MATHEMATICS

W. KUPERBERG

A CYCLIC TWO-DIMENSIONAL COMPACTUM CONTAINING NO IRREDUCIBLY CYCLIC TWO-DIMENSIONAL SUBCOMPACTUM

(Presented by Academician A. N. Tikhonov on 19 III 1968)

Definition 1. An \(n\)-dimensional compactum \(F\) is called cyclic if there exists an abelian group \(\mathfrak A\) and an \(n\)-dimensional true cycle \(\left(^{2}\right)\) with coefficients in \(\mathfrak A\), which is not homologous to zero in \(F\) \(\left(^{1}\right)\).

Definition 2. An \(n\)-dimensional cyclic compactum \(F\) is called irreducibly cyclic (or an \(n\)-dimensional closed Cantor manifold \(\left(^{1}\right)\)) if \(F\) contains no \(n\)-dimensional cyclic proper subcompactum \(F' \subset F\).

Remark. If an \(n\)-dimensional compactum \(F\) is cyclic, then there exists in \(F\) some \(n\)-dimensional convergent cycle \(\left(^{2}\right)\) with coefficients in \(\mathfrak R_1\), which is not homologous to zero (the symbol \(\mathfrak R_1\) here denotes the additive group of all rational numbers reduced modulo 1). This fact is an almost immediate consequence of the so-called convergence theorem \(\left(^{2}\right)\).

The present note solves the following problem posed by P. S. Alexandrov (Problem III in \(\left(^{1}\right)\), p. 227):

Does every \(n\)-dimensional cyclic compactum contain some \(n\)-dimensional closed Cantor manifold?

We shall show that the answer is negative even for \(n=2\).

First take the plane disk
\[ K=\{(x,y): x^2+y^2\leq 1\} \]
and the sequence of circles lying in this disk,
\[ L_k=\left\{(x,y): x^2+y^2=\frac{1}{k}\right\} \]
\((k=1,2,\ldots)\). Let \(\{p_k\}\) be the increasing sequence of all prime numbers.

We now introduce an equivalence relation \(\sigma\) between points of the disk \(K\), defining it as follows: for \(x\ne y\), \(x\sigma y\) if and only if there exists such a \(k\) that \(x\) and \(y\) are vertices of some regular \(p_k\)-gon inscribed in the circle \(L_k\); moreover, of course, \(x\sigma x\) for all \(x\in K\). Obviously, the decomposition of the disk \(K\) into the equivalence classes of \(\sigma\) is lower semicontinuous. Denote by \(F\) the quotient space \(K/\sigma\). Obviously,
\[ \dim F=2. \]

We shall now show that the compactum \(F\) constructed above has the following property:

The two-dimensional subcompactum \(F'\subseteq F\) is cyclic if and only if it is a neighborhood of the point \([O]\) (where \(O=(0,0)\in K\)).

Proof. Let \(\pi:K\to F\) denote the canonical projection \(\pi(x)=[x]\). The set \(F'\subseteq F\) is a neighborhood of the point \([O]\) if and only if \(\pi^{-1}(F')\) is a neighborhood of the point \(O\).

\(1^\circ\). Suppose that \(F'\) is a neighborhood of the point \([O]\). Then there exists a natural number \(m\) such that the disk
\[ K_m=\left\{(x,y): x^2+y^2\leq \frac{1}{m}\right\} \]
lies entirely in \(\pi^{-1}(F')\). To verify the cyclicity of the set \(F'\), it suffices to show that the compactum \(\pi(K_m)\) is cyclic. Consider the two-dimensional polyhedron \(P_m\), which is constructed as follows: on the disk \(K_m\), we identify all vertices of any regular \(p_m\)-gon inscribed in the circ—

ness \(L_m\); in other words, we “wind” the boundary circle of the disk \(K_m\) \(p_m\) times onto itself. It is obvious that the two-dimensional homology group of the polyhedron \(P_m\) with coefficients \(J_{p_m}\) (\(J_{p_m}\) is the group of residues modulo \(p_m\)) is isomorphic to the group \(J_{p_m}\).

Let us now note that, by identifying some points in \(P_m\), we can obtain the compactum \(\pi(K_m)\); more precisely: by identifying in \(K_m\) the points that are identified by the mapping \(\pi|K_m\), but starting with the points lying on \(L_m\), we first obtain a mapping \(K_m \to P_m\), and then \(\pi_1: P_m \to \pi(K_m)\).

On the other hand, by identifying into one all points in \(\pi(K_m)\) that lie in \(\pi(K_{m+1})\), we obtain a mapping \(\pi_2: \pi(K_m) \to P_m\) such that the composition \(\pi_2 \circ \pi_1: P_m \to P_m\) is homotopic to the identity mapping. Consequently, the induced homomorphism
\[ (\pi_2 \circ \pi_1)_*=\pi_{2*}\circ \pi_{1*} \]
is the identity of the (nontrivial) group \(H_2(P_m,J_{p_m})\); in particular, the homomorphism
\[ \pi_{1*}: H_2(P_m,J_{p_m}) \to H_2(\pi(K_m),J_{p_m}) \]
is a monomorphism. It follows that the group \(H_2(\pi(K_m),J_{p_m})\) is nontrivial. Thus it has been proved that every compact neighborhood of the point \([O]\) in \(F\) is cyclic.

Fig. 1

\(2^\circ\). Suppose now that a closed two-dimensional set \(F'\subset F\) is not a neighborhood of the point \([O]\). In order to make sure that the compactum \(F'\) is not cyclic, it is enough to show that for every \(\varepsilon>0\) there exists an \(\varepsilon\)-shift of the compactum \(F'\) into some noncyclic two-dimensional polyhedron. Consider the polyhedron \(\pi(Q_i)\), where
\[ Q_i=\left\{(x,y):\left(\frac{1}{i+1}\right)^2 \le x^2+y^2 \le 1\right\}\subset K,\quad i=1,2,\ldots \]

First, let us show that \(\pi(Q_i)\) is not cyclic. Consider some triangulation \(T\) of the polyhedron \(\pi(Q_i)\) and an arbitrary abelian group \(\mathfrak A\). Take a triangulation \(T'\) of the polyhedron \(Q_i\) such that the mapping \(\pi\) is simplicial. It is obvious that every simplex of the triangulation \(T'\) lies entirely in one of the sets
\[ M_k=\left\{(x,y):\left(\frac{1}{k+1}\right)^2 \le x^2+y^2 \le \left(\frac{1}{k}\right)^2\right\},\quad k=1,2,\ldots,i, \]
and therefore the mapping \(\pi\) does not identify any two two-dimensional simplices of \(T'\). Let
\(\alpha:T\to \mathfrak A\) be an arbitrary two-dimensional cycle lying in \(\pi(Q_i)\), with coefficients in \(\mathfrak A\). Then \(\alpha(\Delta_1)=\alpha(\Delta_2)\) if and only if the two-dimensional simplices \(\Delta_1\) and \(\Delta_2\) lie in the same \(\pi(M_k)\). Let \(a_k\) be the value that the cycle \(\alpha\) takes on all two-dimensional simplices lying in \(\pi(M_k)\). Since \(\partial\alpha=0\), we obtain the equalities:
\[ 2a_1=0,\quad 3a_1=3a_2,\quad 5a_2=5a_3,\ldots,\quad p_i a_{i-1}=p_i a_i,\quad p_{i+1}a_i=0. \]
From these equalities it follows that \(a_1=a_2=\cdots=a_i=0\), since all \(p_k\) are pairwise relatively prime. Hence, in \(\pi(Q_i)\) there is not a single nontrivial two-dimensional cycle.

Fig. 2

Second, let us show that for every \(\varepsilon>0\) there exists an \(\varepsilon\)-shift of the compactum \(F'\) into some \(\pi(Q_i)\). Thus, let \(\varepsilon>0\) be given. For sufficiently

of sufficiently large index \(i\), the diameter of the set \(\pi(K_i)\) is less than \(\varepsilon\). Since the set \(\pi^{-1}(F')\) is compact and is not a neighborhood of the point \(O\), there exists a point \(x_0 \in K_i\) and a neighborhood \(U\) of it that does not meet the set \(\pi^{-1}(F')\). We may assume that \(U\) is a sufficiently small disk meeting none of the circles \(L_k\) (see Fig. 1). We may imagine that Fig. 1 represents the space \(F\). We now carry out a mapping
\(f_1 : F - \pi(U) \to f_1(F - \pi(U))\), “stretching” the hole \(\pi(U)\) (as in Fig. 2), but identical outside the set \(\pi(K_i)\). The point \(x_1\) (Fig. 2) separates the set \(f_1(F - \pi(U))\) into two open sets \(U_1\) and \(U_2\); let \([O] \in U_1\). Define a continuous mapping \(f_2 : f_1(F - \pi(U)) \to F - U_1\) by the formula

\[ f_2(x)= \begin{cases} x_1 & \text{for all } x \in U_1,\\ x & \text{for all } x \notin U_2. \end{cases} \]

The composition \(f_2 \circ f_1\) is an \(\varepsilon\)-shift, since the points lying in \(\pi(K_i)\) do not leave \(\pi(K_i)\), while the remaining points remain in their places; this \(\varepsilon\)-shift moves the set \(F'\) \((F' \subseteq F - \pi(U))\) into some \(\pi(Q_m)\), since \(F - U_1\) is contained in some \(\pi(Q_m)\).

Corollary. The compactum \(F\) contains no two-dimensional closed Cantor manifold, although it is itself two-dimensional and cyclic.

Remark. It can be shown that if an \(n\)-dimensional cyclic compactum \(F\) is an \(ANR\) space, then it contains some closed \(n\)-dimensional Cantor manifold; more precisely, under the conditions \(F \in ANR\), \(\dim F = n\), every carrier \(F' \subseteq F\) of an \(n\)-dimensional cycle non-homologous to zero contains a closed \(n\)-dimensional Cantor manifold.

Warsaw University
Warsaw, PPR

Received
15 III 1968

REFERENCES

1. P. S. Aleksandrow, Math. Ann., 106, 161 (1932).
2. P. S. Aleksandrov, УМН, 4, no. 6, 17 (1949).
3. K. Borsuk, Theory of Retracts, Warszawa, 1967.