A. V. CHERNAVSKII

ON FINITELY MULTIPLE OPEN MAPPINGS OF MANIFOLDS

(Presented by Academician P. S. Aleksandrov, 25 I 1963)

1. Here an outline is given of the proof of several propositions on finitely multiple open mappings of manifolds. Theorems 2 and 3 formulated below generalize the known theorems of Newman and Smith on periodic transformations \((^1)\), while Theorem 1 generalizes Montgomery’s theorem \((^6)\).

Open-and-closed continuous mappings correspond to completely continuous decompositions into closed sets (see, for example, \((^2)\)). Let \(f: X \to Y\) be such a mapping and let \(X\) be locally connected. We shall constantly use the following consideration. If \(x\) is an isolated point of the element of the decomposition containing it, then there is a connected neighborhood \(Ux\) on which \(f\) induces a completely continuous decomposition.

A point \(x\) is called a point of local homeomorphism if it has a neighborhood with which the elements of the decomposition meet in no more than one point. A mapping and the corresponding decomposition are finitely multiple if every element of the decomposition contains a finite number of points, and boundedly multiple if there is a natural number \(k\) such that every element contains no more than \(k\) points. A decomposition is nontrivial if there are elements containing more than one point.

The closure, interior, and boundary of a set \(K\) are denoted, respectively, by \([K]\), \(\operatorname{Int} K\), \(\operatorname{Fr} K\), and the empty set by \(\Lambda\).

1. Theorem 1. An open-and-closed finitely multiple continuous mapping of a connected manifold has bounded multiplicity.

Theorem 2. If \(\theta\) is a completely continuous boundedly multiple decomposition of a connected manifold, then its elements of maximal multiplicity form an open everywhere dense set.

Theorem 3. For every open subset \(H\) with compact closure of a connected manifold there exists a positive number \(\varepsilon\) such that every nontrivial completely continuous decomposition of bounded multiplicity contains elements that meet \(H\) and have diameter greater than \(\varepsilon\).

1. Notation and plan of the proof. Let \(M\) be a connected manifold, \(f: M \to X\) an open-and-closed finitely multiple continuous mapping, and \(\theta\) the corresponding completely continuous decomposition. Denote by \(K_i\) the union of elements of multiplicity not exceeding \(i\), and let \(\Phi_i = K_i \setminus K_{i-1}\), \(E_i = [\operatorname{Int}\Phi_i]\). All \(K_i\), \(\Phi_i\), and \(E_i\) consist of whole elements of the decomposition. On each \(\Phi_i\) the mapping is locally homeomorphic and strictly \(i\)-fold.

It may be assumed that \(M\) has no boundary, since the natural extension of \(f\) to the double of \(M\) reduces all the theorems to this case. Let the dimension of \(M\) be \(n\).

First Theorems 2 and 3 are proved. The proof of Theorem 1 uses Theorem 2. Theorem 2 is proved by induction on the maximal multiplicity.

1. Let first the maximal multiplicity be equal to two. Interchanging the points belonging to one element of the decomposition, we obtain a continuous involution. One can apply the theorems of Newman and Smith \((^{1,3})\), from which it follows that: a) the set of one-point elements of the decomposition is nowhere dense, and b) if the dimension of this set is equal

\(n - 1\), and it separates the manifold, then it separates it into two components, and every element consisting of two points intersects each of them.

Suppose that the maximal multiplicity of the elements of \(f\) is equal to \(k\), and suppose Theorem 2 is true when the maximal multiplicity is less than \(k\).

1. \(K_{k-1}\setminus K_1\) is nowhere dense. Let \(E_i \ne \Lambda\). Take in \(\operatorname{Fr} E_i\), \(1<i<k\), an element of the greatest possible multiplicity \(\xi\). At least one of its points \(x\) will not be a point of local homeomorphism, but it will be such a point relative to \(\operatorname{Fr} E_i\). Take a connected neighborhood \(Ux\) on which there is induced a completely continuous decomposition, whose elements from \(\operatorname{Fr} E_i\) meet in no more than one point each and which \(\operatorname{Fr} E_i\) divides into two nonempty open sets \(V_1\subset E_i\) and \(V_2\), \(V_2\cap E_i=\Lambda\). The decomposition induced on \(U\) is nontrivial. If \(x\) does not coincide with \(\xi\), then the multiplicity of the decomposition induced on \(U\) and, consequently, on each \(V_j\), is less than \(k\), and on at least one \(V_j\) the decomposition is nontrivial. If \(x\) coincides with \(\xi\), then the multiplicity of the decomposition induced on \(V_1\) is equal to \(i\), i.e. the decomposition is nontrivial and the multiplicity is less than \(k\). In both cases, on at least one \(V_j\) there is induced a nontrivial completely continuous decomposition of multiplicity less than \(k\). The points separating \(V_1\) and \(V_2\) serve in \(U\) as one-point elements of the decomposition. Therefore, if on this \(V_j\) we preserve the old decomposition, and declare all points of the other to be one-point elements, we obtain a nontrivial completely continuous decomposition of multiplicity less than \(k\), with the one-point elements filling a nonempty open subset. Since \(U\) is connected, this contradicts the induction hypothesis. It remains to show that \(\operatorname{Int} K_1=\Lambda\).

2. The dimension of \(\Phi_i\), \(1<i<k\), is not greater than \(n-1\). Suppose that for some \(i_0\), \(i<i_0<k\), \(\dim \Phi_{i_0}=n-1\). Let \(x\) be a point of an element \(\xi\) lying in \(\Phi_{i_0}\), and let at it \(\Phi_{i_0}\) locally separate \(M\). Since \(f\) is locally homeomorphic on \(\Phi_{i_0}\), it also locally separates \(M\) at all the other points \(\xi\Phi_{i_0}\). At least one point of \(\xi\), say \(x\), will not be a point of local homeomorphism. As above, take a connected neighborhood \(Ux\) so that on it \(\theta\) induces a completely continuous decomposition, so that the elements from \(\Phi_{i_0}\) meet it in no more than one point and so that \(\Phi_{i_0}\) divides it into two nonempty open sets \(V_1\) and \(V_2\). Preserving on one of them the old decomposition and considering the points of the other as one-point elements of the decomposition, we obtain on \(U\) a completely continuous decomposition of multiplicity less than \(k\). By the induction hypothesis it is trivial. Therefore every element \(\theta\) meets each \(V_j\) in no more than one point, and the multiplicity of the decomposition which \(\theta\) induces on \(U\) is equal to 2. Hence in \(U\) the decomposition is arranged as described in item 4b). Since the image of \(U\) is not locally Euclidean, no point \(\xi\) is a point of local homeomorphism, and the same reasoning is applicable to each of them. Therefore near each point \(\xi\) the decomposition is arranged in the same way. We note that near \(\xi\) there may be either elements of multiplicity \(i_0\), or elements of multiplicity \(2i_0\). Since there must be elements of maximal multiplicity there, \(2i_0=k\). Therefore only one \(\Phi_{i_0}\) can have, at some of its points, dimension \(n-1\), and then \(k\) is even. It is shown that, as described in item 4, the decomposition looks near every point of \(\Phi_{i_0}\) at which \(\Phi_{i_0}\) has dimension \(n-1\). Denote the \((n-1)\)-dimensional part of \(\Phi_{i_0}\) by \(\Phi_{i_0}^{\,n-1}\). It is open in \(K_{k-1}\).

3. Special homologies. As the coefficient domain we take \(Z_p\), where \(p\) is the least prime divisor of \(k\) (if \(\Phi^{\,n-1}\ne\Lambda\), \(p=2\)). Homologies are considered on special coverings (the technique for constructing them is the same as in Smith \((^4)\)). A locally finite covering of a manifold is special if: 1) it is \(n\)-dimensional*; 2) the nerves of \(n\)-dimensional simplexes

* The dimension of a covering is equal to the multiplicity decreased by one; an \(r\)-dimensional nerve of a covering is a nonempty intersection of \(r+1\) elements.

the nerve lie either in \(\Phi_k\), or in \(\operatorname{Int} K_1\), while the \((n-1)\)-dimensional ones either intersect \(K_1 \cup \Phi^{\,n-1}\), but not \(K_{k-1}\setminus (K_1 \cup \Phi^{\,n-1})\), or are contained in \(\Phi_k\); 3) its elements decompose into finite systems, called marked systems, with the following properties: a) the elements of one marked system together form an open set \(G\), consisting of whole elements of the partition, and are its components; b) every element of the partition in \(G\) intersects each of its components, and at least one of them intersects each component in a single point; 4) marked systems that intersect \(\Phi^{\,n-1}\) and do not intersect \(K_{k-1}\setminus \Phi^{\,n-1}\) contain \(k/2\) elements; a marked system lying in \(\Phi_k\) contains \(k\) elements, and if two such systems intersect one another, then two elements of the system \(\Phi^k\) intersect each element of the system \(\Phi^{\,n-1}\).

Special coverings form a cofinal part in the set of all coverings. If a special covering is inscribed in a special one, then there exist special projections carrying marked systems into marked systems.

Together with the elements, the oriented simplexes of the nerve decompose into marked systems.

Define an operator \(\sigma\), analogous to the corresponding Smith operator. If a chain \(C\) of dimension \(n-1\) or \(n\) is considered on the nerve of a special covering, then on simplexes whose carriers do not lie in \(\Phi_k\), the chain \(\sigma C\) takes the value zero, while on simplexes in \(\Phi_k\) the value of the chain \(\sigma C\) is equal to the sum of the values that the chain \(C\) takes on the simplexes belonging with the given one to a single marked system. For an \(n\)-dimensional simplex \(\delta^n\),
\[ \sigma \partial \delta^n = \partial \sigma \delta^n . \]
This is obvious if the carrier of \(\delta^n\) intersects \(K_1\). Suppose the carrier of \(\delta^n\) lies in \(\Phi_k\), and suppose first that the carrier of \(\delta^{n-1}\) intersects \(K_1\). Then \(\delta^{n-1}\) will be a face of all \(k\) simplexes (or of none) from the marked system to which \(\delta^n\) belongs, and \(\bmod p\) the right-hand chain on \(\delta^{n-1}\) is, like the left-hand one, zero. If the carrier of \(\delta^{n-1}\) does not intersect \(K_1\), but intersects \(\Phi^{\,n-1}\), then from the system containing \(\delta^n\) it bounds exactly two simplexes, and \(\bmod 2\) again both chains are equal to zero. If, finally, the carrier of \(\delta^{n-1}\) lies in \(\Phi_k\), then either every simplex of the marked system \(\delta^{n-1}\) serves as the face of exactly one simplex of the system \(\delta^n\), or these systems have no incident simplexes. Accordingly, either both parts of the equality on \(\delta^{n-1}\) are equal to \(\pm 1\), or to zero. From the equality just proved it follows that on special coverings \(\sigma\) carries \(n\)-dimensional cycles into \(n\)-dimensional cycles. The operator \(\sigma\) commutes with the special projections, and its action carries over to the \(n\)-dimensional cycles of \(M\).

1. End of the proof of Theorem 2. If \(\Delta\) is a basic cycle of \(M\), then the carrier of the cycle \(\sigma \Delta\) lies outside \(K_1\), and either \(\sigma \Delta = 0\), or \(K_1\) is nowhere dense. If \(\sigma \Delta = 0\), then on the nerve of every special covering \(\mathcal U\) the sum of the coefficients of the cycle \(\Delta(\mathcal U)\) on the simplexes of one marked system from \(\Phi_k\) is zero. Then it is verified that the boundary of the part of \(\Delta(\mathcal U)\) lying outside \(K_1\) is equal to zero, and \(\Delta\) decomposes into the sum of two cycles \(\Delta_1+\Delta_2\), of which the carrier of \(\Delta_2\) is \(K_1\), and the carrier of \(\Delta_1\) is \([M\setminus K_1]\). But then one of them, obviously \(\Delta_2\), must be zero, and hence \(\Delta\) lies on \([M\setminus K_1]\). Since \(\Delta\) is nonzero, \([M\setminus K_1]=M\) and \(K_1\) is nowhere dense.

2. The proof of Theorem 3 is entirely analogous to the proof of the corresponding assertion of Smith ((1), p. 457), and is therefore omitted here.

3. Proof of Theorem 1. Let \(\theta\) be a completely discontinuous partition of finite, but unbounded, multiplicity. It is proved by a method analogous to the proof of item 5 that the boundary \(E_i\) has the following properties: 1) the set of elements of multiplicity strictly equal to \(i\) is everywhere dense on it, and 2) near each element there are elements of arbitrarily high multiplicity. Consider the closed set
   \[ \Pi = M \setminus \bigcup_i \operatorname{Int} K_i . \]

It consists of whole elements which are characterized by the fact that, near each of them, there lie elements of arbitrarily high multiplicity. From 2) it follows that: 3) \(\bigcup \operatorname{Fr} E_i \subset \Pi\). \(\Pi\) is the sum of its intersections with \(K_i\), and from Baire’s theorem it follows that there is an open subset \(U\) of the manifold, consisting of whole elements of the decomposition and containing elements of \(\Pi\), but the elements of \(\Pi\) in it are bounded in multiplicity. By the property of the elements of \(\Pi\), the multiplicities of the elements in \(U\) will not be bounded. Therefore, in \(U\) there are, for arbitrarily large values of \(i\), elements of \(E_i\), and hence also of \(\operatorname{Fr} E_i\). But, by properties 1) and 3), \(\Pi \cap U\) then contains elements of arbitrarily high multiplicity, which contradicts the choice of \(U\).

Remark. From the proof of Theorem 2 it follows that, if \(f: M \to M'\) is a finite-multiple open-closed mapping of a manifold without boundary onto a manifold without boundary, then the dimension of the set of points which are not points of local homeomorphism does not exceed \(n-2\).

Addendum. In connection with results of Hemmingsen and Church, the following proposition is of interest; it frees one theorem of [5] from unnecessary restrictions.

Theorem. Let \(f: X \to Y\) be an open-closed continuous mapping of a continuum manifold with compact inverse images of points, and let the dimension of the set \(B_f\) of points which are not points of local homeomorphism not exceed \(n-2\). Then either \(f\) has bounded multiplicity, or \(fB_f = fX\), and the sum of the elements of infinite multiplicity forms an everywhere dense set.

The proof uses the same set-theoretic considerations as above and, in addition to Kuratowski’s theorem, to which Hemmingsen and Church refer, also relies on Baire’s theorem.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
24 I 1963

CITED LITERATURE

1. P. A. Smith, Ann. Math., 2nd ser., 42, 446 (1941).
2. C. Kuratowski, Topologie, Varsovie, 1952.
3. P. A. Smith, Ann. Math., 2nd ser., 40, 690 (1939).
4. S. Lefschetz, Algebraic Topology, Appendix B, IL, 1949.
5. P. T. Church, E. Hemmingsen, Duke Math. J. 27, 527 (1960).
6. D. Montgomery, Am. J. Math., 59, 118 (1937).