonals. We give C*(UL(B); L(E)) the tensor product coalgebra structure of UL(B) sL(E). The injection C*(UL(B); L(E)) !' B(UL(B); UL(E)) is a DGC quasi-isomorphism ([9] 6.11). By functoriality of the bar construction and the Cartan-Chevalley- Eilenberg complex with coefficients, finally we get the commutative diagram of coalgebras up to homotopy BC*(f) B(C*(B); C*(E)) ________-BC*(E) _____________-BC*(B) | | | |6 |6 |6 B( ; ) |' B( ) |' B( ) |' Y X | X | Y | | | | | | B(m(f)) | B(T Y ; T X)_____________-B(T X) _______________-B(T Y ) | | | |6 |6 |6 B(;')|~ B(') |~ B() |~ |= |= |= (9.5) | | | | | B(UL(f)) | B(UL(B); UL(E)) ________-B(UL(E)) ____________-B(UL(B)) | | | |6 |6 |6 |' |' |' | | | | | | | | C*L(f) | C*(UL(B); L(E)) __________-C*L(E) _______________-C*L(B) 36 where all the coalgebras up to homotopy are counitary and coassociative exactly except B(T Y ; T X), all the morphisms commute exactly with the diagonals except B(Y ; X ) and B(; '), and where all the vertical maps are quasi-isomorphisms. Define A(f) to be C*L(f) : C*L(B) ! C*L(E) and A(p0) to be the inclusion C*L(E) ,! C*(UL(B); L(E). By dualizing diagram 9.4 and diagram 9.5, we obtain the diagram of 1. By its definition, the CDGA _ A(B) C is a homotopy cofiber of A(f) (Definition 8.5). The CDGA A(F ) := C*(UL(B); L(E)) is also a homotopy cofiber of A(f) := C*(L(f)) (Example 8.6). So A(F ) is weakly CDGA- equivalent to _A(B)C. More precisely diagram 8.4 in the proof of Definition _____ 8.5 gives the diagram of 2 with D3 = C*(UL(B); L(B)) C*L(B)C. |QED_| To construct a factorization of A(f) is quite difficult. As in the rational case, we would rather construct a factorization of a model of A(f): Corollary 9.6 o Let A(f) : A(B) ! A(E) be a CDGA morphism as in Theorem 9.2. Let Y be a Sullivan model of A(B), X a Sullivan model of A(E). Then there is an acyclic CDGA U and a commutative diagram of CDGA's ' Y ______A(B)- || || | |A(f) || || |? |? ' ____-' X _____XoeU A(E) o Let Y ae C !' X be a factorization of : Y ! X such that C is a Y -semifree module. Then the algebra H*(F ) is isomorphic to H*(_Y C). (This isomorphism identifies in homology C*(p0) : C*(E) ! C*(F ) and the quotient map C i _ Y C.) Proof. The first part of this Corollary is just Proposition 7.7 and Remark _____ 7.8 of [14]. The second part is Proposition 8.7 and Theorem 9.2. |QED_| As in the rational case, we can take a factorization of with relative Sullivan models. But mod p, since the pth power of an element of even degree is always a cycle, our relative Sullivan model will have infinitely many generators. We'd rather use a free divided powers algebra V where for v 2 Veven, vp = 0. But now arises the problem of constructing CDGA 37 morphisms from a free divided power algebra to any CDGA where the pth powers are not zero. We give now an effective construction of a factorization of with divided powers algebras. Over Q, this factorization will be just a factorization of through a minimal relative Sullivan model. Lemma 9.7 Let : (Y; d) ! (X; d) be a CDGA morphism between two minimal Sullivan models such that X and Y are concentrated in degree 2. Then there is an explicit factorization of : i ' (Y; d) ae (Y coker' sker'; D) i (X; d) p where o ' is the composite Y ,! Y ! X i X and D is a -derivation, o i is a minimal inclusion of augmented CDGA's such that (Y coker' sker'; D) is (Y; d)-semifree, and o p is a surjective CDGA quasi-isomorphism vanishing on sker'. Proof. We proceed by induction on the degree n 2 N*. Suppose we have constructed the factorization: n n n n ' n (Y ) ; d ae (Y ) (coker' ) s(ker' ); D i (X ); d pn Let w 2 coker'n+1. Define pn+1(w) in obvious way. dpn+1(w) is a cycle of Xn . Since pn is a surjective quasi-isomorphism, there is a cycle z 2 (Y n ) (coker'n ) s(ker'n ) such that pn(z) = dpn+1(w). Define Dw = z. Let v 2 ker'n+1. Since pn+1 is a surjective morphism of graded algebras, there is u 2 2 (Y n coker'n ) such that pn+1(v + u) = 0. Since D(v + u) is a cycle of (Y n ) (coker'n ) s(ker'n ) and pn is a surjective quasi- isomorphism, there is fl 2 (Y n ) (coker'n ) s(ker'n ) such that pn(fl) = 0 and Dfl = D(v + u). Define Dsv = v + u - fl. 38 Now we have the commutative diagram of CDGA's: pn (Y n ) (coker'n ) s(ker'n ); D ___________(Xn-'); d | | | | | | | | | | |? pn+1 |? (Y n+1 ) (coker'n+1 ) s(ker'n+1 ); D ______-(Xn+1 ); d | | | | | | | | | | |? __ ____pn+1 |? (Y n+1) (coker'n+1) s(ker'n+1); D _________-(Xn+1'); 0 Since pn and _____pn+1are quasi-isomorphisms, by comparison of the E2-term * *of the algebraic Serre spectral sequence associated to each column, pn+1 is a _____ quasi-isomorphism. |QED_| Example 9.8 Let f : S2 ,! CP n be the inclusion of CW-complexes with n 2. Applying Corollary 9.6, is ((x2; y2n+1); d) ! ((x2; z3); d) with dy2n+1 = xn+12and dz3 = x22. By Lemma 9.7, factorises through the CDGA ((x2; y2n+1; z3) sy2n+1; D) with Dz3 = x22and Dsy2n+1 = y2n+1- z3xn-12. So H*(F ) ~=z3 sy2n+1 for p 2n. Remark 9.9 The hypotheses of the Theorem 9.2 are necessary: o B must be (r; p)-mild. Indeed even for a path fibration X ,! P X i X, a commutative model of X does not determine the cohomology algebra of the loop space. CP p and S3 _ :: _ S2p+1 just not (2; p)-mild, have a sa* *me commutative model but the cohomology algebras of their loop spaces are not isomorphic. o E and B both (r; p)-mild is not enough: Hrp(f) must also be injective. Take the same example as in Remark 7.5: the suspension of the Hopf map f : S2p-1 ! CP p-1. Remark 9.10 Over Q, replacing A by APL , the Grivel-Thomas-Halperin theorem implies that the CDGA _ A(B) C is weakly DGA equivalent to C*(F ) (Remark 9.3). But over a field of characteristic p, we can't improve Theorem 9.2, by _ A(B) C ~ C*(F ) as DGA's. Indeed, let X be the 2p + 3 skeleton of a K(Z; 4), X is (3; p)-mild and C*(X) is not weakly DGA equiv- alent to a CDGA. 39 Proof. A consequence of Milnor is that there exist two CW-complexes de- noted Y and K(Z; 3) with the same 2p + 2 skeleton, respectively homotopic to X and K(Z; 4). The two morphisms of topological monoids (Y (2p+2)) ! Y and (K(Z; 3)(2p+2)) ! K(Z; 3) induce in homology two algebra morphisms which are isomorphisms in de- gree 2p. Since H*(K(Z; 3)) ~= ff2 as algebras, Y is 1-connected, H2(Y ) = Fpff2 and ffp2= 0. Suppose C*(Y ) is weakly DGA equivalent to a commutative chain algebra A. We can suppose that A is of finite type. The Quillen construction [10, x22 e) and x23 a)] on the coalgebra A_ is a DGL LA equipped with a DGA quasi-isomorphism ULA := (A_) !' C*(Y ). The homology of an universal enveloping algebra of a DGL, ULA is an universal enveloping algebra of a Lie algebra, UE ([14] 8.3). So H*(Y ) admits by the _____ Poincare-Birkoff-Witt Theorem a basis containing ffp26= 0. |QED_| 10 Divided powers algebras The key to the proof of Theorem 9.2 is to apply Anick's Theorem ([2] 5.6). One of the goal of Anick for developing this theorem was to prove a result suggested by McGibbon and Wilkerson "If X is a finite simply-connected CW-complex then for large primes, pthpowers vanish in "H*(X; Fp)." ([17], p. 699). Anick proved precisely that "If X is (r; p)-mild then pthpowers van- ish in "H*(X; Fp)." ([2] 9.1). After Anick, Halperin proved in [14] (Theorem 8.3 and Poincare-Birkoff-Witt Theorem) that in fact: Corollary 10.1 [14] If X is (r; p)-mild then the algebra H*(X) is isomor- phic to sV where V is a minimal Sullivan model of A(X). Proof. Apply Corollary 9.6 to * ! X and see that the homotopy cofiber of (V; d) i (_; 0) given by Lemma 9.7, (_; 0) (V;d)(V sV; D) has a null _____ differential ([14] 2.6). |QED_| Actually, we can show now that Anick's result on pthpowers and Halperin's result on a divided powers algebra structure remain valid if we consider the fiber of any fibration in the Anick range instead of just the loop fibration. But before we need the notion of an admissible CGDA and of a -admissible CGDA. 40 Definition 10.2 A CDGA (respectively -algebra) A is admissible (respec- tively -admissible) if there is a surjective CDGA morphism (respectively -morphism) C i A with C acyclic. Property 10.3 ([15] II.2.6) Let f : A ! B a CDGA morphism (respec- tively -morphism). If f is surjective and A is admissible (respectively - admissible) then so is B. Proposition 10.4 ([15] II.2.7) (i) If f : A ! B is a CGDA morphism with B admissible then we have the commutative diagram of CDGA's A _____________________-B | H H | || H H ||6 | H H |' | H H | |? Hj | A V 0 _______________AoeV' where A ae A V is a relative Sullivan model and A ae A V 0is a -free extension. (ii)In particular, if B is any admissible CDGA, there are CDGA quasi- isomorphisms V 0-' V - '! B where V 0is a -algebra. The essential role of -admissible algebras is that Property 10.5 ([3] 1.3) If A is a -admissible algebra then H(A) is a - algebra (not true if A was only a -algebra!). Lemma 10.6 Let A be a cochain commutative algebra. Assume that for some r 1, A satisfies A = _ {Ai}ir . (i) If Hi(A) = 0; i rp, then A is admissible. (ii)If A is a -algebra and Hi(A) = 0; i rp+p-1, then A is -admissible. 41 Proof. (i) This lemma is just a slight improvement from Lemma 7.6 [14] __odd and the proof is the same: For each a 2 A , construct an obvious_CDGA even morphism oea from the acyclic CDGA (_a _da) to A. For each a 2 A , the cohomology class of lowest degree in (_a _da) is represented by ap. Extend this CDGA to an acyclic Sullivan model of the form (_a _da V ) where V is a graded vector space concentrated in degree rp - 1. Construct a CDGA morphism oea : (_a _da V ) ! A. Now a2A+ oea is a surjective morphism from an acyclic_CDGA to A. odd (ii) For each a 2 A , the cohomology class of lowest degree in the - algebra (_a _da) is represented by flp-1(da)a. After replacing by , the _____ proof is the same as in (i). |QED_| Lemma 10.7 Let A and M be two cochain commutative algebras concen- trated in degrees r + 1. Consider a CDGA morphism A ! M. If Hrp+p (A) = Hrp+p-1 (M) = 0 then TorA (M; _) is a divided powers algebra. Proof. By Lemma 10.6 (i), A and M are admissible. By Proposition 10.4 (ii), there are CDGA quasi-isomorphisms X0 -' X -'! A where X and X0 are concentrated in degrees r + 1. By Proposition 10.4 (i), we get the commutative diagram of CDGA's A ________-M | | |6 |6 ' | |' | | | | | | X ____X- Y @ || @ |' @ || @@R |? X Y 0 42 where Y and Y 0are concentrated in degrees r. By push-out, we have the commutative diagram of CDGA's X ________________X- Y 0 | | | | ' | |' | | | | |? |? X0 _______________-X0 Y 0 where X Y 0-'! X0 Y 0is a CDGA quasi-isomorphism ([9] 2.3(i)) since X Y 0is X-semifree (Property 2.4(iii)). Since push-outs preserve -free extension, X0 ae X0 Y 0is a -free extension. So X0 Y 0is X0-semifree, and by Proposition 8.7, the cohomology algebra of the cofiber Y 0is TorA (M; _). Now since X0 is a -algebra, so is X0 Y 0. Since X0 Y 0is concentrated in degrees r and its cohomology is null in degrees rp + p - 1, by Lemma 10.6(ii), X0 Y 0is -admissible. Since X0 Y 0i Y 0is a surjective -morphism, by Property 10.3, Y 0is a _____ -admissible. So by Property 10.5, H(Y 0) is a -algebra. |QED_| Theorem 10.8 Let p be an odd prime and let f : E i B be a fibration of fiber F such that E and B are both (r; p)-mild with Hrp(f) injective. Then the cohomology algebra H*(F ; Fp) is a (not necessarily free!) divided powers al- gebra. In particular, pth powers vanish in the reduced cohomology H"*(F ; Fp). Proof. By Theorem 9.2, H*(F ; _) ~= TorA(B)(A(E). 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