Mathematics

E. G. SKLYARENKO

ON THE EXTENSION OF HOMEOMORPHISMS

(Presented by Academician P. S. Aleksandrov on 22 VIII 1961)

In this note, from the theorem on the extension of homeomorphisms proved in \((^{3,5})\), there is derived a result, due to K. Kuratowski and S. Eilenberg, on the extension of homeomorphisms to acyclic extensions*. At the same time one obtains a theorem concerning a broader class of spaces and their bicompact extensions.

Theorem 1. Let \(X_1\) and \(X_2\) be connected completely regular spaces; let \(Y_1\) and \(Y_2\) be such bicompact extensions of them that the remainders \(Y_1 \setminus X_1\) and \(Y_2 \setminus X_2\) are punctiform, and \(H^1(Y_1, Z)=0\), \(H^1(Y_2, Z)=0\)*; then every homeomorphism between the spaces \(X_1\) and \(X_2\) extends to a homeomorphism between \(Y_1\) and \(Y_2\).

In other words, a connected space can have at most one bicompact extension, acyclic in dimension 1, with punctiform remainder.

Proof. We use the following theorem (see \((^5)\)):

If in the bicompact extensions \(Y_1\) and \(Y_2\) the remainders are punctiform and at no point of theirs separate \(Y_1\) and \(Y_2\), then every homeomorphism between the spaces \(X_1\) and \(X_2\) extends to a homeomorphism between \(Y_1\) and \(Y_2\).

By virtue of this theorem, it is enough to show that the remainders \(Y_1 \setminus X_1\) and \(Y_2 \setminus X_2\) at no point of theirs separate the bicompacts \(Y_1\) and \(Y_2\). In other words, we have to show that an acyclic in dimension 1 bicompact extension \(Y\) of a connected space \(X\) with punctiform remainder \(Y \setminus X\) is indecomposable (in this case \(Y=\mu X\), see Theorem 3 of \((^5)\)).

Suppose that \(Y \setminus X\) separates \(Y\) at some point \(y \in Y \setminus X\), i.e. the point \(y\) has a neighborhood \(W\) such that \(W \cap X = W' \cup W''\), where \(W'\) and \(W''\) are open in \(X\), \(W' \cap W''=\Lambda\), and \(y \in Y[W'] \cap Y[W'']\). Let \(V\) be such a neighborhood of the point \(y\) that \(Y[V] \subset W\). Then the set
\(N = Y[V] \cap Y[W'] \cap Y[W''] \subset Y \setminus X\)
is closed in \(Y\), zero-dimensional (since it is bicompact and punctiform), separates \(Y\) at the point \(y\), and the space \(X' = Y \setminus N\) is connected.

Choose a neighborhood \(U \subset V\) of the point \(y\), whose boundary does not intersect \(N\). Let \(U' = (U \cap Y[W']) \setminus N\); \(U''\) is defined analogously. Then
\(U \setminus N = U' \cup U''\), \(U' \cap U''=\Lambda\), and
\(y \in Y[U'] \cap Y[U'']\). Let \(A=N \cap U\), \(B=Y \setminus U\); the sets \(A\) and \(B\) are closed in \(Y\) and do not intersect. There exists a mapping \(f\) of the bicompact \(Y\) into the circle \(S^1\) such that the sets \(A\) and \(B\) are mapped to diametrically opposite points of the circle, and the sets \(U'\) and \(U''\) are mapped into the opposite semicircles joining these points. Let \(u\) be a one-dimensional fundamental class

* This result was communicated by K. Kuratowski in his report at the Fourth All-Union Mathematical Congress (Leningrad, July 1961) and by S. Eilenberg at the International Topological Symposium (Prague, September 1961).

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

*** Spectral cohomology is meant.

cohomology of the circle; we shall show that \(f^*u \ne 0\), contrary to the assumption that \(Y\) is acyclic.

One may assume that \(f\) is a barycentric map to the nerve of some covering \(\alpha\). Let \(\beta\) be an arbitrary covering inscribed in \(\alpha\). It suffices to show that \(\pi_\beta^\alpha u \ne 0\), where \(\pi_\beta^\alpha\) is the cohomology mapping corresponding to the projection of the nerve of the covering \(\beta\) to the nerve of the covering \(\alpha\) (i.e. to \(S^1\)). Choose a neighborhood \(V_0\) of the point \(y\), contained in \(U\) and in one of the elements of the covering \(\beta\). Let \(z\) be an arbitrary point in \(B\). Since \(X'\) is connected, in \(Y\) one can find a chain of open sets \(V_1,\ldots,V_m\) such that: 1) \(V_i \cap N = \Lambda,\ i=1,\ldots,m\); 2) each \(V_i\) is contained in some element of the covering \(\beta\); 3) \(V_0 \cap V_1 \ne \Lambda,\ V_{i-1}\cap V_i\ne \Lambda,\ i=2,\ldots,m,\ z \in V_m\); 4) \(V_1 \subset U'\); 5) if some \(V_i\subset U\), then either \(V_i\subset U'\), or \(V_i\subset U''\). There also exists a chain \(W_1,\ldots,W_n\) satisfying the preceding conditions with \(U'\) in condition 4) replaced by \(U''\). Let \(\gamma\) be the covering obtained by adjoining to \(\beta\) the sets \(V_0,V_1,\ldots,V_m,W_1,\ldots,W_n\). In the nerve of the covering \(\gamma\) we have a cycle which projects to \(S^1\) with degree 1. Therefore \(\pi_\gamma^\alpha u\ne 0\), and since the covering \(\gamma\) is inscribed in \(\beta\), still more so \(\pi_\beta^\alpha u\ne 0\). The theorem is proved.

We shall show that in fact the requirement that the extensions be acyclic in the theorem on extension of homeomorphisms can be replaced by a weaker condition.

Theorem 2. Let \(X_1\) and \(X_2\) be connected completely regular spaces; let \(Y_1\) and \(Y_2\) be their bicompact extensions with punctiform remainders, and suppose that the natural mappings
\[ H^1(Y_i,\mathbf Z)\to \check H^1(X_i,\mathbf Z),\quad i=1,2, \]
are monomorphisms (or, equivalently, the mappings of homology groups with bicompact coefficients are epimorphisms); then every homeomorphism between the spaces \(X_1\) and \(X_2\) extends to a homeomorphism between \(Y_1\) and \(Y_2\). *

Proof. It is enough to show that if, for a bicompact extension \(Y\) of a connected space \(X\) with punctiform remainder, \(Y\setminus X\), the mapping
\[ \pi^*: H^1(Y,\mathbf Z)\to H^1(\beta X,\mathbf Z) \]
is monomorphic, then the extension \(Y\) is perfect (see the footnote).

We shall give two proofs of this fact.

First proof is obtained by a small addition to the preceding argument. Assuming that the extension \(Y\) is not perfect, we construct, as above, a mapping \(f:Y\to S^1\), determining a nonzero cohomology class \(f^*u\in H^1(Y,\mathbf Z)\). We shall show that \(\pi^*f^*u=0\), contrary to the assumption that the mapping \(\pi^*\) is monomorphic. To this end we prove that the mapping \(f\circ\pi:\beta X\to S^1\) is homotopic to a constant. Let
\[ F=B\cap X,\quad V'=U'\cap X,\quad V''=U''\cap X. \]
The set \(F\) is closed in \(X\) and separates \(X\) into the sets \(V'\) and \(V''\). Since the extension \(\beta X\) is perfect, \(\beta X[F]\) separates \(\beta X\) into \(O\langle V'\rangle\) and \(O\langle V''\rangle\) (see (5), Theorem 1). The mapping \(f\circ\pi\) sends the set \(\beta X[F]\) to a point, and the sets \(O\langle V'\rangle\) and \(O\langle V''\rangle\) to semicircles. Therefore the mapping \(f\circ\pi\) is contractible to a point.

Second proof ** does not depend on the preceding arguments

* For a non-bicompact space \(X\) we consider the cohomology groups \(\check H^*(X,\mathbf Z)\), defined by means of finite normal coverings, in other words, the groups canonically isomorphic to the cohomology groups of the Čech extension \(\beta X\). Note that the requirement that the mapping \(H^1(Y,\mathbf Z)\to \check H^1(X,\mathbf Z)\) be monomorphic is weaker than the requirement that the mapping \(H^1(Y,\mathbf Z)\to H^1(X,\mathbf Z)\) be monomorphic, since the latter is represented as the composition
\[ H^1(Y,\mathbf Z)\to \check H^1(X,\mathbf Z)\to H^1(X,\mathbf Z) \]
(\(\check H^*(X,\mathbf Z)\) denotes cohomology defined by means of all coverings). The mapping
\[ i^*:H^1(Y,\mathbf Z)\to \check H^1(X,\mathbf Z) \]
is identical with the mapping
\[ \pi^*:H^1(Y,\mathbf Z)\to H^1(\beta X,\mathbf Z), \]
induced by the natural projection \(\pi:\beta X\to Y\). Therefore, in what follows, \(H^1(\beta X,\mathbf Z)\) and \(\pi^*\) are used instead of \(\check H^1(X,\mathbf Z)\) and \(i^*\).

** As is clear from the arguments given below, the requirement that \(X\) be connected can be replaced by a weaker condition; it is enough to assume that the mapping
\[ i^*:H^0(Y,\mathbf Z)\to \check H^0(X,\mathbf Z) \]
is an epimorphism onto.

arguments, but uses elements of sheaf theory. By virtue of Theorem 2 from (⁵), it suffices to show that the mapping \(\pi\) is monotone. Let \(Z\) be the constant sheaf isomorphic to the group of integers (which we shall consider on the bicompacta \(\beta X\) and \(Y\)). Let \(R^0\pi(Z)\) be the direct image of the sheaf \(Z\) under the mapping \(\pi\) (see, for example, (⁴)). The stalk of the sheaf \(R^0\pi(Z)\) over a point \(y \in Y\) is the group \(H^0(\pi^{-1}y, Z)\). The mapping \(\pi\) is monotone if and only if, for every point \(y\), \(H^0(\pi^{-1}y, Z)=Z\), i.e. if the canonical embedding \(Z \to R^0\pi(Z)\) is an isomorphism of sheaves. Consider the exact cohomology sequence

\[ 0 \to H^0(Y,Z) \to H^0\bigl(Y,R^0\pi(Z)\bigr) \to H^0\bigl(Y,R^0\pi(Z)/Z\bigr) \]
\[ \to H^1(Y,Z) \to H^1\bigl(Y,R^0\pi(Z)\bigr) \to \ldots , \tag{*} \]

corresponding to the exact sequence of sheaves

\[ 0 \to Z \to R^0\pi(Z) \to R^0\pi(Z)/Z \to 0. \]

Since the spaces under consideration are connected, the mapping

\[ H^0(Y,Z) \to H^0\bigl(Y,R^0\pi(Z)\bigr)=H^0(\beta X,Z) \]

is an isomorphism onto. On the other hand, consider the mappings

\[ H^1(Y,Z) \to H^1\bigl(Y,R^0\pi(Z)\bigr) \to H^1(\beta X,Z). \]

Since, by hypothesis, the composite mapping is a monomorphism, all the more so the mapping \(H^1(Y,Z)\to H^1\bigl(Y,R^0\pi(Z)\bigr)\) is a monomorphism. Therefore, from the exactness of the sequence \((*)\) we conclude that
\[ H^0\bigl(Y,R^0\pi(Z)/Z\bigr)=0. \]
We shall show that then \(R^0\pi(Z)/Z=0\). Since the mapping \(\pi\) is the identity at the points of the space \(X\), the sheaf \(R^0\pi(Z)/Z\) is concentrated on the remainder \(Y\setminus X\). Suppose that at some point \(y\in Y\setminus X\)
\[ \bigl(R^0\pi(Z)/Z\bigr)_y \ne 0. \]
Let \(s\) be a section of the sheaf \(R^0\pi(Z)/Z\), defined over some neighborhood \(U\) of the point \(y\), and such that \(s_y\ne 0\). Since the sheaf \(R^0\pi(Z)/Z\) vanishes at the points of the space \(X\), the set \(N\) of those points at which the section \(s\) is different from zero is closed in \(U\) and is contained in the remainder. Since the remainder is punctiform, \(\operatorname{ind} N=0\); consequently, at the point \(y\) there exists a neighborhood \(V\subset U\) whose boundary does not meet the set \(N\). Define a global section \(\bar{s}\) of the sheaf \(R^0\pi(Z)/Z\) by putting \(\bar{s}_x=s_x\) if \(x\in V\), and \(\bar{s}_x=0\) if \(x\in Y\setminus V\). The section \(\bar{s}\ne 0\), contrary to the fact that
\[ H^0\bigl(Y,R^0\pi(Z)/Z\bigr)=0. \]
Thus it has been proved that the extension \(Y\) is perfect.

Remark. We have derived Theorem 2 from the theorem on extension of homeomorphisms cited above from (⁵). In fact, for connected spaces these two theorems are equivalent, as the following proposition shows.

If a bicompact extension \(Y\) of a completely regular space \(X\) is perfect, then the mapping
\[ i^*: H^1(Y,Z)\to \check{H}^1(X,Z) \]
is a monomorphism.

Indeed, the mapping \(\pi:\beta X\to Y\) is monotone (see (⁵), Theorem 2), i.e. the inverse images of points are acyclic in dimension 0. Therefore our assertion follows from the Vietoris–Begle theorem (¹, ²).

Moscow State University
named after M. V. Lomonosov

Received
22 VIII 1961

References

¹ E. G. Begle, Ann. Math., 51, 3, 534 (1950).
² E. G. Begle, Michigan Math. J., 3, No. 2, 179 (1955–1956).
³ R. Duda, Indagationes Math., 22, 2, 132 (1960).
⁴ A. Grothendieck, On Certain Questions of Homological Algebra, IL, 1961.
⁵ E. G. Sklyarenko, DAN, 137, No. 1, 39 (1961).