Adjunction of n-equivalences and triad connectivity

Publicacions Matem`atiques, Vol 39 (1995), 367–377. ADJUNCTION OF n-EQUIVALENCES AND TRIAD CONNECTIVITY P. J. Witbooi Abstract We prove a new adjunction theorem for n-equivalences. This theorem enables us to produce a simple geometric version of proof of the triad connectivity theorem of Blakers and Massey. An important intermediate step is a study of the collapsing map S∨X→S, Sbeing a sphere. Since the invention of the concept of quasifibration by Dold and Thom, their basic globalization theorem [7, Satz 2.2] has been applied widely. Some of the applications make use of an adjunction theorem of Hardie [9, Theorem 0.2]. This adjunction theorem, when applicable, is more convenient to use than the original theorem of Dold and Thom. The work [10] of May provides a new approach to quasifibrations. In Part I of this article we prove an adjunction theorem, Theorem 7, which merges the work of Hardie and that of May. As an application of this theorem, in Part II we provide a quite straightforward geometric proof of the triad connectivity theorem of Blakers and Massey [4]. Following the original proof due to Blakers and Massey, several alternative proofs have emerged. Moore [11] gives a proof based on the Serre spectral sequence of a fibration, and Namioka’s proof [12] uses the Hurewicz isomorphism theorem. There are several other versions of the proof such as in [6],[8], and [14], which do not make use of homology. In these three citations the proofs follow by geometric arguments. Using algebraic methods of a completely different nature, Brown and Loday [3] give yet another proof of the triad connectivity theorem (in fact they prove a stronger result, the homotopy excision theorem of [5]). In cases where homology is not used in the arguments, we can deduce simple alternative proofs of the Freudenthal suspension theorem and the Hurewicz isomorphism theorems as shown in [8] and [14]. A special case of the homotopy excision theorem follows from Theorem 13, and the more general homotopy excision theorem can be deduced 368 P. J. Witbooi from our result by making use of CW-approximation. We do not include this latter step here since the arguments are similar to those in [8]. Our method also supports proofs of the suspension theorem and the Hurewicz theorems. The merit of the proof of Theorem 13 is its simplicity. PART I: Adjunction of k-equivalences. Definition 1. Let kbe a positive integer and let p:(X, A)→(Y,B) be a map. (a) The map p:X→Yof spaces is said to be a 0-equivalence if it induces a surjective function on path components. For k>0, the map pis said to be a k-equivalence if it induces a bijection on path components, and if, for every x∈X, the function p∗: πr(X, x)→πr(Y,x) with x=p(x), is bijective whenever r<k and surjective for r=k. (b) The map p:(X, A)→(Y,B) of pairs of spaces is said to be a 0-equivalence if condition (1) below holds. If also condition (2) holds, then the map of pairs is said to be a k-equivalence (k>0). (1) Im[π0(A)→π0(X)] = p−1 ∗Im[π0(B)→π0(Y)]. (2) For every a∈A, and b=p(a), the function p∗:πr(X, A, a)→ πr(Y,B,b) is bijective whenever r<kand surjective for r= k. (c) A map of spaces or of pairs of spaces is said to be a weak equivalence if it is a k-equivalence for all k>0. Definition 1(a) appears in textbooks such as [8] or [14], while Definition 1(b) is due to May [10]. For Definition 1(b), the condition (1) on path components is automatically fulfilled if A=p−1(B). We note that the composition of two k-equivalences is again a k-equivalence. The proof of the following proposition is an easy exercise on the level of set theory and we omit it. Proposition 2. Suppose that we have a commutative triangle of maps of spaces as below. If αis a (k−1)-equivalence and δis a k-equivalence, then βis a k-equivalence. A α   δ B−−−−→ β C −−−−→ −−− By Top2we shall mean the category of which the objects are maps of spaces, that is to say we take the morphisms of Top as objects. The Triad connectivity 369 morphisms in Top2from an object qto an object p, is a pair of maps (g,f) such that f◦q=p◦g, see Diagram (A) below. We note that the mapping path fibration construction is an endofunctor of Top2. We observe also that, given a Top2-morphism (g,f):q→p, then for every point b∈B, there is an induced map of the homotopy fibre of qover binto the homotopy fibre of pover f(b). If every such induced map is a k-equivalence, then the Top2-morphism (g,f) is said to be a k-equivalence of homotopy fibres. If (g,f):q→pis a Top2-morphism, note also that there is an obvious map r:Zg→Zffrom the mapping cylinder of ginto the mapping cylinder of f. (A) Ag −−−−→X q      p B−−−−→ f Y Proposition 3. Consider the commutative Diagram (A). (a) The following conditions are equivalent. (1) The map r:(Zg,A)→(Zf,B)of mapping cylinders is a k-equivalence. (2) The Top2-morphism (g, f):q→pis a (k−1)-equivalence of homotopy fibres. (b) If the maps gand fare inclusions of subspaces, then the following condition, (3), is equivalent to condition (2) in (a) above: (3) The map p:(X, A)→(Y,B)is a k-equivalence. (c) Suppose that gand fare inclusion maps, and that A→Bis a k-equivalence. Then p:(X, A)→(Y,B)is a k-equivalence if and only if X→Yis a k-equivalence. Proof: (c) This follows easily by the five-lemma applied to the ladder formed by the homotopy sequences of the pairs (X, A) and (Y,B), together with the homomorphisms arising from the map (X, A)→(Y,B). (b) We consider the mapping path fibration factorization of pand q, respectively. Xp0 −−−−→Wp1 −−−−→YA q0 −−−−→Eq1 −−−−→B. In each case it is a homotopy equivalence followed by a fibration. The maps p0and q0are embeddings and we regard them as inclusions. Let 370 P. J. Witbooi F=q−1 1(∗), F=p−1 1(∗) and E=p−1 1(B). Then E⊂E,F⊂F, and q1(x)=p1(x) for every x∈E. We first assume that (3) holds. Then p1:(W, E)→(Y,B)isakequivalence. Since p1is a fibration, the map p1:(W, E)→(Y,B) is a weak equivalence. Thus the injection (W, E)→(W, E)isakequivalence. Similar to the argument for (c), it follows that the inclusion E→Eis a (k−1)-equivalence. The pull-back of p1over the inclusion B⊂Y, is (a fibration and thus) a weak equivalence (E,F)→(B,∗). Similarly, q1:(E,F)→(B,∗) is a weak equivalence. Thus the injection (E,F)→(E,F) is a weak equivalence. Since the inclusion E⊂Eis a (k−1)-equivalence, it follows that also the inclusion F⊂Fis a (k−1)-equivalence. Therefore condition (2) of (a) follows. This proves one of the implications claimed in (b). The other implication claimed in (b) can be proved by simply reversing the previous argument. (a) This follows by (b). The theorem which we quote without proof below, appears (in a stronger form) in a paper by May [10, Theorem 1.2], and results from a reworking of the fundamental theory of quasifibrations in the pioneering paper [7] by Dold and Thom. Theorem 4. Let Ybe a space with open subspaces B1and B2such that Y=B1∪B2.LetB0=B1∩B2.Letp:X→Ybe a map and let Ai=p−1(Bi),i=0,1,2. If for each j=1,2, the map (Aj,A 0)→(Bj,B 0)is a k-equivalence, then for each j=1,2, the map (X, Aj)→(Y,Bj)is a k-equivalence. From Theorem 4 we deduce an adjunction theorem, Theorem 7, generalizing the adjunction theorem for quasifibrations [9, Theorem 0.2] of Hardie. We consider the commutative Diagram (B) in Top to be a cotriad in the category Top2. (B) E1 g1 ←−−−−E0 g2 −−−−→E2 p1       p0    p2 B1←−−−− f1 B0−−−−→ f2 B2 The push-out of this Top2-cotriad is a map p:E→B, where Eand B are the spaces obtained as the push-outs (in Top) of the cotriads sitting in the top row and bottom row of Diagram (B). There is a similar map of double mapping cylinders. We denote this map by p:E→B, and Triad connectivity 371 refer to it as the double mapping cylinder of the Top2-cotriad. Firstly we note the following fact regarding the natural map B→Bbetween the double mapping cylinder and the push-out of a Top-cotriad, (C) B1 f1 ←−−−−B0 f2 −−−−→B2. Proposition 5 is a weaker form of the result [2, 7.5.4 on p. 275] in the book of Brown. A proof in an axiomatic setting appears in Baues’s book [1]. We state it without proof. Proposition 5. If in the Top-cotriad of Diagram (C), f1is a cofibration, then the natural map B→Bis a homotopy equivalence. The homotopy fibres approach to the study of Top2-cotriads can be observed in the work of Puppe [13]. Theorem 6 supplements and generalizes Puppe’s work. Theorem 6. Suppose that in Diagram (B), for each j=1,2, the Top2-morphism (gj,f j)is a k-equivalence of homotopy fibres, and g1 and f1are cofibrations. Then for each j=1,2, the Top2-morphism pi→pto the map of double mapping cylinders is a k-equivalence of homotopy fibres. Proof: The double mapping cylinder Bhas open subsets V0,V1and V2satisfying the following three conditions. (1) V0=V1∩V2and V1∪V2=B. (2) For the pull-back qi:Ui→Viof pover the inclusion Vi⊂B, there are homotopy equivalences i:Ui→Eiand βi:Vi→Bi, such that for each i=0,1,2, pi◦ i=βi◦qi. (3) The Top2-morphisms ( i,β i), resulting from (2) above, fits into a commutative diagram in Top2as shown below. (D) q1←−−−−q0−−−−→q2 (1,B1)  (0,β0)  (2,B2)   p1←−−−−p0−−−−→p2 Due to the homotopy equivalences of (2) and the conditions of the theorem, it follows that for each j=1,2,the Top2-morphism q0→qj is a k-equivalence of homotopy fibres. By Proposition 3(a) then, each map (Uj,U 0)→(Vj,V 0)isa(k+ 1)-equivalence. Thus by Theorem 4, each map (E,U j)→(B,V j)isa(k+1)-equivalence. Again by Proposition 3(a), each Top2-morphism qj→pis a k-equivalence of homotopy fibres. 372 P. J. Witbooi Theorem 7. Suppose that for the commutative Diagram (B) we have a subset Tof B0such that the inclusion T⊂B0is a surjection of path components. Suppose that for each x∈T, there are subsets Fx 0⊂p0 −1(x)and Fx j⊂pj −1(fj(x)). We assume that Fx j=Fy jwhenever fj(x)=fj(y). It is also assumed that for each j=1,2,gj(Fx 0)⊂Fx jso that there is an induced map hx j:Fx 0→Fx j. Suppose further that for each x∈S, the following conditions hold (k≥0): (1) p0:(E0,Fx 0)→(B0,x)is a k-equivalence, (2) pj:(Ej,Fx j)→(Bj,f j(x)) is a (k+1)-equivalence for each j= 1,2, (3) hx j:Fx 0→Fx jis a k-equivalence for each j=1,2. Then for each j=1,2, the map of double mapping cylinders is a (k+1)- equivalence, p:(E,E j)→(B,B j). Proof: Fix any x∈Sand j∈{1,2}. In Diagram (E) below, Hx 0and Hx jare, respectively, the fibres of the mapping path fibration of p0and pjover xand fj(x). The vertical arrows are inclusions. The map β is the induced map due to functoriality of the mapping path fibration construction. (E) Fx 0 hx j −−−−→Fx j α  α    Hx 0−−−−→ β Hx j Due to condition (1), αisa(k−1)-equivalence and due to (2), α is a k-equivalence. In view of condition (3), it follows that α◦hx jis a k-equivalence. We put δ=α◦hx jand apply Proposition 2, by which β is a k-equivalence. Since T⊂B0is a surjection of path components, it follows that the Top2-morphism (gj,f j)isak-equivalence of homotopy fibres (for both values of j). Thus by Theorem 6, for both values of j, the Top2-morphism pj→pis a k-equivalence of homotopy fibres. Our result follows from Proposition 3(b). Corollary 8. We assume the conditions of Theorem 7 together with the requirement that f1and g1are cofibrations. Then for each j=1,2, the map of push-outs is a (k+1)-equivalence p:(E,Ej)→(B,Bj). Triad connectivity 373 Proof: We deduce this result from Theorem 7 as follows. By Proposition 5, the canonical map E→Eis a homotopy equivalence, and hence a weak equivalence. By the five-lemma applied to the ladder formed by the homotopy sequences of the pairs (E,E j) and (E,Ej) and the homomorphisms arising from the map (E,E j)→(E,Ej), it follows that the map of pairs is a weak equivalence. Similarly, (B,B j)→(B,Bj) is a weak equivalence. The assertion now follows from Theorem 7. PART II: Relative homeomorphisms. In the sequel, we assume spaces to have a base point, denoted by the symbol ∗. The definition of k-equivalence remains as for free spaces. Definition 9. Let p:(X, A)→(Y,B) be a map of pairs of spaces. Then pinduces maps p1and p2as in Diagram (F). The map of pairs pis said to be a relative homeomorphism if Diagram (F) is a push-out square. & & (F) A−−−−→X p1       p2 B−−−−→Y Proposition 10. Suppose that f:V→Ais a map with mapping cone X, and p:(X, A)→(Y,B)is a relative homeomorphism. Suppose further that p0:A→Bis a k-equivalence (k>0). Then each of the maps (X, A)→(Y,B)and X→Yis a k-equivalence. Proof: In Diagram (G) below, qis the identity map and thus the diagram is commutative. The map of double mapping cylinders of this Top2-cotriad is precisely our map p. (G) ∗←−−−−Vf −−−−→A q      qp 0    ∗←−−−−V−−−−→ p0◦f B The Top2-morphism q→qis a weak equivalence of homotopy fibres. Since the homotopy fibres of p0are (k−1)-connected, q→p0is a (k−1)-equivalence of homotopy fibres. Thus by Theorem 7, the Top2morphism p0→pis a (k−1)-equivalence of homotopy fibres. From Proposition 3(b), it follows that (X, A)→(Y,B)isak-equivalence, and then from Proposition 3(c), X→Yis a k-equivalence. 374 P. J. Witbooi Lemma 11. Suppose that Gis a subspace of a space Fsuch that the inclusion G⊂Fis an m-equivalence (m>0).LetWnbe a bouquet of n-dimensional spheres, and let Cnbe the subset F×∗∪G×Wnof F×Wn. Then the restriction of the projection map F×Wn→Wn,isan(m+ n)-equivalence of pairs rn:(Cn,F ×∗)→(Wn,∗). Proof: We proceed by induction on n. The case n= 0 is obviously true. Now let us assume the statement to be true for all nsuch that 0≤n≤t−1, where t≥1, and show that it is also true for n=t. We apply Corollary 8 to Diagram (H) below. Vis the corresponding bouquet of the cones of the spheres of Wt−1. The map qis the restriction of the projection map F×Wt−1→Wt−1, and his the restriction of the projection map F×Wt−1→F. (H) F×∗∪G×V←−−−−F×∗∪G×Wt−1 h −−−−→F q   α      V←−−−−Wt−1−−−−→∗ * *−−−−−−−−−−−−−−− The diagram is commutative and the push-out of the Top2-cotriad is a map of the form rt:Ct→Wt. Note that there is a one to one correspondence between the spheres in Wt−1and those in Wt. In accordance with Theorem 7, we must choose a set T⊂Wt−1.If t= 1, then we choose T=Wt−1, otherwise we choose T={∗}. The subsets Fx irequired in Theorem 7 are chosen to be the complete inverse images (of the relevant point with respect to the relevant vertical arrow in Diagram (H)). For the horizontal arrows pointing to the left, every induced map between fibres is a homeomorphism. For the horizontal arrows pointing to the right, the induced map between fibres over ∗is a homeomorphism, and otherwise (only relevant in the case t=1)it is an m-equivalence. So, the conditions of Theorem 7 can be seen to be fulfilled for k=m+t−2, using the induction assumption. Thus the push-out of the Top2-cotriad is a (m+t−1)-equivalence. This completes the induction and hence the proof of the lemma. Proposition 12. Let p0:A→Bbe an m-equivalence (m>1).Let (X, A)be a relative CW-complex having cells of dimension nonly, and let p:(X, A)→(Y,B)be a relative homeomorphism. Then p:(X, A)→(Y,B)is an (m+n−1)-equivalence. Proof: We first prove the result assuming that p0:A→Bis a fibra- Triad connectivity 375 tion, and thereafter we deduce the general case. So let us assume that p0:A→Bis a fibration. Then the fibres of p0are (m−1)-connected. Let g:W→Abe the attaching map for the cells of X, where Wis a bouquet of (n−1)-spheres. The cone on Wis denoted by V. Let F=p−1 0(∗) be the fibre of p0over the base point of B. In Diagram (I), qand qare relative homeomorphisms which collapse the subspace F. The map ghas restrictions g|W=gand g|F is a homeomorphism onto the subspace Fof A. Thus Diagram (I) is commutative. (I) F∨V←−−−−F∨Wg −−−−→A q   q   p0    V←−−−−W−−−−→ g◦p0 B * * By Lemma 11, q:(W∨F, F)→(W, ∗)isan(m+n−2)-equivalence (we choose the space Grequired in Lemma 11 to be the one-point set {∗}). Furthermore, we have weak equivalences q:(F∨V,F)→(V,∗), and p0:(A, F )→(B,∗). By Corollary 8 it follows that p:(X, A)→(Y,B) is an (m+n−1)-equivalence, and the special case is proved. We now turn to the general case. For the mapping path fibration factorization of p0below, the inclusion is a homotopy equivalence and f is a fibration. A−−−−→Ef −−−−→B & This factorization induces relative homeomorphisms between relative CW-complexes, (X, A)h −−−−→(Z, E)f −−−−→(Y,B), and the composition of the two maps coincides with p. By Proposition 10, the map h:(X, A)→(Z, E) is a weak equivalence. From the special case of Proposition 12 that we have already proved, it follows that f: (Z, E)→(Y,B)isan(n+m−1)-equivalence. Thus f◦h:(X, A)→ (Y,B)isan(m+n−1)-equivalence. Theorem 13. Let p0:A→Bbe an m-equivalence, m>0, and (X, A)a relative CW-complex having only cells of dimension nand higher. Then the relative homeomorphism p:(X, A)→(Y,B)is an (m+n− 1)-equivalence.