E. G. SKLYARENKO

ON THE DEFINITION OF COHOMOLOGICAL DIMENSION

(Presented by Academician P. S. Aleksandrov, 23 X 1964)

The following definition of the cohomological dimension of a topological space, due to P. S. Aleksandrov \((^{1})\), is well known.

Definition 1. The cohomological dimension \(\dim_G X\) of a space \(X\) with respect to an Abelian group \(G\) is the greatest integer \(n\) for which there exists in \(X\) a closed subset \(A\) such that
\[ H^n(X,A;G)\ne 0. \]

In recent papers Okuyama \((^{3})\) used another definition.

Definition 2. The cohomological dimension \(D(X;G)\) of a space \(X\) with respect to an Abelian group \(G\) is the least nonnegative integer \(n\) such that, for every set \(A\) closed in \(X\), the mapping
\[ H^m(X;G)\to H^m(A;G) \]
is epimorphic for all \(m\ge n\).

An analogous definition was used in \((^{2})\) for locally bicompact spaces.

In the present note the equivalence of these two definitions is proved for paracompact spaces*.

The inequality \(D(X;G)\le \dim_G X\) is, obviously, a consequence of the exact cohomology sequence
\[ \cdots \to H^m(X,A;G)\to H^m(X;G)\to H^m(A;G)\to H^{m+1}(X,A;G)\to\cdots . \tag{*} \]

Let us prove the reverse inequality. Suppose that \(D(X;G)\le n\). Since the mapping
\[ H^m(X;G)\to H^m(A;G) \]
is epimorphic for \(m\ge n\), it follows from the exact sequence \((*)\) that, for every subset \(A\) closed in \(X\), the group \(H^{m+1}(X,A;G)\) is isomorphic to a subgroup of the group \(H^{m+1}(X;G)\). Therefore it is sufficient to prove that the group \(H^{m+1}(X;G)\) is trivial.

For the proof of this we shall need the following lemma.

Lemma 1. Let \(h\) be an arbitrary element of the group \(H^{m+1}(X;G)\). Suppose that the space \(X\) can be represented as the union of a finite number of closed sets \(X_1,\ldots,X_r\), on each of which the restriction of the element \(h\) is equal to zero. Then \(h=0\), if \(D(X;G)\le m\).

Proof can be carried out by induction on the number of sets \(r\). For \(r=1\) the assertion of the lemma is trivial. Suppose that the lemma is true if the number of sets \(X_i\) is \(k-1\); we prove the assertion for \(r=k\). Let
\[ X'=\bigcup_{i=2}^{k}X_i. \]
Since \(X'\) is closed in \(X\), we have \(D(X';G)\le m\). Let \(h'\) be the restriction of the element \(h\) to the set \(X'\). By the induction hypothesis we conclude that \(h'=0\). Next consider the exact Mayer–Vietoris cohomology sequence
\[ \cdots \to H^m(X_1;G)+H^m(X';G)\xrightarrow{i} H^m(X_1\cap X';G)\to \]
\[ \to H^{m+1}(X;G)\xrightarrow{j} H^{m+1}(X_1;G)+H^{m+1}(X';G)\to\cdots . \]

* Independently, and by another method, this result was also obtained by I. A. Shvedov.

Since \(D(X;G)\le m\), the mapping \(i\) is an epimorphism. Consequently, \(\delta=0\). Thus the mapping \(j\) is a monomorphism, and since the restriction of the element \(h\) to the sets \(X_1\) and \(X'\) is zero, it follows that \(h=0\). The lemma is proved.

Thus, let us prove that \(H^{m+1}(X;G)=0\). Let \(h\) be an arbitrary element of the group \(H^{m+1}(X;G)\). For each point \(x\in X\) there exists a neighborhood \(U_x\) such that the restriction of the element \(h\) to the set \(U_x\) is equal to zero. The sets \(U_x\) form an open covering \(u\) of the space \(X\). Since the space \(X\) is paracompact, one can inscribe in the covering \(u\) a locally finite covering \(v\) consisting of closed sets \(V_\lambda\). Moreover, in [4] it is in fact shown that the covering \(v\) can be chosen in such a way that it splits into a countable sequence of subsystems
\[ v^i=\{V_\lambda^i\}, \]
each of which is discrete (a system of subsets of a space \(X\) is called discrete if at each point of the space \(X\) there exists a neighborhood intersecting no more than one of the sets of the system). Let
\[ A_i=\bigcup_\lambda V_\lambda^i. \]
Since the covering \(v\) is inscribed in \(u\), the restriction of the element \(h\) to each of the sets \(V_\lambda^i\) is zero. Since for each \(i\) the set \(A_i\) is the union of a discrete system of closed sets \(V_\lambda^i\), the restriction of the element \(h\) to each of the closed sets \(A_i\) is also zero. Moreover,
\[ X=\bigcup_{i=1}^{\infty} A_i. \]

Consider the following system of sets in the space \(X\). Since the space \(X\) is paracompact, for the set \(A_1\) there exists a neighborhood \(B_1\) such that the restriction of the cohomology class \(h\) to the closure of this neighborhood
\[ C_1=X[B_1] \]
is zero. Suppose that sets \(B_i,C_i\), \(i\le k-1\), have already been constructed, such that each \(B_i\) is open,
\[ A_i\cup C_{i-1}\subset B_i,\qquad C_i=X[B_i], \]
and the restriction of the element \(h\) to each \(C_i\) is zero. By the lemma, the restriction of the element \(h\) to the set \(C_{k-1}\cup A_k\) is also zero. Therefore, since the space \(X\) is paracompact, for the set \(C_{k-1}\cup A_k\) there exists a neighborhood \(B_k\) such that the restriction of the element \(h\) to the closed set
\[ C_k=X[B_k] \]
is zero. Thus one may assume that the sets \(B_i,C_i\) have been constructed for all \(i=1,2,\ldots\).

Since \(A_i\subset B_i\subset C_i\), we have
\[ X=\bigcup_{i=1}^{\infty} B_i=\bigcup_{i=1}^{\infty} C_i. \]
By construction,
\[ B_i\subset C_i \]
and
\[ C_i\subset B_{i+1}. \]
Let
\[ L_i=C_{i+1}\setminus B_i. \]
The sets \(L_i\) are closed and form a locally finite covering of the space \(X\), with each \(L_i\) able to intersect only the two elements \(L_{i-1}\) and \(L_{i+1}\) of this covering. Since \(L_i\subset C_{i+1}\), the restriction of the element \(h\) to each of the sets \(L_i\) is zero. Let, further, \(X_1\) be the union of the sets \(L_i\) with even indices, and \(X_2\) the union of all the remaining ones. The sets \(X_1\) and \(X_2\) are closed; since each of them is the union of a discrete system of sets \(L_i\), the restriction of the element \(h\) to each of these sets is zero. Finally, since
\[ X=X_1\cup X_2, \]
it follows from the lemma that \(h=0\). Since as \(h\) one may take any element of the group \(H^{m+1}(X;G)\), it follows that
\[ H^{m+1}(X;G)=0. \]
Thus,
\[ \dim_G X\le n. \]
The assertion is proved.

Moscow State University
named after M. V. Lomonosov

Received
23 VI 1964

CITED LITERATURE

1. P. Alexandroff, Proc. Roy. Soc., A, 189, 11 (1947).
2. H. Cohen, Duke Math. J., 21, No. 2, 209 (1954).
3. A. Okuyama, Proc. Japan Acad., 38, No. 8, 489 (1962); 38, No. 9, 655 (1962).
4. A. H. Stone, Bull. Am. Math. Soc., 54, 977 (1948).