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A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950's.


Introduction
It is well-known (see [10,12]) that every n-dimensional compactum is weakly homotopy equivalent to an (n + 1)-dimensional cell-like compactum (i.e. a compactum with the trivial shape). Therefore there exist nonaspherical cell-like simply connected compacta in all dimensions ≥ 3.
It was heretofore unknown whether such compacta also exist in dimension 2. In this paper we give the affirmative answer to this question. We show that the space SC(S 1 ) which we constructed in our earlier paper [9], is in fact, a nonaspherical cell-like 2-dimensional simply connected Peano continuum (i.e. locally connected continuum).
We also modify our original construction of the space SC(S 1 ) and show that the modified construction gives a space which has the homotopy type of the classical well-known space [11] from the 1950's, which is a non-simply connected one-point union of two contractible spaces.
Our main result concerns SC(X) for a non-simply connected path-connected space X. To analyze the sigular homology H 2 (SC(X)), we use infinitary words and a result from [5]. Although infinitary words have already been introduced in [1], they may not be a familiar notion. In the special case X = S 1 , we can prove the result only by using finitary words -we present it at the end of Section 3.

Preliminaries
We recall the construction of the space SC(S 1 ) from [9]. Consider the so-called Topologist sine curve T and embed T into the square I 2 = I × I as in Figure 1, i.e. T is embeded as the union of A 1 B 1 A 2 B 2 · · · and AB. Let S 1 be the circle and s 0 any of its points which we consider as the base point. Consider the topological sum of I 2 and T × S 1 . The space SC(S 1 ) is now defined as the quotient space of this sum, obtained by identification of the points (t, s 0 ) with t ∈ T ⊂ I 2 , and by identification of each set {t} × S 1 with t, when t ∈ {0} × I. For an arbitrary compactum X, one defines the space SC(X) by replacing S 1 everywhere above by X. For the details of the definition of SC(X) we refer the reader to [9].
The subspace H = ∞ m=1 {(x, y) : (x − 1/m) 2 + y 2 = 1/m 2 } of the Euclidean plane R 2 is called the Hawaiian earring. Denote θ = (0, 0) ∈ H and let C(H) be the cone over H. We consider H as the subspace of C(H). A space G is then defined as the one-point union of two copies of C(H), obtained by identifying two copies of θ at the point θ. This space is a well-known example of a non-contractible space which is a one-point union of contractible spaces -Griffiths was the first to investigate this kind of spaces [11, p.190], where he also acknowleges ideas by James. The fact that G is aspherical was proved in [8]. For further information of this space and its generalizations we refer the reader to [4,6,7].
Throughout the paper, we shall denote the singular homology with integer coefficients by H * ().
3. On nonasphericity of SC(S 1 ) and SC(X) Obviously, SC(S 1 ) is a cell-like Peano continuum. It was shown in [9] that this space is simply connected. Therefore it suffices to show that SC(S 1 ) is nonaspherical. In order to prove this it certainly suffices to verify that there exists a nontrivial 2-dimensional singular cycle in SC(S 1 ). We shall prove this as a corollary of the following general result -Theorem 3.1 below -in the sense of [9]. Our notation for SC(X) is the same as in [9].
Consider Figure 1: the piecewise linear line A 1 B 1 A 2 B 2 · · · with the segment AB in this figure is the PL Topologist sine curve which was used to build SC(X), i.e. along which we attached the "infinite tube".
Theorem 3.1. Let X be any path-connected space. Then the following assertions hold: (1) If X is not simply connected, then H 2 (SC(X)) is not trivial; and (2) If π 1 (X) and π 2 (X) are trivial, then H 2 (SC(X)) is also trivial.
For the proof of Theorem 3.1, we recall a notion of the free σ-product of groups and a lemma from [5]. Let (X i , x i ) be any family of pointed spaces such that X i ∩ X j = ∅, for i = j. The underlying set of a pointed space ( i∈I (X i , x i ), x * ) is the union of all X i s obtained by identifying all x i to a point x * and the topology is defined by specifying the neighborhood bases as follows: The point x * has a neighborhood base, each element of which is of the form: In particular I = N, We also need basic descriptions of paths and loops. A loop f : . For loops f, g with the same base point, the concatenation f g is a loop defined by: f g(t) = f (2t) for 0 ≤ t ≤ 1/2 and f g(t) = g(2t − 1) for 1/2 ≤ t ≤ 1. We denote the homotopy class relative to end points of a loop f by [f ] and the homology class of f by [f ] s . Proof of Theorem 3.1. Let p be the natural projection of SC(X) onto I 2 which we consider as a subspace of the plane R 2 . Let Consider the following Mayer-Vietoris homology exact sequence: We now present the proof of property (1) above. We first observe that noninjectivity of h implies that H 2 (SC(X)) is non-trivial. Since Choose a point x # ∈ X and form a one point union (X, x # ) ∨ (I, 0) under the identification of x # and 0. Let X n s be copies of (X, x # ) ∨ (I, 0) and x n s copies of 1 ∈ I. Then the space p −1 (I×{0}) has the same homotopy type Y = n∈N (X n , x n ).
Hence (X n , x n ) is locally simply connected and first countable at x n . Lemma 3.3 implies that π 1 (Y ) ≃ × × n∈N π 1 (X n , x n ).
Since X is not simply connected, we can find an essential loop f in X whose base point is (These infinite concatenations make sense, since the ranges of loops converge to C.) Observe that r 0 * Since we have homotopies from f − An+1 f An+1 to the constant A and the images of the homotopies converge to A, it follows that r 0 * It suffices to show that [g 0 g 1 ] s is non-zero, i.e. that [g 0 g 1 ] does not belong to the commutator subgroup of π 1 (Y 0 ∩ Y 1 ). The isomorphism from π 1 (Y 0 ∩ Y 1 ) to n∈N (X n , x n ) maps [g 0 g 1 ] to c 1 c −1 2 c 3 c −1 4 · · · c −1 1 c 2 c −1 3 c 4 · · · , where c n is the letter corresponding to [f Cn ]. To show the conclusion by contradiction, suppose that c 1 c −1 2 c 3 c −1 4 · · · c −1 1 c 2 c −1 3 c 4 · · · belongs to the commutator subgroup. Then, by [5,Lemma 4.11] there exist non-empty reduced words U 1 , · · · , U 2m such that c 1 c −1 2 c 3 c −1 4 · · · c −1 1 c 2 c −1 3 c 4 · · · = U 1 · · · U 2m , where U 1 , · · · , U 2m is of the canonical commutator form, i.e. there are j l , k l such that {j 1 , · · · j m } ∪ {k 1 , · · · , k m } = {1, · · · , 2m}, U j l = U −1 k l and the reduced word c 1 c −1 2 c 3 c −1 4 · · · c −1 1 c 2 c −1 3 c 4 · · · is obtained by multiplying the rightmost elements U i and the leftmost elements of U i+1 at most (2m − 1)-times. Therefore, W 2m is of infinite length and is well-ordered from the left to the right, and hence there exists U i which is of infinite length and is well-ordered from the right to the left. But this is impossible, because is well-ordered from the left to the right. Next we show the second statement (2). Suppose that π 1 (X) and π 2 (X) are trivial. Consider another part of the Mayer-Vietoris sequence: Similarly, H 2 (Y 1 ) = 0 and H 2 (Y 0 ) ⊕ H 2 (Y 1 ) = {0}. Now the above exact sequence implies that H 2 (SC(X)) is trivial.
We denote the commutator aba −1 b −1 by [a, b]. Alternative proof of Corollary 3.2. For the case X = S 1 we take c n as the generator of the fundamental group of X Cn , which is isomorphic to Z. As in the preceding proof of Theorem 3.1, it suffices to show that the element c = c 1 c −1 2 c 3 c −1 4 · · · c −1 1 c 2 c −1 3 c 4 · · · does not belong to the commutator subgroup of the group π 1 (Y 0 ∩ Y 1 ). To prove this by contradiction, suppose that c belongs to the commutator subgroup, i.e. c is a product of m commutators for some m.

A PL Model for SC(S 1 ) and Some Related Constructions
In this section we demonstrate piecewise linear constructions which are similar to SC(S 1 ), using parameters for oscillations of a tube. Actually we prove in Theorem 4.3 that they are homotopy equivalent to the point, SC(S 1 ), or G depending on their parameters.
For 0 ≤ y ≤ 1 and ε ≥ 0 with 0 < y + ε ≤ 1, we construct a space S(y, ε) ⊆ R 3 as follows. Consider the following points on I 2 for n ∈ N (see Figure 2), where we regard I 2 ⊆ R 2 as I 2 × {0}: Let E n and F n be points on the plane {(z, x, y) ∈ R 3 | z = 1 2 x} the projections of which to the plane R 2 are points E n and F n respectively, i.e., Let H 2n−1 be the convex hull of the points A n , B n , C n , D n , E n and F n and H 2n the convex hull of the points A n+1 , B n , C n+1 , D n , F n and E n+1 .
Proof. It is easy to see that S(y, ε) and S(y, 0) are homeomorphic and so we only need to prove that S(y, 0) for 0 < y < 1/2 and S(1/2, 0) are homotopy equivalent. (Without any loss of generality we may assume that y = 1/3.) Since there might be some confusion regarding the homotopy equivalence, we explain this first. Let A n , B n , C n , D n ,... be the notation for S(1/2, 0) and C ′ n , D ′ n ,... be the corresponding notation for S(1/3, 0). If we remove {0} × I from S(1/2, 0) and S(1/3, 0), then the resulting spaces are homeomorphic, that is, S(1/2, 0) \ {0} × I and S(1/3, 0) \ {0} × I are homeomorphic. However, this homeomorphism cannot be extended over to S(1/2, 0), since the homeomorphism maps C n to C ′ n and D n to D ′ n , that is, upwards for C n and downwards for D n , with respect to the y-coordinate. Conversely, if we construct a homotopy on S(1/2, 0) \ {0} × I or S(1/3, 0) \ {0} × I, whose projection to the y-coordinate only depends on the y-coordinate on the domain, it extends on SC(1/2, 0) or SC (1/3, 0).
and ψ(x, y, z) = (ψ 0 (x, y, z), ψ 1 (y), ψ 2 (x, y, z)), where ψ 2 (x, y, 0) = 0 and ψ 2 (x, y, z) > 0, for z > 0 and ψ 0 (x, y, z) is defined as we explain using Figure 3 in the sequel.  First we explain the map ψ. The two shadowed triangles are mapped to C n+1 or D n , respectively. Accordingly, the segments B ′ n C ′ n+1 and D ′ n A ′ n+1 are mapped onto B n C n+1 and D n A n+1 respectively. The segments N ′ n D n and C ′ n+1 M ′ n are mapped bijectively to C n+1 D n .
Next we explain the map ϕψ. The two shadowed triangles are mapped to ϕ(C n+1 ) or ϕ(D n ), which are the dotted point. The two bending segments are mapped onto C ′ n+1 B ′ n or A ′ n+1 D ′ n . Last we explain the map ψϕ. The two shadowed triangles are mapped to C n+1 or D n . The two segments having slope greater than 1 are mapped to C n+1 B n or A n+1 D n .
To show this, we introduce some notation. Since the cone C(X) over the space X is the quotient space of X × I, obtained by identifying X × {1} to a point, we let p : X × I → C(X) be the canonical projection.