MATHEMATICS

V. KUZ’MINOV

AN EXAMPLE OF A DIMENSIONALLY DEFICIENT COMPACTUM

(Presented by Academician P. S. Aleksandrov on 16 VI 1961)

In this note, for each prime number (q) a three-dimensional compactum (P_3(q)) is constructed, for which (\dim (P_3(q)\times P_3(p))=4) when (p\ne q) and (\dim (P_3(q)\times P_3(q))=6). In a certain sense this compactum generalizes the compactum (P_2(q)) of L. S. Pontryagin ((^{1,3})).

Definitions. Let a three-dimensional simplicial complex (K) be given in the Euclidean space (R^7). A polyhedron (T) is called a tube in the complex (K) if the following conditions are satisfied: a) there exists a homeomorphism (\Phi) mapping the product (E\times I) of the disk (E) of radius 1 and the segment (I=[0,1]) onto the polyhedron (T); b) the polyhedron (T) lies in (\widetilde K) and does not intersect the one-dimensional skeleton of the complex (K); c) if the intersection of (T) and a two-dimensional simplex (t) of the complex (K) is nonempty, then there exist numbers (s_i,\ i=1,\ldots,n), such that (0\le s_i\le 1) and

[
T\cap t=\bigcup_i \Phi(E\times s_i).
]

The sets (\Phi(E\times 0)) and (\Phi(E\times 1)) will be called the bases of the tube (T). By definition, the tubes (T_1,\ldots,T_q) have a common base if there exist numbers (\varepsilon_1,\ldots,\varepsilon_q), equal to 0 or 1, such that the homeomorphisms (\varphi_i(x)=\Phi_i(x,\varepsilon_i)) of the disk (E) coincide. We shall say that the tubes (T_1,\ldots,T_q) connect the two-dimensional complexes (L_1) and (L_2) if these tubes have two common bases, one of which lies in (L_1) and the other in (L_2).

The compactum (P_3(q)) will be obtained as the inverse limit of the spectrum of polyhedra ({K_n,\mathfrak D_n^{\,n+1}}).

Construction of the complexes (K_n) and the projections (\mathfrak D_n^{\,n+1}). Let (K_0) be some triangulation of a three-dimensional simplex. Find pairwise disjoint tubes (T_1,\ldots,T_{s_1}), whose bases lie on the boundary of the simplex (K_0), such that every two-dimensional complex of the complex (K_0) intersects at least one of the tubes (T_1,\ldots,T_{s_1}). Let (A) be the open disk of radius (1/2), concentric with the disk (E), and let (S) be its boundary. The homeomorphisms (\Phi_i) could have been chosen so that the sets (\Phi_i(A\times I)) are bodies of open subcomplexes of some subdivision of the complex (K_0).

Define a mapping (identification) of the complex

[
K_0\setminus \bigcup_i \Phi_i(A\times I)
]

in the following way:

I. (F_1(x)=F_1(y)) if and only if (x=\Phi_i(a\times t)), (y=\Phi_i(b\times t)), where the points (a) and (b) belong to (S) and can be obtained from one another by a rotation of the circle (S) through an angle that is a multiple of (2\pi/q). The subdivision of the complex (K_0) could have been chosen so that the identification (F_1) of the complex

[
K_0\setminus \bigcup_i \Phi_i(A\times I)
]

was a simplicial mapping onto some complex (K'_1), which we shall assume embedded in Euclidean space.

Construct the mapping

II. (\mathfrak D_0^{\,1}: K'_1\to K_0).

Let (\mathfrak D_0^{\,1}(x)=F_1^{-1}(x)), if (x\in \overline{\bigcup_i F_1(T_i)}),

[
\mathfrak D_0^{\,1}\bigl(F_1(\Phi_i(a\times t))\bigr)=\Phi_i(b\times t),
]

where (b) is the point dividing the radius of the disk (E), passing through ...

point (a), in the same relation in which the point (a) divides the segment of this radius lying in the ring (E \setminus A). Choose a sufficiently fine subdivision (K'_1) of the complex (K_1) so that the diameters of the images of the simplices of the complex (K_1) under the mapping (\mathfrak{D}_0^1) are less than (1/2). Denote by (Q_i), (i=1,\ldots,s_1), the subcomplexes (F_1(\Phi_i(S \times I))), and by (N_j) the subcomplexes (F_1(t_j)), (j=1,\ldots,r_1), where (t_j) is an arbitrary closed simplex of a triangulation (K_0).

To construct the complex (K_2), find in the complex (K_1) a system of tubes
[
T_{s_1+1},\ldots,T_{s_2},
]
satisfying the following conditions:

1. The base of each tube lies either on a two-dimensional simplex that is a face of only one three-dimensional simplex, or on a simplex from (\bigcup_i Q_i). In the latter case there will be (q) tubes having this base in common.

2. Two tubes may intersect only in common bases.

3. Each two-dimensional simplex of the complex (K_1) intersects at least one of the tubes (T_{s_1+1},\ldots,T_{s_2}).

4. Any two two-dimensional simplices lying in the subcomplex
   [
   \bigcup_{i=1}^{s_1} Q_i \cap N_j
   ]
   are joined by (q) tubes in the subcomplex (N_j).

If the system of tubes satisfies these conditions, then the subcomplex
[
\bigcup_{s_1+1}^{s_2} \Phi_i(A \times I)
]
of some subdivision of the complex (K_1) will be open. The identification I, applied to the complex (K_1), defines a simplicial mapping (F_2) of the complex
[
K_1 \setminus \bigcup_{s_1+1}^{s_2} \Phi_i(A \times I)
]
onto a certain complex (K'_2), which we shall assume embedded in Euclidean space (R^7). Formulae II define a mapping
[
\mathfrak{D}_1^2 : K'_2 \to K_1.
]

Choose such a subdivision (K_2) of the complex (K'_2) that the diameters of the images of the three-dimensional simplices of the complex (K_2) under the mappings (\mathfrak{D}_1^2) and (\mathfrak{D}_0^2=\mathfrak{D}_1\mathfrak{D}_1^2) are less than (1/2^2).

III. Let
[
Q_i=F_2(Q_i)
]
for (i \leq s_1), and
[
Q_i=F_2(\Phi_i(S \times I))
]
for (s_1+1 \leq i \leq s_2). Let (N_j=F_2(N_j)) for (j \leq r_1), and (N_j=F_2(t_j)) for (r_1+1 \leq j \leq r_2) ((t_j) is an arbitrary closed three-dimensional simplex of the complex (K_1)).

Suppose that the complexes (K_i), the mappings (\mathfrak{D}{i-1}^i) for (i \leq n), and the subcomplexes (Q_k) and (N_l) for (k \leq s_n) and (l \leq r_n) have already been constructed. Find in the complex (K_n) a system of tubes
[
T,},\ldots,T_{s_{n+1}
]
satisfying conditions 1–3 and the following condition:

(4'). For (s_i<j\leq s_{i+1}), any two two-dimensional simplices of the complex
[
N_j \cap \bigcup_{k=s_i+1}^{s_n} P_k
]
are joined by (q) tubes in the complex (N_j). Just as in I and II, we define a mapping (F_{n+1}) of the complex
[
K_n \setminus \bigcup_{s_n+1}^{s_{n+1}} \Phi_i(A \times I)
]
onto a complex (K'{n+1}), and a mapping
[
\mathfrak{D}_n^{\,n+1}: K'\to K_n.
]
Let (K_{n+1}) be a subdivision of the complex (K'{n+1}) for which the diameters of the images of the three-dimensional simplices under the mappings
[
\mathfrak{D}_i^{\,n+1}=\mathfrak{D}_i^{\,i+1}\ldots \mathfrak{D}_n^{\,n+1}
]
are less than (1/2^{\,n+1}). As in III, we define the system of subcomplexes (Q_i, N_j) for (i \leq s).}), (j \leq r_{n+1

We denote by (P_3(q)) the inverse-limit space of the spectrum
[
{K_n,\mathfrak{D}_n^{\,n+1}}.
]

Remark. The construction of the compactum (P_3(q)) is not unique, so that we have constructed not a single compactum, but a class of compacta (P_3(q)). The compacta (\mathfrak{D}_n^{-1}N_j) also belong to the class of compacta (P_3(q)).

Computation of the cohomological dimensions of the compacta (P_3(q)). The coefficient domains are as follows: (Z) is the group of integers; (Z_p) is the group of residues modulo (p); (Q) is the group of rational numbers; (Q_p) is the group of rational numbers of the form (a/p^\alpha), reduced modulo 1; (R_p) is the group of rational numbers whose denominator is not divisible by (p).

Let (P=P_3(q)), (\dot P=\widetilde{\omega}_0^{-1}(\dot K_0)), and (\dot K_n=(\widetilde{\omega}_0^n)^{-1}\dot K_0), where (\dot K_0) is the boundary of the simplex (K). It is easy to see that (H^3(P,\dot P;Z)\approx Z_q). Hence ((^2)), (\dim P=cd_Z P=3). To compute the Čech homology group (H_2(P,\dot P;G)), the following lemma is needed.

Lemma. Any two-dimensional cycle (z) of the complex ((K_n,\dot K_n)) is homologous in ((K_n,\dot K_n)) to a cycle lying on the subcomplex
[
M_n=\bigcup_{i=1}^{s_n} Q_i .
]

Proof. Suppose the lemma has been proved for the complexes (K_i) with (i\leq n). We prove it for the complex (K_{n+1}). Let
[
L=\bigcup_{i=s_n+1}^{s_{n+1}}\Phi_i(A\times I),
]
(\overline L) be the closure of (L), and (\dot L) the boundary of (L). Represent the cycle (z) as a sum of chains
[
z=z_1+z_2,
]
where (z_1) lies in (K_{n+1}\setminus F_{n+1}(\dot L)) and (z_2) in (F_{n+1}(\dot L)). Since the mapping (F_{n+1}) onto (K_n\setminus \overline L) was a homeomorphism, there is a chain (x_1) in (K_n\setminus \overline L) for which (F_{n+1}(x_1)=z_1). Then
[
F\Delta x_1=\Delta F x_1=-\Delta z_2.
]
It is not difficult to see that any one-dimensional cycle (z) in (\dot L) such that (F_{n+1}(z)\sim 0) in (F_{n+1}(\dot L)) is itself homologous to zero in (\dot L). Thus there exists a chain (y_1) in (\dot L) for which
[
\Delta x_1=\Delta y_1.
]
The cycle (x_1-y_1) is homologous in (K_n) to some cycle (y_2) from (M_n), i.e. there exists a chain (X^3) for which
[
\Delta X^3=x_1-y_1-y_2.
]

Represent the chain (X^3) as a sum of chains:
[
X^3=X_1^3+X_2^3,
]
where (X_1^3) lies on (K_n\setminus \overline L) and (X_2^3) on (\overline L). The chain
[
\Delta X_2^3+y_1+y_2
]
lies in (M_n\cup \overline L), and
[
F x_1-F\Delta X_1^3=F(\Delta X_2^3+y_1+y_2).
]
Therefore
[
z-\bigl(z_2+F(\Delta X_2^3+y_1+y_2)\bigr)=\Delta F X_1^3 .
]
This is the homology we need. The lemma is proved.

We now prove that
[
H_2(P,\dot P;G)=0
]
if the group (G) admits unique division by (q). Indeed,
[
H_2(P,\dot P;G)\approx \lim_{\longrightarrow}{H_2(K_n,\dot K_n;G);(\widetilde{\omega}n^{\,n+1})}.
]
Let
[
\alpha={a_n}\in \lim_{\longrightarrow}{H_2(K_n,\dot K_n;G);(\widetilde{\omega}n^{\,n+1})}.
]
The element (a_{n+1}) contains a cycle (z_{n+1}) lying on the subcomplex (M_{n+1}). This cycle takes equal values on all simplices of the subcomplex
[
\bigcup_{i=1}^{s_n} Q_i .
]
Indeed, any two simplices (\tau_i) and (\tau_j) of this subcomplex in (K_{n+1}) are joined by (q) tubes (Q_{l_1},\ldots,Q_{l_k}), (s_n<l_m\leq s_{n+1}). Computing the value of the boundary of the cycle (z_{n+1}) on the common bases of these tubes, we obtain
[
q\cdot z_{n+1}(\tau_i)=\sum_{m=1}^{q} z_{n+1}(Q_{l_m})=qz_{n+1}(\tau_j)
]
(the cycle (z_{n+1}) takes equal values on all simplices of the subcomplex (Q_{l_m})). But the cycle (\widetilde{\omega}n^{\,n+1}z) (hence values equal to one another). Therefore the cycle}) is homologous to a cycle (z_n) taking, on the simplices of (Q_i) for (i\leq s_n), the same values as the cycle (z_{n+1
[
\widetilde{\omega}n^{\,n+1}(z)\sim 0
]
in (K_n), (a_n=0), and (\alpha=0).

Theorem. Let (cd_G X) be the cohomological dimension of a compactum (X) over the coefficient domain (G). Then
[
cd_Z P_3(q)=cd_{R_q}P_3(q)=cd_{Z_q}P_3(q)=3,
]
[
cd_{Q_q}P_3(q)=2,\qquad
cd_{Z_p}P_3(q)=cd_{R_p}P_3(q)=cd_QP_3(q)=1
\quad\text{for }p\ne q.
]

We shall prove only that (cd_{Z_p}P_3(q)=1) (the remaining assertions are proved analogously).

Let (cd_{Z_p}P_3(q)=l), (l>1). Then the compactum (P_3(q)) contains an (l)-dimensional subcompactum (F), each point (x) of which has the following property: there is a neighborhood (U) of the point (x) such that for any neighborhood (x\in V\subset U)

the group (H^l(\bar V,\dot V;Z_p)) is nontrivial (7). Let (K_n^2) be the two-dimensional skeleton of the complex (K_n). The subspace (\bigcup_{i=0}^{\infty}\omega_i^{-1}K_i^2) of the compactum (P_3(q)) is the sum of a countable number of compacta homeomorphic to L. S. Pontryagin’s compactum (P_2(q)). Since (\operatorname{cd}{Z_p}P_2(q)=1), by the sum theorem for cohomological dimension (6) the compactum (F) contains a point (x) not belonging to (\bigcupK_i^2).}^{\infty}\omega_i^{-1

For any neighborhood (U) of the point (x) there exist a number (n) and an open three-dimensional simplex (t) of the complex (K_n) such that (x\in\omega_n^{-1}t\subset U). It is not difficult to prove that the closure of the neighborhood (\omega_n^{-1}t) coincides with (\omega_n^{-1}(\bar t)), and its boundary with (\omega_n^{-1}(\dot t)). But the compactum (\omega_n^{-1}(\bar t)) belongs to the class of compacta (P_3(q)). Thus (H^i(\omega_n^{-1}(\bar t),\omega_n^{-1}(\dot t);Z_p)=0) for (i=2,3). Therefore (l<2) and (\operatorname{cd}_{Z_p}P_3(q)=1). The theorem is proved.

From Bokshtein’s formulas (4,5), expressing the dimension of a product of compacta in terms of the cohomological dimensions of the factors computed by us, it follows that (\dim(P_3(q)\times P_3(p))=4) for (p\ne q) and (\dim(P_3(q)\times P_3(q))=6).

In conclusion I express my sincere gratitude to V. G. Boltyanskii for valuable advice and comments and to Yu. M. Smirnov for constant support and assistance in editing this note.

Moscow State University
named after M. V. Lomonosov

Received
14 VI 1961

CITED LITERATURE

1. L. Pontriagin, C. R., 190, 1105 (1930).
2. P. S. Aleksandrov, UMN, 4, 6, 17 (1949).
3. V. Bokshtein, UMN, 6, 3, 99 (1951).
4. M. F. Bokshtein, DAN, 63, No. 3, 221 (1948).
5. E. Dyer, Fund. Math., 47, 141 (1959).
6. H. Cohen, Duke Math. J., 21, 209 (1954).
7. V. Kuzminov, DAN, 139, No. 1 (1961).