DOLD THEOREMS IN SHAPE THEORY

Transcription

1 Volume 9, 1984 Pages DOLD THEOREMS IN SHAPE THEORY by Harold M. Hastngs and Mahendra Jan Topology Proceedngs Web: Mal: Topology Proceedngs Department of Mathematcs & Statstcs Auburn Unversty, Alabama 36849, USA E-mal: ISSN: COPYRIGHT c by Topology Proceedngs. All rghts reserved.

2 TOPOLOGY PROCEEDINGS Volume Research Announcement DOLD THEOREMS IN SHAPE THEORY 1 Harold M. Hastngs 2 and Mahendra Jan3 1. Introducton D. Coram and P. Duvall [5] ntroduced approxmate fbratons by generalzng the lftng property of celllke maps. S. Marde~c and T. B. Rushng further extended the concept of Hurewcz fbraton by generalzng approxmate fbratons for compact ANR's to shape fbratons for compacta. Ths appears to be the approprate concept of fbratons n the shape theory of compact metrc spaces [9,10,12,14,17]. The homotopy theory of fbratons of ANR's extends well beyond ther lftng property. In partcular, Dold [6] proved that a map of such fbratons over a common, compact, path-connected base s a fbered homotopy equvalence whenever ts restrcton to a sngle fber s a homotopy equvalence. Our purpose s to announce a Dold theorem and related results for shape fbratons. Detals and proofs wll appear n [9]. Recently H. Kato [12] proved strong versons of Dold's theorems. Kato used the T. A. Chapman's [4] complement theorem to defne strong shape. We shall gve drect geometrc proofs of these theorems usng closed model categores [19,7]. lbased on a talk "Equvalences of shape fbratons" by M. Jan. 2partally supported by NSF grant MCS partally supported by a research grant from Wllam Paterson College of New Jersey.

3 360 Hastngs and Jan 2. Prelmnares We shall use the followng categores: TOP, the usual category of topologcal spaces and contnuous maps; CM, the full subcategory of compact metrc spaces; PL, the subcategory of fnte polyhedra and pecewse lnear maps; TOP/B, the usual category of topologcal spaces and contnuous maps over B. For any category C, pro-c shall denote the category of nverse systems over C [1]. To each compact metrc space X, we assocate the category of fnte polyhedra under X, X ~ PL + PL J followng A. Calder and H. M. Hastngs [3]. Ths yelds a strong shape functor CM + pro-pl + pro-top. A map s a strong shape equvalence f t nduces an somorphsm n the strong pro-homotopy category Ho(pro-TOP) [7]. shape category s the category of fractons CM -1 shape equvalences). The strong (strong We recall the crtera of S. Mardesc and T. B. Rushng for a shape fbraton n terms of ts lftng propertes. Defnton [16]. A 1eve1wse map (P): {E } ~ {B } s sad to have the homotopy lftng property (HLP) f for each n there s an m ~ n, such that for any commutatve sold-arrow dagram the ndcated fller exsts. X x a j X x I h E m 1 _-- ~ H ) r ) E --~ )' B m ) B n

4 TOPOLOGY PROCEEDINGS Volume Theopem (Mardesc and Rushng [16]). A (contnuous) map p: E +B of compact metrc spaces s a shape fbraton f and only f for each representaton of B as the nverse lmt of a tower {B } of compact ANR's, there s a smlar representaton {E } of E, and a levelwse map {P}: {E } + {B } wth HLP. In partcular, the nverse lmt of a sequence of Hurewcz fbratons of compact ANR's s a shape fbraton. However, n general, we cannot requre that each map P: E + B be a Hurewcz fbraton wthout volatng the requrement of compactness. Thus shape fbratons are natural geometrc analog of pro-fbratons. In [11] M. Jan has proved Dold-theorems for shape fbratons under some addtonal movablty assumptons, usng shrnkable open covers. 3. Man Theorems We now defne fbered strong shape equvalence n the strong shape cateogry. All the spaces consdered are compact metrc spaces. Defnton 1. A map f: E + E ' of spaces over B s called a fbered strong shape equvalence f for each map C + B, the pullback of f over C s a strong shape equvalence (over C). Theorem 1. Let p: E + B and p: E' + B be shape fbratons, and let f: E + E' be a map over B and a strong shape equvalence. Then f s a fbered strong shape equvalence.

5 362 Hastngs and Jan Sketch of proof. By Mardesc and Rushng theorem of secton 2, we can assume that p and p are nverse lmts of maps of sequences E: ~ ~ Band 1: ~I ~ ~ respectvely, each wth the HLP. Wthout loss of generalty, we Can also assume that f s the nverse lmt of a strong prohomotopy equvalence!: ~ ~ E I We frst replace p and p by fbratons p and p n the sngular model structure on pro-top [3,7]. We then use propertes of fbratons n pro-top [3]. Corollary 1. A shape fbraton whch s albo a strong shape equvalence s a fbered strong shape equvalence. We now descrbe our versons of Dold's theorems. Let p: E ~ B and p: E ' ~ B be shape fbratons. Let f: E ~ E ' be a map over B. We shall show that f s a fbered strong shape equvalence under a varety of addtonal hypotheses. Theorem 2. Suppose that E~ E ' and B have fnte shape dmensons and all spaces and fbers are ponted contnua. If the restrcton of f to p-l(*) s a ponted shape equvalence then f s a (ponted) strong shape equvalence over B. Sketch of proof. Apply the fve-lemma to the longexact sequence of pro-homotopy groups [17] nduced by the shape fbratons p and pl. ~ pro-iin+l(b) -+ pro-ii (F) ~ pro-iin(e) ~ pro-iin(b) ~ pro"'iin(f)~ I n I ;; ~ -to -to pro-i1 + l {B) pro-iin (F I) -to pro-iin (E I ) -to pro-iin (B) pro-iin {F I)-to n f* -

6 TOPOLOGY PROCEEDINGS Volume By usng the results of [10] and by nducton, we prove the followng: Theorem 3. If B admts a fnte closed cover {B}~ = 1,2,,N such that for each ~ the pullback of f over B s a strong shape equvalence then f s a fbered strong shape equvalence. Combnng the theorem 3 wth the result [10,9] that the pullbacks of a shape fbraton va two strong shape equvalent maps are fbered shape equvalent, we prove that Theorem 4. Suppose B admts a fnte shrnkable closed cover, and for one pont {*} n each strong shape path component of B~ the pullback of f over {*} s a st~ong shape equvalence then f s a fbered strong shape equvalence. Corollary to Theorem 3. If B admts a fnte shrnkable closed aover~ then every CE-shape fbraton s a fbered strong shape equvalence. Note that there s an nverse sequence of fbratons E: ~ ~ B whose 'fber' s a pro-trval yet whose nverse lmt s not a shape equvalence. However B beng an nfnte product of spheres, t does not admt fnte shrnkable closed cover. 4. Open Problems Several nterestng and harder problems reman open. (I) Foremost s the classfcaton problem for shape

7 364 Hastngs and Jan fbratons. A key step along the way nvolves a glueng lemma, extendng Brown and Heath [2]. (II) The 'best' defnton of 'prncpal shape bundle' remans open. The defnton should be general enough to nclude CE maps but restrcted enough to permt a relatvely straghtforward proof of a classfcaton theorem for such maps. See [8] for classfcaton of open prncpal fbratons. (III) Fnally, the theory of coverng maps appears much rcher n geometrc content than that of Hurewcz fbratons. It should be very appealng to have a comparable theory of shape coverng maps. References 1. M. Artn and B. Mazur, Etale homotopy, Lecture Notes n Math. 100, Sprnger-Verlag, Berln-Hedelberg-New York (1969). 2. R. Brown and P. Heath, Coglueng homotopy equvalences, Math. Zeet. 113 (1970), A. Calder and H. M. Hastngs, Realzng strong shape equvalences, J. Pure Appl. Alg. 20 (1981), T. A. Chapman, On some applcatons of nfnte dmensonal topology to the theory of shape, Fund. Math. 76 (1972), D. S. Coram and P. F. Duvall, Approxmate fbratons, Rocky Mt. J. of Math. 7 (1977), A. Dold t Parttons of unty n the theory of fbratons, Annals of Math. (2) 78 (1963), D. A. Edwards and H. M. Hastngs, Cech and Steenrod homotopy theory wth applcatons to algebrac topology, Lecture Notes n Math. 542, Sprnger-Verlag, Berln Hedelberg-New York (1976). 8., Classfcatons of open prncpal fbratons, Trans. AMS, 240 (1978),

8 TOPOLOGY PROCEEDINGS Volume H. M. Hastngs and M. Jan, Equvalences of shape fbratons, G1asnk Mat. (to appear). 10. M. Jan, Induced shape fbratons and fber shape equvalences, Rocky Mt. J. of Math. 12 (1982), , Cell-lke shape fbratons whch are fber shape equvalences, Topology Proceedngs 7 (1982), H. Kato, Shape fbratons and fbre shape equvalences I, II, Tsukuba J. Math. 5 (1982), , Fbre shape categores, Tsukuba J. Math. 5 (1982), J. Keeslng and S. Mardesc, A shape fbraton wth dfferent shapes, Pacfc J. Math. (2) 84 (1979), S. Mardesc, On the Whtehead theorem n shape theory I, II, Fund. Math. 91 (1976), and T. B. Rushng, Shape fbratons I, General Topology and Apples. 9 (1978), , Shape fbratons II, Rocky Mt. J. Math. 9 (1979), M. Moszynska, The Whtehead theorem n the theory of shapes, Fund. Math. 80 (1973), D. G. Qullen, Homotopcal algebra, Lecture Notes n Math. 43, Sprnger-Verlag, Ber1n-Hede1berg-New York (1967). 20. L. Sebenmann, Infnte smple homotopy theory, Indag. Math. 32 (1970), Hofstra Unversty Hempstead, New York and Wllam Paterson College Wayne, New Jersey 07470