ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 7, Number 2, 1977 APPROXIMATE FIBRATIONS D. S. CORAM AND P. F. DUVALL, JR.l 1. Introduction and basic lemmas. The several concepts of a fibra- tion have been important tools in the study of maps. Thus, the de­ fining property, called the homotopy lifting property, is a valuable property for a map to have. The works of Lacher [ 11], [ 12], and also Armentrout and Price [ 1], and Kozlowski [ 10], suggest that an ap­ proximate homotopy lifting property might be almost as valuable while applying to a larger class of maps. This paper intends to show that this is indeed the case. In particular, we prove analogues to the follow­ ing theorems about fib rations: the existence of a path lifting function (Proposition 1.3), the property that point inverses are absolute neigh­ borhood retracts (Corollary 2.5), the homotopy equivalence of point inverses (Theorem 2.12), and the exact homotopy sequence of a fibra- tion (Corollary 3.5). Our conclusions, of course, are weaker in that they give shape theoretic, rather than homotopy theoretic, information. We use the following terminology and notation. If A C X, a topologi­ cal space, a neighborhood of A is a set containing A in its interior; Int A denotes the interior of A; CI A denotes the closure of A; Fr A denotes the frontier, or topological boundary, of A. On the other hand, the boundary of a topological manifold M is denoted Bd M. A map is a continuous function. For any positive integer q, Iq is the q-fold product of intervals; I1 = I; and other intervals are denoted by [a, b]. If H : X X I —» Y is a homotopy, then Ht : X —» Y is the map defined by Ht(x) = H(x, t). For all metric spaces, d(x, y) is the distance between points x and y, and N(x,e) = {y | d(x, y) < €}. Let /: X —> Y and g : X —» Y be maps and 8 be a cover of Y. We say that / and g are 8-close if for each x G X f(x) and g(x) are contained in some member of 8. Also / and g are 8-homotopic iff and g are homotopic by a ho­ motopy h such that h({x} X I) is contained in some member of 8 for each x G X. Such a homotopy is called a 8-homotopy. If ô is a positive number,/and g are 8-close (8-homotopic) if they are close (homotopic) relative to the cover {N(y, 8/2) \y G Y}. If Y is an ANR (that is, an absolute neighborhood retract for metric spaces), then for every cover € of Y there is a cover 8 such that any two 8-close maps from a metric Received by the editors on May 15, 1975, and in revised form on February 7, 1976. Research partially supported by the College of Arts and Sciences, Office of Research and Graduate Studies, Oklahoma State University, and by NSF Grant MPS 75-07084. Copyright © 1977 Rocky Mountain Mathematics Consortium 275 276 D. S. CORAM AND P. F. DUVALL space into Y are €-nomotopic by a homotopy which is fixed on the set where /= g [9]. The symbol = stands for isomorphism or homeomor- phism depending on the context; — stands for homotopy. The homot­ opy class of a map is denoted [f]. The 9-th homotopy group of a based space or pair of spaces is denoted irq. DEFINITION. A surjective map p : E-+ B between metric spaces has the approximate homotopy lifting property with respect to a space X provided that, given a cover e of B and maps g : X —» E and H : X X I —» B such that pg = H0, there exists a map G : X X / -» E such that G0 = g and pG and H are €-close. The map G is said to be an e-lift of H. If € is a number, we define an e-lift as above. Furthermore, G is stationary with H if for each x0EX such that H(x0, i) is constant as a function of t, the function G(x0, t) of t is constant also. The map p : E —» B is said to have the regular approximate homotopy lifting property with respect to X if the €-lift G can always be chosen to be stationary with H. The above definition, of course, generalizes the usual homotopy lifting property, the definition of which is the same except that pG = H is required rather than that pG, H be e-close. Thus, the approximate homotopy lifting property holds for a larger set of maps. However, we should point out that, although shape theory is used in the latter parts of this paper, this generalization of the homotopy lifting property is probably not an appropriate generalization to the shape category. The reason for this, roughly stated, is that two maps in the shape category are "similar" if they are merely homotopic in some set, whereas we require that they be close. This paper shows that much of the theory of Hurewicz fibrations (maps with the homotopy lifting property with respect to all spaces) carries over, little changed, to this larger set of maps. We will also make some reference to the theory of Serre, or weak, fibrations (maps with the homotopy lifting property with respect to all n-cells, or equivalently, all polyhedra). The first proposi­ tion provides a tool for constructing examples of maps which satisfy the approximate homotopy lifting property with respect to metric spaces, but which do not have the homotopy lifting property for some space. PROPOSITION 1.1. Let E and B be ANR's. Suppose p : E —» B is a surjection with the property that for each cover 8 of B there is a map p8: E —> B such that p8 is 8-close to p and p8 has the homotopy lifting property with respect to a metric space X. Then p has the approximate homotopy lifting property with respect to X. APPROXIMATE FIBRATIONS 277 PROOF. Given €, g, H as in the definition, choose 8 such that any two ô-close maps into B are -^-nomotopic, where r) twice star refines e. Let p8 be the map given by the hypothesis, and let K : X X [ — 1,0] —» B be an rj-homotopy such that K(x, — 1) = psg(x) and K(x,0) = pg(x). Define a homotopy * by *(x, *) = K(x, t)ift^ 0, and <I>(x, i) = H(x, f) if £ è 0. There exists a homotopy ifß : X X [ — 1,1] —> E such that i/r(x, — 1) = g(x) and p6^ = 4>. Now choose a map q : X —» (0,1) such that H({x} X [0, q(x)] ) is contained in some member of 17 for each x. Define f *(x,2ty(x)-l), tfOStScWÄ G(x, 0 = < ^(x, 2t - q(x)), if q(x)l2 ^ * ^ </(x), I *(x,f), ifç(x)gtSl. Then G(x, 0) = *(x, - IH g(*)- If 0 ^ *^ qf(x)/2, then t/ = 2tlq(x) - 1 G [ — 1,0], so pG(x, t) = pi/f(x, y). Then pG(x, £) is 17-close to pôt/f(x, y) which is 17-close to <ï>(x, 0) = H(x, 0) which is in turn 17-close to H(x, t) by our choice ofq(x). Since 17 twice star refines €, pG(x, f) is e-close to H(x, £). The case q(x)l2 §f§ g(x) is verified similarly. A slight change in the above proof also gives PROPOSITION 1.2. Let p : E —» B be a map between ANR's. Ifp has the approximate homotopy lifting property with respect to a space X, then for any cover e of B there is a cover 8 of B such that whenever g : X —• E and H : X X / -» B are maps such that g is a 8-lift of H0, there is a map G : X X I -+ E such that G0 = g and G is an e-lift ofH. 2 EXAMPLE. Let W be the "Warsaw circle" in R ; that is, W = WY U B, where Wl = {(0, f)|-l^^l}U {(x, Simrlx)} and B is an arc which meets Wx only in its endpoints (0,0) and (1,0). Let x0 be a base point in the 1-sphere S1, and let IT : S1 X S1 —> S1 be the projection map onto the second factor. Clearly there is a compactum A C S1 X S1 such that A is homeomorphic to W and such that there is a homeo- morphism 1 1 1 1 h : (S X S ) - A -» S X (S - {x0}). Then the map p:S]X S^S1 given by l 1 rnh(x)9 xŒ(S X S )- A (X)= P ix0, xGA l is continuous, and p has the property that p~ (x0) = A and p~\y) is an l l 1 essential copy of S in S X S for each y ^ x0. 278 D. S. CORAM AND P. F. DUVALL We claim that p can be uniformly approximated by fibrations. To see this, let e > 0 be given and let U be an open interval in S1 such 1 l that x0 G U C N(x0, (l/2)e). Then, since p-^S - U) and p~ (Cl U) are homeomorphic to S1 X I, it is easy to see that p | p_1(S1 — U) ex­ 1 1 1 tends to a map p€ : S X S —> S which is topologically equivalent to 77.