CORRECTING TAYLOR S CELL-LIKE MAP. Katsuro Sakai University of Tsukuba, Japan

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1 GLSNIK MTEMTIČKI Vol. 46(66)(2011), CORRECTING TYLOR S CELL-LIKE MP Katsuro Sakai University of Tsukuba, Japan bstract. J. L. Taylor constructed a cell-like map of a compactum X onto the Hilbert cube I N such that X is not cell-like. In this note, we point out a defect in the construction and show how to fix it. Usingdams examplein[1], J.L.Taylorconstructedin[7]acell-likemap of a compactum X onto the Hilbert cube I N such that X is not cell-like. This Taylor s example is very important in Shape Theory and its related theories. In fact, it was widely used by many authors for various counterexamples. In this note, we point out a defect in the construction and show how to fix it. In [1], J. F. dams constructed a compact polyhedron with a map : Σ r from the r-fold suspension of onto such that every composition Σ r Σ (i 1)r : Σ ir is not null-homotopic. Taylor defined the compactum X as the inverse limit of the inverse sequence: Σ r Σ r Σ 2r Σ 2r Then, the inverse limit projection of X to is not null-homotopic, which implies X is not cell-like. Observe that Σ ir is homeomorphic to ( ) the quotient space I ir /{ {z} z I ir }. The Hilbert cube I N can be regarded as the inverse limit of the sequence: I r I 2r I 3r, 2010 Mathematics Subject Classification. 54C10, 54C56, 55P40, 57N25. Key words and phrases. Taylor s cell-like map, Hilbert cube, n-fold suspension. 483

2 484 K. SKI where : I (i+1)r = I ir I r I ir is the projection. For each space Y and n N, the following definition is adopted: (1) Σ n Y = I n Y / {{z} Y z I n }. Let q n Y : In Y Σ n Y be the quotient map. s is easily observed, the projection pr I ir : I ir I ir induces the map f i : Σ ir I ir. Taylor s map f : X I N is defined by the following commutative diagram: Σ r Σ r Σ 2r Σ 2r Σ 3r Σ 3r ( ) f 1 f 2 f 3 I r I 2r I 3r In the above diagram ( ), the map Σ ir : Σ ir Σ r Σ ir is induced by the mad : I ir Σ r I ir, that is, the following diagramcommutes: I ir id I ir I ir Σ r Σ r Σ ir Σ ir Σ r. Σ ir It should be remarked that Σ ir Σ r Σ (i+1)r but Σ ir Σ r Σ (i+1)r with respect to our definition of n-fold suspension (1). In fact, observe Σ ir Σ r = I ir Σ r /{ {z} Σ r z I ir} = I ir I r /{ {z} I r, {y} z I ir,y (0,1) ir I r} Thus, the commutativity of the diagram ( ) depends on how to identify Σ ir Σ r with Σ (i+1)r. In the following diagram, the outside pentagon is commutative but we have to find a homeomorphism θ : Σ (i+1)r Σ ir Σ r making the bottom rectangle ( ) commutative: I ir I r I (i+1)r id q r id q r I ir id I ir I ir Σ r q (i+1)r pr I ir Σ ir f i Σ ir Σ r Σir Σr ( ) θ Σ (i+1)r f i+1 I ir I (i+1)r. pr I (i+1)r

3 CORRECTING TYLOR S CELL-LIKE MP 485 ssume that such a homeomorphism θ exists and take a point y I ir. Since Σ r ({y} Σr ) is a singleton in Σ ir Σ r, it follows that θ ( q (i+1)r ({(y,z)} ) ) = qσ ir ({y} Σ r ) for each z I r. r If z z (0,1) r then q (i+1)r ({(y,z)} ) and q (i+1)r ({(y,z )} ) are distinct singletonsin Σ (i+1)r. Thisisacontradictionbecauseθ isabijection. Therefore, we cannot identify Σ ir Σ r with Σ (i+1)r so that the diagram ( ) is commutative. In the rest of this note, we shall show how to fix this defect. To this end, we now adopt the following definition: (2) Σ n Y = B n Y /{ {z} Y z S n 1 }, where B n is the unit closed ball of R n and S n 1 (= B n ) is the unit sphere. Let q n Y : Bn Y Σ n Y be the quotient map. In this definition (2), we have Σ ir Σ r = B ir Σ r /{ {z} Σ r z S ir 1 } = B ir B r /{ {z} B r, {y} z S ir 1, y (B ir \S ir 1 ) S r 1}. Of course, Σ (i+1)r Σ ir Σ r. For each i N, let : B (i+1)r B ir be the restriction of the projection of R (i+1)r = R ir R r onto R ir, and define a map ϕ i : B ir B r B (i+1)r as follows: ϕ i (y,z) = { (y, 1 y 2 z) if z 0, (y,0) if z 0. Then, we have the following commutative diagram: B ir B r pr B ir ϕ i B ir B (i+1)r. For each z S ir 1, ϕ({z} B r ) is a singleton with ϕ 1 (ϕ({z} B r )) = {z} B r. The restriction ϕ i (B ir \S ir 1 ) B r is injective and ϕ 1 (S (i+1)r 1 ) = (S ir 1 B r )((B ir \S ir 1 ) S r 1 ).

4 486 K. SKI s is easily observed, ϕ i id induces the homeomorphism ϕ i : Σ ir Σ r Σ (i+1)r which makes the following diagram commutative: B ir B r id q r id q r ϕ i id B ir id B ir B ir Σ r B (i+1)r Σ r q (i+1)r pr B ir Σ ir Σ ir Σir Σr ϕ i Σ (i+1)r pr B (i+1)r f i f i+1 B ir B (i+1)r. Identifying each Σ ir Σ r with Σ (i+1)r by this homeomorphism ϕ i, we have the following commutative diagram: Σ r Σ r Σ 2r Σ 2r Σ 3r Σ 3r f 1 f 2 f 3 B r B 2r B 3r Let Y be the inverse limit of the bottom sequence above. Then, we can obtain the map f : X Y induced by maps f i, i N. Just as in the proof given in [7], it can be proved that f is a cell-like map. It should be noticed that Y can be regarded as an infinite-dimensional compact convex set in the Fréchet space 1 R N. Indeed, R N can be regarded the inverse limit of the top sequence in the following commutative diagram: R r R 2r R 3r B r B 2r B 3r, where : R (i+1)r = R ir R r R ir is the projection. Then, we can apply the classical result of Keller ([4]) to show Y I N. Indeed, every infinitedimensional compact convex set in a Fréchet space is affinely homeomorphic to aninfinite-dimensionalcompactconvexsetinhilbertspacel 2 ([2,ChapterIII, Proposition 3.1]), which is homeomorphic to the Hilbert cube I N by Keller s Theorem [4] (cf. [2, Chapter III, Theorem 3.1]). space. 1 completely metrizable locally convex topological linear space is called a Fréchet

5 CORRECTING TYLOR S CELL-LIKE MP 487 Remark 1. To show Y I N, we can also apply Toruńczyk s characterization of the Hilbert cube [8] (cf. [6], [9]). In fact, it is easy to verify that Y has the disjoint cells property. Remark 2. To see that Y is an R, since every is a fine homotopy equivalence, we can also apply Theorem 6.3 in [3]. cknowledgements. The author would like to express his thanks to Katsuhisa Koshino who found the gap of the proof in [7]. References [1] J. F. dams, On the groups J(X). IV, Topology 5 (1966), 21 71; errata, ibid. 7 (1968), 331. [2] C. Bessaga and. Pe lczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne 58, PWN Polish Scientific Publishers, Warsaw, [3] Z. Čerin, Characterizing global properties in inverse limits, Pacific J. Math. 112 (1984), [4] O.-H. Keller, Die Homoiomorphie der kompakten konvexen Mengen im Hilbertschen Raum, Math. nn. 105 (1931), [5] S. Mardešić and J. Segal, Shape theory. The inverse system approach, North-Holland Mathematical Library 26, North-Holland Publishing Co., msterdam-new York, [6] J. van Mill, Infinite-dimensional topology. Prerequisites and introduction, North- Holland Mathematical Library 43, North-Holland Publishing Co., msterdam, [7] J. L. Taylor, counterexample in shape theory, Bull. mer. Math. Soc. 81 (1975), [8] H. Toruńczyk, On CE images of the Hilbert cube and characterization of Q-manifolds, Fund. Math. 106 (1980), [9] J. J. Walsh, Characterization of Hilbert cube manifolds: an alternate proof, in: Geometric and algebraic topology, Banach Center Publ. 18, PWN Polish Scientific Publishers, Warsaw, 1986, K. Sakai Institute of Mathematics University of Tsukuba Tsukuba, Japan sakaiktr@sakura.cc.tsukuba.ac.jp Received: Revised: