MATHEMATICS

E. G. SKLYARENKO

ON THE VIETORIS–BEGLE THEOREM

(Presented by Academician P. S. Aleksandrov, 9 X 1962)

The classical Vietoris–Begle theorem \((^{2,3})\) asserts that if, under a continuous mapping of a bicompactum \(X\) onto a bicompactum \(Y\), the inverse images of points are acyclic in dimensions \(0, 1, \ldots, N-1\), then the induced mappings of cohomology
\(H^i(Y) \to H^i(X)\) are isomorphisms for \(i \le N-1\), and the mapping \(H^N(Y) \to H^N(X)\) is monomorphic. In the present paper the behavior of cohomology under a continuous mapping is studied in the case when the inverse images of points are not necessarily acyclic, but non-acyclicity occurs only on sets of sufficiently small dimension.

Let \(M\) be a subset in a topological space \(X\). Following P. S. Aleksandrov \((^1)\), we shall call the relative dimension \(\operatorname{rd}_X M\) of the set \(M\) in \(X\) the greatest of the dimensions \(\dim E\) of the sets \(E\) closed in \(X\) and contained in \(M\). If the set \(M\) is empty, we shall assume that \(\operatorname{rd}_X M = -\infty\).

Theorem 1. Let \(f\) be a closed mapping of a paracompactum \(X\) onto a paracompactum \(Y\), \(L\) a sheaf of abelian groups over \(Y\), \(L^*\) the inverse image of the sheaf \(L\) over \(X\). Let

\[ M_0=\{y\in Y \mid H^0(f^{-1}y; L^*)\ne L_y\}, \]

\[ M_k=\{y\in Y \mid H^k(f^{-1}y; L^*)\ne 0\}, \qquad k \ge 1 \; **. \]

Let \(d_k=\operatorname{rd}_Y M_k\); put

\[ n = 1+\max_{0\le k\le N-1}(d_k+k), \]

where \(N\) is some natural number or \(+\infty\). If \(n<N\), then the mapping
\(H^i(Y;L)\to H^i(X;L^*)\) is epimorphic for \(i=n\), isomorphic for \(n<i<N\), and monomorphic for \(i=N\) ***.

In the case when all the sets \(M_k\) are empty and \(L\) is a constant sheaf, Theorem 1 turns into the Vietoris–Begle theorem. If, as \(L\), one takes a sheaf that is constant on some open set \(U\subset Y\) and equal to zero on \(Y\setminus U\), then we obtain the Vietoris–Begle theorem for relative cohomology. Let us note some consequences of Theorem 1.

Corollary 1. Let \(f\) be a closed mapping of a paracompactum \(X\) onto a paracompactum \(Y\), \(L\) a sheaf of abelian groups over \(Y\), \(L^*\) the inverse

* A paracompactum is a paracompact Hausdorff space.
* \(L_y\) denotes the stalk of the sheaf \(L\) over the point \(y\in Y\). A sheaf over the whole space and the sheaves induced by it on subsets are denoted by one and the same symbol.
** In particular, as \(L\) one may take an arbitrary group of coefficients \(G\); in this case \(G^*=G\), and \(M_k\) is the set of those points in \(Y\) whose full inverse images are non-acyclic over \(G\) in dimension \(k\).

is an image of the sheaf \(L\) over \(X\). Let \(A_0\) be the set of points \(y \in Y\) whose full preimages are disconnected,

\[ A_k=\{y\in Y \mid \dim f^{-1}y \geq k\}, \qquad k=1,2,\ldots \]

Let \(d_k=\operatorname{rd}_Y A_k\); put

\[ n=1+\max_{k\geq 0}(d_k+k). \]

Then the map \(H^n(Y;L)\to H^n(X;L^*)\) is an epimorphism, and the maps \(H^i(Y;L)\to H^i(X;L^*)\), \(i>n\), are isomorphisms.

In particular, if the map \(f\) is zero-dimensional, then \(n=1+d_0\), and, consequently, the map \(H^{1+d_0}(Y;L)\to H^{1+d_0}(X;L^*)\) is an epimorphism. The following special case of this assertion plays an essential role in the theory of bicompact extensions of peripherally bicompact spaces.

Let \(Y_1\) and \(Y_2\) be bicompact extensions of a completely regular space \(X\) with punctiform remainders \(Y_1\setminus X\) and \(Y_2\setminus X\), and suppose that the extension \(Y_2\) follows the extension \(Y_1\), i.e. there exists a map \(f:Y_2\to Y_1\) that is the identity on the points of the space \(X\). Then the map \(H^1(Y_1;G)\to H^1(Y_2;G)\) is an epimorphism* (\(G\) is an arbitrary coefficient group).

Under the assumption that the remainders are closed, this assertion was proved in \((^7)\).

Corollary 2. Let \(f\) be a closed mapping of a paracompactum \(X\) onto a paracompactum \(Y\), increasing dimension, i.e. \(\dim Y>\dim X\). Let \(A_0\) be the set of points \(y\in Y\) whose full preimages are disconnected,

\[ A_k=\{y\in Y\mid \dim f^{-1}y\geq k\}, \qquad k=1,2,\ldots \]

Let \(d_k=\operatorname{rd}_Y A_k\). If \(\dim Y=m<\infty\) and \(d_k+k+1<m\), \(k=1,2,\ldots\), then \(\operatorname{rd}_Y M_0\geq m-1\).

In the case when the map \(f\) is zero-dimensional, \(A_0\) coincides with the set of those points of the space \(Y\) whose preimages consist of more than one point, while the sets \(A_k\) for \(k\geq 1\) are empty. Therefore Corollary 2 becomes the well-known theorem of V. Gurevich \((^5)\).

Corollary 3. Let \(f\) be a closed mapping of a paracompactum \(X\) onto a paracompactum \(Y\); let \(M_0\) be the set of points \(y\in Y\) whose full preimages are disconnected,

\[ M_k=\{y\in Y\mid H^k(f^{-1}y;Z)\neq 0\}, \qquad k=1,2,\ldots \]

Let \(d_k=\operatorname{rd}_Y M_k\). If \(\dim Y=m<\infty\) and \(1+d_k+k<m\), \(k=0,1,2,\ldots\), then \(\dim Y\leq \dim X\), i.e. the map \(f\) does not increase dimension.

In the case when all the sets \(M_k\) are empty, we obtain the well-known theorem of Dyer that a map for which the preimages of all points are acyclic does not increase dimension \((^6)\).

In the author’s paper \((^8)\) the following proposition was proved (although formulated only for a special case), which may be regarded as the converse of the Vietoris–Begle theorem for \(N=1\).

Let \(f\) be a closed mapping of a paracompactum \(X\) onto a paracompactum \(Y\) such that the full preimages of all points \(y\in Y\setminus M\) are connected, where \(M\) is some subset of \(Y\) of relative dimension zero. If the induced map \(H^0(Y;Z)\to H^0(X;Z)\) is an isomorphism, and the map \(H^1(Y;Z)\to H^1(X;Z)\) is a monomorphism, then the map \(f\) is monotone, i.e. the full preimages of all points \(y\in Y\) are connected.

* A space is called punctiform if every connected bicompact subset of it consists of one point.

** We note that the kernel of this epimorphism is a torsion-free group if \(G\) is a torsion-free group.

An analogous converse is also admitted by Theorem 1.

Theorem 2. Let \(f\) be a closed mapping of a paracompactum \(X\) onto a paracompactum \(Y\), \(L\) a sheaf of Abelian groups over \(Y\), and \(L^*\) the inverse image of the sheaf \(L\) over \(X\). Let

\[ M_0=\{y\in Y\mid H^0(f^{-1}y;L^*)\ne L_y\}, \]

\[ M_k=\{y\in Y\mid H^k(f^{-1}y;L^*)\ne 0\},\qquad k\geqslant 1, \]

and \(d_k=\operatorname{rd}_Y M_k\). Suppose that

\[ 1+\max_{\substack{d_k>0\\ 0\leq k\leq N-1}}(d_k+k)<N, \]

where \(N\) is some natural number or \(+\infty\). Let \(n\) be an integer such that

\[ 1+\max_{\substack{d_k>0\\ 0\leq k\leq N-1}}(d_k+k)\leqslant n<N, \]

or, if all \(d_k\leqslant 0\) for \(k\leqslant N-1\), then \(0\leqslant n<N\). If the mapping \(H^i(Y;L)\to H^i(X;L^*)\) is epimorphic for \(i=n\), isomorphic for \(n<i<N\), and monomorphic for \(i=N\), then for all points \(y\in Y\) we have \(H^i(f^{-1}y;L^*)=0\) for \(n\leqslant i<N\), \(i>0\); if \(n=0\), then, in addition, for all points \(y\in Y\) we have \(H^0(f^{-1}y;L^*)=L_y\).

The proposition formulated above from \((^8)\) follows from Theorem 2 for \(n=0\) and \(N=1\). If in this proposition one additionally requires that the mapping \(f\) be zero-dimensional, then we obtain that the mapping \(f\) is homeomorphic. In particular, the following assertion for bicompact extensions is obtained.

Let \(Y_1\) and \(Y_2\) be bicompact extensions of a completely regular space \(X\) with pointlike remainders, with the extension \(Y_2\) following the extension \(Y_1\); let \(f:Y_2\to Y_1\) be the corresponding mapping. If the mapping \(H^1(Y_1;Z)\to H^1(Y_2;Z)\) is monomorphic (and, consequently, isomorphic), and the mapping \(H^0(Y_1;Z)\to H^0(Y_2;Z)\) is isomorphic, then the extensions \(Y_1\) and \(Y_2\) coincide, i.e. \(f\) is a homeomorphism.

For the case of closed remainders this assertion was proved in \((^7)\).

Białynicki-Birula in \((^4)\) generalizes the Vietoris–Begle theorem in another direction. He considers a triple of spaces \(X,Y,T\) and mappings \(f:X\to Y\), \(g:Y\to T\), \(h=gf:X\to T\), and proves that the assertion of the Vietoris–Begle theorem remains valid if, instead of acyclicity of the inverse images of points under the mapping \(f\), one requires that, for every point \(t\in T\), the mappings \(H^i(g^{-1}t)\to H^i(h^{-1}t)\), \(i=0,1,\ldots,N-1\), be isomorphic, and the mapping \(H^N(g^{-1}t)\to H^N(h^{-1}t)\) be monomorphic. Białynicki-Birula also proves the converse of this theorem. The following two theorems include both Theorems 1 and 2 and the theorems of Białynicki-Birula.

Theorem 3. Let \(X,Y,T\) be paracompacta, \(f:X\to Y\), \(g:Y\to T\) closed superjective\(^*\) mappings, and \(h=gf:X\to T\). Let \(L\) be a sheaf of Abelian groups over \(Y\), and \(L^*\) the inverse image of the sheaf \(L\) over \(X\). Let \(M_k\) be the set of points \(t\in T\) for which the mapping

\[ H^k(g^{-1}t;L)\to H^k(h^{-1}t;L^*) \]

is not an isomorphism, and let \(d_k=\operatorname{rd}_T M_k\). Put

\[ n=1+\max_{0\leqslant k\leqslant N-1}(d_k+k), \]

\(^*\) A mapping \(\alpha:A\to B\) is called superjective if \(\alpha(A)=B\).

where \(N\) is some natural number or \(+\infty\). If \(n<N\), then the mapping \(f^*: H^i(Y;L)\to H^i(X;L^*)\) is epimorphic for \(i=n\) and isomorphic for \(n<i<N\). If, moreover, for every point \(t\in T\) the mapping \(H^N(g^{-1}t;L)\to H^N(h^{-1}t;L^*)\) is monomorphic, then the mapping \(f^*:H^N(Y;L)\to H^N(X;L^*)\) is also monomorphic.

Theorem 4. Let \(X,Y,T\) be paracompacts, \(f:X\to Y\), \(g:Y\to T\) closed surjective mappings, and \(h=gf:X\to T\). Let \(L\) be a sheaf of Abelian groups over \(Y\), and let \(L^*\) be the inverse image of the sheaf \(L\) over \(X\). Let \(M_k\) be the set of points \(t\in T\) for which the mapping
\(H^k(g^{-1}t;L)\to H^k(h^{-1}t;L^*)\) is not an isomorphism, and let \(d_k=\operatorname{rd}_T M_k\). Suppose that

\[ 1+\max_{\substack{d_k>0\\ 0\leq k\leq N-1}}(d_k+k)<N, \]

where \(N\) is some natural number or \(+\infty\). Let \(n\) be such an integer that

\[ 1+\max_{\substack{d_k>0\\ 0\leq k\leq N-1}}(d_k+k)\leq n<N, \]

or, if all \(d_k\leq 0\) for \(k\leq N-1\), then \(0\leq n<N\). Suppose that the mapping \(H^i(Y;L)\to H^i(X;L^*)\) is epimorphic for \(i=n\), isomorphic for \(n<i<N\), and monomorphic for \(i=N\). Then for every point \(t\in T\) the mapping \(H^i(g^{-1}t;L)\to H^i(h^{-1}t;L^*)\) is epimorphic for \(i=n\) and isomorphic for \(n<i<N\). If, moreover, the set of points for which the mapping \(H^N(g^{-1}t;L)\to H^N(h^{-1}t;L^*)\) is not monomorphic has relative dimension \(\leq 0\), then this mapping is monomorphic for all points \(t\in T\).

If \(Y=T\) and \(g\) is the identity mapping, then Theorems 3 and 4 become, respectively, Theorems 1 and 2. If in Theorem 3 all the sets \(M_k\) are empty for \(0\leq k\leq N-1\), then this theorem becomes the first theorem of Bialynicki-Birula. The essential difference between Theorem 4 and the second theorem of Bialynicki-Birula consists in the fact that in his theorem it is additionally assumed that all the sets \(M_k\) for \(0\leq k\leq N-1\) are contained in one and the same set \(T_1\) of relative dimension zero and that for every point \(t\in T\setminus T_1\) the groups \(H^i(g^{-1}t)\) and \(H^i(h^{-1}t)\), \(0<i<N\), are not only isomorphic but also equal to zero.

The proofs of Theorems 1–4 are based on the Leray spectral sequence.

Moscow State University
named after M. V. Lomonosov

Received
4 X 1962

CITED LITERATURE

\(^{1}\) P. Alexandroff, Proc. Roy. Soc., 189, 11 (1947).
\(^{2}\) E. G. Begle, Ann. Math., 51, 3, 534 (1950).
\(^{3}\) E. G. Begle, Michigan Math. J., 3, No. 2, 179 (1955–1956).
\(^{4}\) A. Bialynicki-Birula, Fund. Math., 51 (1963).
\(^{5}\) W. Hurewicz, J. f. reine u. angew. Math., 169, 71 (1932).
\(^{6}\) E. Dyer, Ann. Math., 63, 1, 15 (1956).
\(^{7}\) K. Kuratowski, S. Eilenberg, Fund. Math., 50, No. 5, 515 (1962).
\(^{8}\) E. G. Sklyarenko, DAN, 141, No. 5, 1045 (1961).