MATHEMATICS

R. L. Frum-Ketkov

ON THE BEHAVIOR OF CYCLES UNDER CONTINUOUS MAPPINGS OF COMPACTA

(Presented by Academician P. S. Aleksandrov on 13 II 1957)

§ 1. Let (M^n) and (M_1^n) be two (n)-dimensional closed orientable manifolds. If (f) is a continuous mapping of (M^n) into (M_1^n) of degree zero, then (z^n), the fundamental (n)-dimensional cycle of the manifold (M^n), is mapped to zero. In particular, this holds for a mapping of (M^n) into (n)-dimensional Euclidean space.

P. S. Aleksandrov posed the problem: to prove that for every (k), (0 \leq k \leq n-1), there exists in (M^n) an essential (k)-dimensional cycle (z^k) that is mapped to zero. A positive answer to the question posed, in a more general formulation, is given in this note. The group of coefficients is taken to be the group of rational numbers.

An essential cycle is a cycle not (\sim 0) on some carrier of its own. If a cycle is essential, then its essential carrier is such a carrier on which the cycle is not (\sim 0). We shall say that a (k)-dimensional essential cycle (z^k) is mapped to zero by means of (f) (written (f(z^k)=0)) if some essential carrier (\Phi) of the cycle (z^k) is mapped either into a set containing no nonzero (k)-dimensional cycles, or into a (k)-dimensional orientable manifold (M^k), and the cycle (f(z^k)\sim 0) in (M^k). In what follows, by the words carrier of an essential cycle we shall always mean an essential carrier.

§ 2. In Theorems 1, 2, 3 the cycle (z^k), which is mapped to zero, (0 \leq k \leq n-1), is a cycle arising as the result of a “cutting” of a ((k+1))-dimensional cycle (z^{k+1}). This means the following. There is a theorem, proved by P. S. Aleksandrov and called by him the “Fragma—Brouwer theorem.”

Let a compactum (B), lying in Euclidean space (R^m) and being an essential carrier of the cycle (z^{k+1}), be represented as the sum of two compacta: (B=B_1 \cup B_2). If (B_1) and (B_2) are such that a cycle (y^q), linked with (z^{k+1}) and lying in (R^m \setminus B), is homologous to zero in (R^m \setminus B_1) and in (R^m \setminus B_2), then in (B_1 \cap B_2) there lies a (k)-dimensional cycle (z^k), not homologous to zero in (B_1 \cap B_2). This cycle (z^k \sim 0) in (B_1) and in (B_2) and is the boundary of the chain formed by all simplices of the cycle (z^{k+1}) that have at least one vertex in (B_1). Of such a cycle (z^k) we shall say that it is formed as the result of “cutting” the cycle (z^{k+1}).

Theorem 1. Let a compactum (F), lying in Euclidean space (R^m), be an essential carrier of the (n)-dimensional cycle (z^n). Let (f) be such a continuous mapping of the compactum (F) into an (n)-dimensional closed orientable manifold (M^n) that the cycle (z^n) is mapped to zero. Then for every (k), (0 \leq k \leq n-1), in (F) there exists a (k)-dimensional

nontrivial cycle (z^k), homologous to zero in (F), which is mapped to zero.

If (F) is taken to be an (n)-dimensional manifold, then one obtains the answer to the problem posed in § 1.

The proof of Theorem 1 is obtained by successive application of Theorem 2.

Theorem 2. Let a compact set (F), lying in (R^m), be an essential carrier of an (n)-dimensional cycle (z^n), and let (f) be such a continuous mapping of (F) into an (n)-dimensional closed orientable manifold (M^n) that (f(z^n)=0). Then in (F) there exists an ((n-1))-dimensional cycle (z^{n-1}), homologous to zero in (F), and such an essential carrier (F_1) of the cycle (z^{n-1}) that the set (f(F_1)) lies in a polyhedron (S^{n-1}), homeomorphic to an ((n-1))-dimensional sphere, and the cycle (f(z^{n-1})\sim 0) in (S^{n-1}), i.e. (f(z^{n-1})=0).

Proof. Let (y^q) be a polyhedral cycle in (R^m\setminus F), linked with (z^n), (q+n=m-1). Number all (n)-dimensional simplexes of (M^n) in such a sequence (T_1,T_2,\ldots,T_N) that, for any (k>1), the simplex (T_k) adjoins at least one of the simplexes (T_1,T_2,\ldots,T_{k-1}) along an ((n-1))-dimensional face.

Suppose first that none of the sets (f^{-1}(\overline T_j)), (j=1,2,\ldots,N), contains a set linked with (y^q) (1), i.e.

[
y^q\sim 0\ \text{in } R^m\setminus f^{-1}(\overline T_j),\qquad j=1,2,\ldots,N.
\tag{(\alpha)}
]

Denote by (s) such a natural number that

[
y^q\nsim 0\ \text{in } R^m\setminus f^{-1}!\left(\bigcup_{j=s}^{N}\overline T_j\right),
]

but

[
y^q\sim 0\ \text{in } R^m\setminus f^{-1}!\left(\bigcup_{j=s+1}^{N}\overline T_j\right).
\tag{1}
]

Such an (s) will be found, since, according to ((\alpha)), we have (y^q\sim 0) in (R^m\setminus f^{-1}(\overline T_N)) and (y^q\nsim 0) in
[
R^m\setminus f^{-1}!\left(\bigcup_{j=1}^{N}\overline T_j\right)=R^m\setminus F,
]
and, consequently, among the numbers (1,2,\ldots,N-1) there will be one satisfying condition (1).

Case (s=1). Put
[
B_1=\overline T_1,\qquad B_2=\bigcup_{j=2}^{N}\overline T_j,\qquad A_1=f^{-1}(B_1),
]
[
A_2=f^{-1}(B_2).
]
According to ((\alpha)), (y^q\sim 0) in (R^m\setminus A_1) and, by virtue of (1), (y^q\sim 0) in (R^m\setminus A_2), but (y^q\nsim 0) in (R^m\setminus(A_1\cup A_2)=R^m\setminus F).

According to the theorem of Fragmaen–Brouwer(^1), the set (A=A_1\cap A_2) contains such an ((n-1))-dimensional cycle (z^{n-1}) that (z^{n-1}\nsim 0) on (A) and (z^{n-1}) bounds a chain (x_1) consisting of all simplexes of the cycle (z^n) having at least one vertex in (A_1). We have (f(A)\subset B_1\cap B_2), i.e. (f(A)) is contained in the boundary of the simplex (T_1), which we denote by (S^{n-1}). Therefore
[
f(z^{n-1})=c z_1^{\,n-1},
]
where (z_1^{\,n-1}) is the fundamental cycle on (S^{n-1}).

The chain (f(x_1)=f(z^n)\cap T_1) is the part of the cycle (f(z^n)) lying on (T_1); since (f(z^n)=0), (f(x_1)) covers the simplex (T_1) with degree zero. Further, since
[
\Delta f(x_1)=f(\Delta x_1)=f(z^{n-1})=c z_1^{\,n-1},
]
it follows that (c=0). The cycle (z^{n-1}) and (F_1=A) are the desired ones in this case.

Case (s>1). Put
[
B_1=\overline T_s,\qquad B_2=\bigcup_{j=s+1}^{N}\overline T_j,\qquad A_1=f^{-1}(B_1),\qquad A_2=f^{-1}(B_2).
]
As in the case (s=1), we obtain that (A_1\cap A_2) is an essential carrier of an ((n-1))-dimensional cycle (z^{n-1}). The set (B_1\cap B_2) lies in the part

boundary of the simplex (T_s), which does not contain the interiors of those ((n-1))-dimensional faces along which (T_s) is adjacent to the simplexes (T_1,T_2,\ldots,T_{s-1}). Therefore (B_1\cap B_2) contains no nonzero ((n-1))-dimensional cycles, and (f(z^{n-1})=0), since its carrier (F_1=A_1\cap A_2) is mapped into (B_1\cap B_2). It remains to consider the case when ((\alpha)) is not fulfilled. We have, for some (j_0), (1\le j_0\le N), (y^q\sim 0) in (R^m\setminus f^{-1}(\overline{T}{j_0})). Represent (T); denote the closed half-spaces containing the points (f(p_1)) and (f(p_2)), respectively, by (E_1^n) and (E_2^n). The sets}) as an (n)-dimensional cube (J^n). The set (f^{-1}(T_{j_0})) contains a set (B) linked with (y^q). If the set (f(B)) is a single point, then, since (B) contains a compactum (F_1) that is an essential carrier of an ((n-1))-dimensional cycle (z^{n-1}), homologous to zero in (B) (((^{1})), 1.90), the cycle (z^{n-1}) is the one sought—its carrier (F_1) is mapped to the point (a). In the contrary case, take two points (p_1) and (p_2) in (B) such that (f(p_1)\ne f(p_2)). Regard the cube (J^n) as situated in (n)-dimensional Euclidean space and separate the points (f(p_1)) and (f(p_2)) by an ((n-1))-dimensional hyperplane (E^{n-1
[
B_1=[f^{-1}(E_1^n\cap J^n)]\cap B
]
and
[
B_2=[f^{-1}(E_2^n\cap J^n)]\cap B
]
are proper subsets of (B); hence (y^q\sim 0) in (R^m\setminus B_1) and in (R^m\setminus B_2). The set (B=B_1\cup B_2); by the theorem of Phragmén—Brouwer, (B_1\cap B_2=f^{-1}(E^{n-1}\cap J^n)) is the carrier of a cycle (z^{n-1}), homologous to zero in (B_1) and in (B_2). Since (E^{n-1}\cap J^n) contains no ((n-1))-dimensional cycles, the cycle (z^{n-1}) is the one sought. The theorem is proved.

From the proof just given of Theorem 1 one can obtain Theorem 3, whose direct proof is considerably simpler.

Theorem 3. Let a compactum (F), lying in Euclidean space (R^m), be an essential carrier of an (n)-dimensional cycle (z^n). Let (f) be a continuous mapping of (F) into (n)-dimensional Euclidean space (R^n), or let (f) be such a continuous mapping of (F) into an (n)-dimensional orientable manifold (M^n) that (f(F)) does not contain some point (a) of the manifold (M^n). Then for every (k), (0\le k\le n-1), there exists in (R^n) or in (M^n) a set homeomorphic to (k)-dimensional Euclidean space, whose preimage is an essential carrier of a (k)-dimensional cycle (z^k), homologous to zero in (F), i.e. (f(z^k)=0).

§ 3. Under a mapping of an (n)-dimensional orientable manifold or an (n)-dimensional cycle into (n)-dimensional Euclidean space, the following theorems hold.

Theorem 4. Let (f) be a continuous mapping of an (n)-dimensional closed orientable manifold (M^n) into Euclidean space (R^n). Let (q<n-1), and let (\zeta^q) be a (q)-dimensional cycle of the manifold (M^n), homologous to zero in (M^n), such that the cycle (f(\zeta^q)\sim 0) on (f(\Phi)), where (\Phi) is an essential carrier of (\zeta^q). Then in (R^n) there exists a polyhedron (Q), containing no nonzero (p)-dimensional cycles, (p+q=n-1), whose preimage is an essential carrier of a (p)-dimensional cycle (z^p), homologous to zero in (M^n). Moreover, either (z^p) is linked with a (q)-dimensional cycle (\zeta_1^q) obtained from (\zeta^q) by an (\varepsilon)-modification, or the polyhedron (Q) is the sum of a finite number of simple polygonal lines having no common points.

Theorem 5. Let (f) be a continuous mapping of an (n)-dimensional polyhedral cycle (M^n) into (R^n), and let (q<n-1) and (\zeta^q) be a (q)-dimensional cycle in (M^n), lying entirely in one of the (n)-dimensional simplexes of some triangulation of (M^n). If in (M^n) there exists such an essential carrier (\Phi) of the cycle (\zeta^q) that the cycle (f(\zeta^q)\sim 0) on (f(\Phi)), then in (R^n) there exi

there exists a polyhedron (Q), containing no nonzero (p)-dimensional cycles, (p+q=n-1), whose preimage is an essential carrier of a (p)-dimensional cycle (z^p), homologous to zero in (M^n). In this case either (z^p \sim 0) in (M^n \setminus \Phi_1), where (\Phi_1) is the carrier of the (q)-dimensional cycle (z_1^q) obtained from (\zeta^q) by means of an (\varepsilon)-modification, or the polyhedron (Q) is a sum of a finite number of simple polygonal lines having no points in common.

In conclusion I express my gratitude to P. S. Aleksandrov and K. A. Sitnikov for the great assistance rendered in solving the problem.

Moscow State University
named after M. V. Lomonosov

Received
13 II 1957

REFERENCES

1. P. S. Alexandroff, Dimensions Theorie, 1932, pp. 106, 161.