MATHEMATICS

V. EGOROV

ON UNIFORMLY CONTINUOUS MAPPINGS OF UNIFORM COMPLEXES INTO A SPHERE

(Presented by Academician P. S. Aleksandrov, 11 VI 1960)

The present note is devoted to a generalization of the well-known classification theorem of Hopf for mappings of \(n\)-dimensional polyhedra into the \(n\)-sphere. Below we set forth facts that make it possible to classify uniformly continuous mappings of uniform triangulations* of dimension \(n\) into the \(n\)-sphere.

Basic definitions. A chain \(x=\sum \alpha_i T_i\) of a complex \(K\) will be called uniform if, for every \(i\), the inequality \(\alpha_i<\alpha\)** holds, where \(\alpha\) is a constant independent of \(i\). A \(\nabla\)-cycle will be called uniformly homologous to zero if it is the \(\nabla\)-boundary of some uniform chain. Uniform \(\nabla\)-cycles, as well as uniform \(\nabla\)-cycles uniformly homologous to zero (of the same dimension), form groups. The quotient group of the uniform \(\nabla\)-cycles by the subgroup of cycles uniformly homologous to zero will be called the uniform \(\nabla\)-group of the complex \(K\) and denoted by the symbol \(\nabla_u^n(K)\). Uniformly continuous mappings \(f_0\) and \(f_1\) of a set \(A\) into a set \(B\) will be called uniformly homotopic if, in the direct product \((A,I)\) of the set \(A\) with the unit interval \(I\), there exists a uniformly continuous mapping \(F\) into \(B\), for every \(a\in A\), satisfying the condition \(F(a,0)=f_0(a)\), \(F(a,1)=f_1(a)\)***. The set of all uniformly continuous mappings of a set \(A\) into a set \(B\) decomposes into uniformly homotopic classes.

Main theorem. Between the set of elements of the group \(\nabla_u^n(\widetilde K^n)\) of a uniform complex \(K^n\) and the set of uniformly homotopic classes of the set of uniformly continuous mappings of the triangulation \(\widetilde K^n\) into the sphere \(S^n\) there exists a one-to-one correspondence.

Everything that follows is devoted to the proof of this fact.

Lemma 1. For every uniformly continuous mapping \(f\) of a uniform triangulation \(\widetilde K_1\) into a uniform triangulation \(\widetilde K_2\), there exist simplicial mappings (simplicial approximations) into \(\widetilde K_2\) of barycentric subdivisions of corresponding order of the triangulation \(\widetilde K_1\), uniformly homotopic to the mapping \(f\).

* A triangulation (or complex) is called uniform if: 1) the lengths of all its one-dimensional simplexes are bounded above; 2) the distances between any two simplexes of this triangulation that do not have common vertices are bounded below by some constant \(h>0\) (see the fundamental work on the theory of uniform complexes \((^1)\)).

** In this work all chains and, consequently, all cycles are assumed to be integral; thus all \(\alpha_i\) are integers.

*** By the symbol \((A,\theta)\) we denote the “layer” of the set \((A,I)\) corresponding to the numerical coordinate \(\theta\) \((0\leq \theta\leq 1)\); analogously, \((a,\theta)\in(A,I)\) is the point with coordinates \(a\in A\) and \(\theta\) \((0\leq \theta\leq 1)\).

This lemma follows from the following facts:

I. The stars of vertices (principal stars) of a uniform triangulation \(\widetilde K\) form a Lebesgue covering of the set \(\widetilde K\)* (see \((^2)\)).

II. Every simplicial mapping of a uniform triangulation \(\widetilde K_1\) into a uniform triangulation \(\widetilde K_2\) is uniformly continuous.

III. The barycentric subdivision of a uniform triangulation is itself a uniform triangulation.

IV. Suppose that on the “bases” \((A,0)\) and \((A,1)\) of the set \((A,I)\) there are defined mappings, respectively, \(f_0\) and \(f_1\) into \(E^N\), and that
\[ \rho[f_0(a,0); f_1(a,1)]<l \]
(\(l\) is a constant and \(a\) is any point of \(A\)). If \(f_0\) and \(f_1\) are uniformly continuous, then the mapping \(F\) of the set \((A,I)\), defined as follows, is also uniformly continuous:
\[ \overrightarrow{OF}(a,\theta)=\overrightarrow{Of_0}(a,0)+\theta\,[\overrightarrow{Of_1}(a,1)-\overrightarrow{Of_0}(a,0)] \]
(\(O\) is some fixed point of the space \(E^N\)).

Lemma 2. For every uniformly continuous mapping \(f\) of a uniform triangulation \(\widetilde K^n\) into the sphere \(S^n\), there exists a mapping of the triangulation \(\widetilde K^n\) into \(S^n\), uniformly homotopic to the mapping \(f\), which sends the \((n-1)\)-dimensional skeleton of the triangulation \(\widetilde K^n\) to a single point of the sphere \(S^n\).

We shall call such mappings special.

Let \(\varphi\) be a special uniformly continuous mapping of \(\widetilde K^n\) onto the oriented boundary \(S^n\) of some \((n+1)\)-dimensional simplex, and let \(\widetilde K^{\prime n}\) be a barycentric subdivision of the triangulation \(\widetilde K^n\) of such order that the image of each principal star of the triangulation \(\widetilde K^{\prime n}\) under the mapping \(\varphi\) is contained in some principal star of the triangulation \(S^n\). Define on \(\widetilde K^{\prime n}\) a simplicial approximation \(g\) uniformly homotopic to the mapping \(\varphi\). Under the mapping \(g\), the image of each oriented \(n\)-simplex \(T_i^n\in \widetilde K^{\prime n}\) will be a cycle \(\gamma_i S^n\) of the sphere \(S^n\). The \(\nabla\)-cycle
\[ Z_\varphi=\sum_i \gamma_i T_i^n \]
of the complex \(K^n\) is called the degree of the mapping \(\varphi\). Let
\[ x^n=\sum_i \alpha_i T_i^n \]
be a chain of the complex \(K^n\). The number
\[ \gamma_\varphi(x^n)=\sum_i \alpha_i\gamma_i \]
is called the degree of the mapping of the chain \(x^n\)***.

Theorem 1. The degree of a uniformly continuous mapping of a uniform triangulation \(\widetilde K^n\) into the sphere \(S^n\) is a uniform \(\nabla\)-cycle.

Theorem 2. Every uniform \(\nabla\)-cycle \(Z^n\) of a uniform complex \(K^n\) determines a special uniformly continuous mapping of the triangulation \(\widetilde K^n\) into the sphere \(S^n\), and the cycle \(Z^n\) itself is the degree of this mapping.

Lemma 3. Let uniformly homotopic mappings \(f_0\) and \(f_1\) into the sphere \(S\) be defined on a subset \(B\) of a set \(A\). If one of these mappings, for example \(f_0\), has a uniformly continuous extension \(F_0\) to the set \(A\), then the mapping \(f_1\) also extends to some mapping \(F_1\) of the set \(A\) into the same sphere \(S\), uniformly homotopic to the mapping \(F_0\).

Theorem 3. If special mappings \(\varphi_0\) and \(\varphi_1\) of a uniform triangulation \(\widetilde K^n\) into the sphere \(S^n\) are uniformly homotopic, then the degrees of these mappings are uniformly homologous.

* A covering \(\lambda\) is called a Lebesgue covering if there exists a covering \(\{c_i\}\) such that the covering \(\{O(\eta,c_i)\}\) is inscribed in \(\lambda\), \(\eta>0\).

** The reality of such an assumption is guaranteed by the uniformity of the triangulation \(\widetilde K^n\) and the uniform continuity of the mapping \(\varphi\).

*** \(\gamma_i\) does not depend on the choice of subdivisions (see, for example, \((^3)\)).

**** Only those chains are meant for which the degree defined in this way exists.

Indeed, represent the set \((\widetilde K^n,I)\) as a prism \(\Pi \widetilde K^n\) over the triangulation \(\widetilde K^n\), and on its bases \((\widetilde K^n,0)\) and \((\widetilde K^n,1)\) define respectively the mappings \(\varphi_0\) and \(\varphi_1\). Lemma 3 permits us to assume that the mapping \(\Phi\), which realizes a uniform homotopy of the mappings \(\varphi_0\) and \(\varphi_1\), sends the entire \((n-1)\)-dimensional skeleton of the prism \(\Pi \widetilde K^n\) to one point of the sphere \(S^n\). We define the \((n-1)\)-dimensional chain \(x^{n-1}\), which realizes a uniform homology of the degrees of the mappings \(\varphi_0\) and \(\varphi_1\), on each simplex \(T^{n-1}\in \widetilde K^n\) to be the degree of the mapping \(\gamma_\Phi(\Pi,T^{n-1})\) of the prism \(\Pi T^{n-1}\). Note that the uniform discontinuity of the mapping \(\Phi\) implies the uniformity of the chain constructed.

Theorem 4. If the degrees of uniformly continuous special mappings \(\varphi_0\) and \(\varphi_1\) of the uniform triangulation \(\widetilde K^n\) into \(S^n\) are uniformly homologous, then \(\varphi_0\) and \(\varphi_1\) are uniformly homotopic.

Proof. Consider the mapping of the sum of the upper and lower bases, coinciding with \(\varphi_0\) on the lower base and with \(\varphi_1\) on the upper. Next we construct a uniformly continuous extension to the prism over the \((n-1)\)-dimensional skeleton of the triangulation \(\widetilde K^n\). We extend the resulting mapping to the entire \(n\)-dimensional skeleton of the prism \(\Pi \widetilde K^n\) (denote this extension by \(\Phi\)). In this case the boundary of each simplex \(T_i^{n+1}\) of the prism \(\Pi \widetilde K^n\) is mapped into \(S^n\) inessentially, and the mapping \(\Phi\) sends the entire \((n-1)\)-dimensional skeleton of the prism \(\Pi K^n\) to one point. Construct for \(\Phi\) a simplicial approximation \(G\), based on a certain suitable barycentric subdivision of the \(n\)-dimensional skeleton of the prism \(\Pi \widetilde K^n\). Let \(T^n\) be some (for example, regular) complex whose boundary is subdivided by a barycentric subdivision of the same order as the \(n\)-dimensional skeleton of the prism \(\Pi \widetilde K^n\), and let \(T_i^{n+1}\) be an arbitrary simplex of the prism \(\Pi \widetilde K^n\). Assigning to the vertices of the simplex \(T_i^{n+1}\) bijectively the vertices of the simplex \(T^{n+1}\), to the center of gravity of the simplex \(T_i^{n+1}\) the center of gravity of the simplex \(T^{n+1}\), to the centers of gravity of the faces of the simplex \(T_i^{n+1}\) the centers of gravity of the corresponding faces of the simplex \(T^{n+1}\), and step by step doing the same for barycentric subdivisions of higher orders, we define linearly a simplicial homeomorphism \(u_i\) of the simplex \(T_i^{n+1}\) onto the simplex \(T^{n+1}\). Consider the system of mappings \(g_i=Gu_i^{-1}\) of the boundary of the simplex \(T^{n+1}\). This system splits into a finite number of classes, each of which contains identically coinciding mappings. The inessentiality of the mappings \(g_i\) permits each mapping \(g_i\) to be extended to a mapping \(\overline G_i\) into \(S^n\) of the entire simplex \(T^{n+1}\), and the system \(\{\overline G_i\}\) (just as \(\{g_i\}\)) contains only a finite number of essentially distinct classes, and therefore (just as \(\{g_i\}\)) is a uniformly continuous system of mappings. Putting, for every point \(a\in T_i^{n+1}\), \(F(a)=\overline G_i[u_i(a)]\), we obtain the required extension of the mapping \(G\) to the entire prism \(\Pi \widetilde K^n\). Applying Lemma 3, we uniformly continuously extend to \(\Pi K^n\) also the mapping \(\Phi\). The theorem is proved.

Thus, by assigning to each homotopy class the element of the group \(\nabla_u^n(K^n)\) containing the degree of some mapping from the given homotopy class, we construct the required one-to-one correspondence.

We express our gratitude to Prof. Yu. M. Smirnov, and also to V. Kuzmin, for the assistance rendered in writing this work.

Received
11 VI 1960

REFERENCES CITED

1. Yu. M. Smirnov, Matem. sbornik, 40, No. 2, 137 (1956).
2. V. I. Egorov, Matem. sbornik, 48, No. 2, 227 (1959).
3. V. G. Boltyanskii, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 47 (1955).