UDC 513.83

MATHEMATICS

Academician P. ALEKSANDROV

ON THE FUNDAMENTAL THEOREM OF HOMOLOGICAL DIMENSION THEORY

1. In this note the fundamental theorem of homological dimension theory—in a form directly adjoining the one originally given by me in 1930 (see (¹))—is proved for bicompacta (and consequently also for arbitrary normal spaces)*. The novelty consists in the fact that, with maximum generality, the formulation proposed here is extremely simple and transparent: I use only the concepts of a cycle and of homology on a finite simplicial complex and dispense with any inverse limits whatever. The proof rests only on the elementary case of Hopf’s theorem on mappings of an \(n\)-dimensional polyhedron onto an \(n\)-dimensional complex (see, for example, (²), p. 70).

2. By a covering in this paper is always meant a finite open covering. The nerve of a covering \(\omega\) is denoted by \(N_\omega\). If \(\Phi \subset X\), then \(N_{\omega\Phi}\) denotes the subcomplex of the nerve \(N_\omega\) consisting of all simplices all vertices of each of which correspond to elements of the covering \(\omega\) having common points with the set \(\Phi\). By the dimension of a space \(X\) we shall always mean the dimension \(\dim X\), defined by means of coverings.

The following is proved.

Fundamental theorem (Theorem 1). If a bicompactum \(X\) has (finite) dimension \(\dim X = n > 0\), then there exist a closed set \(\Phi \subset X\) and a covering \(\omega\) of the space \(X\) such that for every covering \(\omega'\) inscribed in \(\omega\) the following assertions may be made:

(a) On the nerve \(N_{\omega'}\) there is an \(n\)-dimensional relative cycle \(z_{\omega'}^n \bmod N_{\omega'\Phi}\) with respect to some (depending on \(\omega'\)) modulus \(m=m_{\omega'}\), whose projection \(\delta_\omega^{\omega'} z_{\omega'}^n\) into the nerve \(N_\omega\) is not homologous to zero on the complex \(N_\omega, \bmod N_{\omega\Phi}\).

(b) On the subcomplex \(N_{\omega'\Phi}\) of the nerve \(N_{\omega'}\) there is an \((n-1)\)-dimensional cycle \(z_{\omega'}^{\,n-1}\) modulo \(m_{\omega'}\), homologous to zero on \(N_{\omega'}\), and possessing the property that its projection \(\delta_\omega^{\omega'} z_{\omega'}^{\,n-1}\) is a cycle (of the complex \(N_{\omega\Phi}\)) not homologous to zero on \(N_{\omega\Phi}\).**

Finally:

(c) For \(r>n\), for every covering \(\omega\) of the space \(X\) there is a covering \(\omega'\) inscribed in \(\omega\) (of the space \(X\)) such that, for any closed \(\Phi \subset X\), every \(r\)-dimensional relative cycle \(z_{\omega'}^{\,r}\) on \(N_{\omega'} \bmod N_{\omega'\Phi}\) and every \((r-1)\)-dimensional cycle \(z_{\omega'}^{\,r-1}\) on \(N_{\omega'\Phi}\), homologous to zero on \(N_{\omega'}\), are equal to zero.

Assertion (c) is obvious: it suffices to take as \(\omega'\) any covering inscribed in \(\omega\) of multiplicity \(n+1\).

1. We begin the proof of the main part of the theorem with some preliminary remarks.

By \(\bar Q\) we shall denote once and for all an \(n\)-dimensional closed complex with center of gravity \(c\) and interior \(Q\), lying in \(n\)-dimensional Euclidean space \(R^n\), in turn embedded in some \(R^{2n+1}\). Under

* The passage from normal spaces \(X\) to bicompacta is effected automatically by means of the maximal bicompact extension \(\beta X\).

** From what has been said it evidently follows that \(\delta_\omega^{\omega'} z_{\omega'}^{\,n-1} \sim 0\) on \(N_\omega\) and, on the other hand, \(z_{\omega'}^{\,n-1}\) is not \(\sim 0\) on \(N_{\omega'\Phi}\) (just as \(z_\omega^n\) is not \(\sim 0\) on \(N_\omega, \bmod N_{\omega\Phi}\)).

By a “smaller” simplex \((\bar Q_0, \bar Q_1,\) etc.) we shall always mean a simplex with the same center of gravity as, and homothetic to, the basic simplex \(\bar Q\) (hence lying strictly inside \(Q\)).

Let \(\varphi\) be a continuous mapping of a once-and-for-all given \(n\)-dimensional bicompactum \(X\) into the simplex \(\bar Q\). Let \(\omega=\{O_1,\ldots,O_s\}\) be a covering of the bicompactum \(X\) of multiplicity \(n+1\). If \(\varepsilon>0\) is given, then by \(N_\omega\) we shall denote the nerve of the covering \(\omega\), realized in the following way as a triangulation lying in \(R^{2n+1}\supset R^n\supset \bar Q\). For each \(i=1,2,\ldots,s\) choose a point \(p_i\in O_i\), and, moreover, if a closed set \(\Phi\subset X\) has been distinguished, then for all \(O_i\in\omega\) with \(O_i\cap\Phi\ne\Lambda\) choose \(p_i\in O_i\cap\Phi\). We choose the vertices \(e_1,\ldots,e_s\) of the nerve \(N_\omega\) in \(R^{2n+1}\) in general position so that
\[ \rho(e_i,\varphi p_i)<\varepsilon \]
(for \(i=1,2,\ldots,s\)). A realization of the nerve \(N_\omega\) satisfying these conditions will be called canonical (relative to the mapping \(\varphi\) and the number \(\varepsilon\)).

Assigning to each vertex \(e_i\) the point \(\varphi p_i\), we obtain a simplicial mapping \(g_\omega\) of the nerve \(N_\omega\) into \(\bar Q\), which we shall call the canonical \(\varepsilon\)-shift (of the complex \(N_\omega\)).

If the covering \(\omega\) is normal (i.e. consists of open \(F_\sigma\)-sets), then by \(\mu_\omega\) we denote the barycentric mapping of the bicompactum \(X\) into the body \(\widetilde N_\omega\) of the nerve \(N_\omega\).

4. Lemmas.

Lemma 1. Let a mapping \(\varphi\to\bar Q\) and a number \(\varepsilon>0\) be given. Then there exist an \(\varepsilon'>0\) and a covering \(\omega=\{O_1,\ldots,O_s\}\) of the bicompactum \(X\) such that, for any canonical realization of \(N_\omega\) (relative to \(\varphi\) and \(\varepsilon'\)) and the corresponding canonical \(\varepsilon'\)-shift \(g_\omega:N_\omega\to\bar Q\), for every normal covering \(\omega'\) inscribed in \(\omega\), the mapping \(f=g_\omega \delta_\omega^{\omega'}\mu_{\omega'}:X\to\bar Q\) satisfies the inequality
\[ \rho(\varphi x,fx)<\varepsilon \quad \text{for all } x\in X . \]

Indeed, take \(\varepsilon'<\varepsilon/3\) and open sets \(V_1,\ldots,V_\nu\) in the space \(R^n\), covering the simplex \(\bar Q\) and having diameters \(<\varepsilon'\). Put \(U_j=\varphi^{-1}V_j\) for \(j=1,2,\ldots,\nu\), and take a covering \(\omega=\{O_1,\ldots,O_s\}\) of the bicompactum \(X\), of multiplicity \(n+1\), inscribed in the covering \(\{U_1,\ldots,U_\nu\}\). Inscribe in \(\omega\) any normal covering \(\omega'=\{O'_1,\ldots,O'_{s'}\}\) of multiplicity \(n+1\).

Let the nerve \(N_\omega\) be canonically realized at the vertices \(e_1,\ldots,e_s\), and the nerve \(N_{\omega'}\) at the vertices \(e'_1,\ldots,e'_{s'}\) (both realizations are relative to the mapping \(\varphi\) and the number \(\varepsilon'\)).

Take an arbitrary point \(x\in X\); suppose it is contained in
\[ O'_{k_0},\ldots,O'_{k_r} \]
and only in these elements of the covering \(\omega'\). Put
\[ e_{i_\lambda}=\delta_\omega^{\omega'} e'_{k_\lambda},\quad \lambda=0,\ldots,r. \]
Then
\[ \mu_{\omega'}x\in |e'_{k_0}\cdots e'_{k_r}|\in N_{\omega'},\qquad \delta_\omega^{\omega'}|e'_{k_0}\cdots e'_{k_r}|=|e_{i_0}\cdots e_{i_r}|, \]
so that
\[ \delta_\omega^{\omega'}\mu_{\omega'}x\in |e_{i_0}\cdots e_{i_r}|. \]
Further,
\[ x\in O'_{k_\lambda}\subset O_{i_\lambda}\subset U_{j_\lambda}. \]
Hence
\[ \rho(\varphi x,\varphi p_{i_\lambda})<\varepsilon',\qquad \rho(\varphi x,e_{i_\lambda})<2\varepsilon' \]
(all this for \(\lambda=0,\ldots,r\)). Since
\[ \delta_\omega^{\omega'}\mu_{\omega'}x\in |e_{i_0}\cdots e_{i_r}|, \]
we have
\[ \rho(\varphi x,\delta_\omega^{\omega'}\mu_{\omega'}x)<2\varepsilon', \]
and therefore
\[ \rho(\varphi x,g_\omega\delta_\omega^{\omega'}\mu_{\omega'}x)<3\varepsilon'<\varepsilon, \]
which proves Lemma 1.

Lemma 2. Let \(\varphi:X\to\bar Q\) be an essential mapping. Then, for any smaller simplex \(\bar Q_0\subset Q\), one can find a covering \(\omega=\{O_1,\ldots,O_s\}\) of the bicompactum \(X\) and a number \(\varepsilon>0\) such that, for an arbitrary normal covering \(\omega'=\{O'_1,\ldots,O'_{s'}\}\), of multiplicity \(n+1\), inscribed in \(\omega\), and for the canonical \(\varepsilon\)-shift \(g_\omega:N_\omega\to\bar Q\), the simplicial mapping
\[ \psi=g_\omega\delta_\omega^{\omega'}:\widetilde N_{\omega'}\to\bar Q \]
essentially covers the simplex \(\bar Q_0\) (i.e. the mapping of the polyhedron \(\psi^{-1}\bar Q_0\) onto \(\bar Q_0\) is essential).

The proof is based on the following slight modification of the main lemma for the theorem on essential mappings (see, for example, (³), Ch. 6, p. 217), whose proof we leave to the reader:

Lemma 2₀. With the notation adopted above, there is an \(\varepsilon > 0\) such that every continuous mapping \(f: X \to \bar Q\) differing from \(\varphi\) by less than \(\varepsilon\) essentially covers the simplex \(\bar Q_0\).

But by Lemma 1 there exists a covering \(\omega\) of the bicompactum \(X\) such that for every \(\omega'\) inscribed in \(\omega\), for sufficiently small \(\varepsilon'\), and for the corresponding canonical \(\varepsilon'\)-shift \(g_\omega: N_\omega \to \bar Q\), we have the inequality

\[ \rho(\varphi x,\; g_\omega \widetilde{\omega}^{\omega'}\mu_{\omega'}x)<\varepsilon \quad \text{for all } x\in X. \]

Hence the mapping \(f=g_\omega\widetilde{\omega}^{\omega'}\mu_{\omega'}:X\to\bar Q\) essentially covers \(\bar Q_0\). We show that then also the simplicial mapping \(\psi=g_\omega\widetilde{\omega}^{\omega'}:\widetilde N_{\omega'}\to\bar Q\) essentially covers \(\bar Q_0\). Let \(X_0=f^{-1}\bar Q_0=\mu_{\omega'}^{-1}\psi^{-1}Q_0\).

Put \(Y=\psi^{-1}\bar Q_0\subseteq \widetilde N_{\omega'}\); then \(X_0=\mu_{\omega'}^{-1}Y\). Denoting by \(S_0\) the boundary of the simplex \(Q_0\), put further \(\Psi=\psi^{-1}S_0\subset Y\). It is necessary to prove that the mapping \(\psi:Y\to\bar Q_0\) is essential. Otherwise there is a mapping \(\psi_1:Y\to S_0\) for which \(\psi_1y=\psi y\) for every \(y\in\Psi\).

We consider the barycentric mapping \(\mu_{\omega'}:X_0\to Y\) and define \(\varphi_1=\psi_1\mu_{\omega'}:X_0\to S_0\). For \(x\in f^{-1}S_0\) we have \(S_0\ni fx=g_\omega\widetilde{\omega}^{\omega'}\mu_{\omega'}x=\psi\mu_{\omega'}x\), i.e. \(\mu_{\omega'}x\in\psi^{-1}S_0=\Psi\), and, consequently, by the definition of \(\psi_1\) we have \(\psi_1\mu_{\omega'}x=\psi\mu_{\omega'}x\), while by the definition of \(\varphi_1\) we have \(\varphi_1x=\psi_1\mu_{\omega'}x=\psi\mu_{\omega'}x=fx\). Thus we have a mapping \(\varphi_1:X_0\to S_0\) coinciding with \(f\) on \(f^{-1}S_0\), contrary to the essentiality of the mapping \(f\). Lemma 2 is proved.

5. Proof of assertion (a) of Theorem 1. Keeping the notation of Lemmas 1 and 2, take smaller simplexes \(\bar Q_1\subset Q_0\) and \(\bar Q_2\subset Q_1\). Put \(\Phi=\varphi^{-1}(\bar Q\setminus Q_1)\subset X\). Suppose that \(\varepsilon\) in Lemma 2 is less than half the distance from each smaller simplex to its complement (in \(R^n\)) in the larger one. Then

\[ f^{-1}S_0\subseteq \Phi . \tag{1} \]

Otherwise we would have a point \(x_0\in X\) for which \(fx_0\in S_0\), but \(\varphi x_0\in Q_1\), and hence \(\rho(\varphi x_0,fx_0)>\varepsilon\).

We derive from formula (1) the formula

\[ \psi^{-1}S_0\subseteq N_{\omega\Phi}\cap Y;\qquad N_{\omega\Phi}\subseteq \psi^{-1}(Q\setminus Q_2). \tag{2} \]

We prove the first inclusion in (2). If* \(y\in\psi^{-1}S_0\) and \(x\in\mu_{\omega'}^{-1}y\), then \(fx=\psi\mu_{\omega'}x\in S_0\), i.e. \(x\in f^{-1}S_0\subseteq\Phi\), and every \(O_k'\in\omega'\) containing the point \(x\) intersects \(\Phi\).

Let \(x\) be contained in \(O_{k_0}',\ldots,O_{k_r}'\), and only in these elements of the covering \(\omega'\). Then \(y=\mu_{\omega'}x\in |e_{k_0}\ldots e_{k_r}|\in N_{\omega\Phi}\). Thus, \(\psi^{-1}S_0\subseteq N_{\omega\Phi}\). Since, moreover, \(\psi^{-1}S_0\subseteq Y\), the first inclusion in formula (2) is proved.

We prove the second inclusion. Let \(y\in |e_{k_0}\ldots e_{k_r}|\in N_{\omega\Phi}\). We must reduce to a contradiction the inclusion \(\psi y\in Q_2\). For this we take some \(x\in\mu_{\omega'}^{-1}y\). The inclusion \(\psi y\in Q_2\) means \(fx\in Q_2\). Since \(\rho(\varphi x,fx)<\varepsilon\), it follows that for every \(O_k'\subset\omega'\) containing \(x\), we have (recalling that \(\operatorname{diam}\varphi O_k'<\varepsilon\)) the inclusions

\[ \varphi O_k'\subset O(\varphi x,\varepsilon)\subseteq O(fx,2\varepsilon)\subseteq Q_1, \]

and hence \(O_k'\subseteq\varphi^{-1}Q_1\). On the other hand, the inclusion \(y\in |e_{k_0}\ldots e_{k_r}|\in N_{\omega\Phi}\) means that \(x\) is contained in \(O_{k_0}',\ldots,O_{k_r}'\) and only in these elements of the covering \(\omega'\), and that each of these \(O_{k_0}',\ldots,O_{k_r}'\) intersects \(\Phi=\varphi^{-1}(\bar Q\setminus Q_1)\). The required contradiction has been obtained.

* We may always assume (replacing, if necessary, the covering \(\omega'\) by an inscribed covering subordinate to it) that \(\mu_{\omega'}:X\to N_{\omega'}\) is a mapping onto \(N_{\omega'}\). In this case \(\omega'\) is called irreducible if there is no covering inscribed in it whose nerve is a proper subcomplex of the nerve \(N_{\omega'}\).

Remark. Let us prove the formula

\[ g_\omega \widetilde N_{\omega\Phi}\subseteq \overline Q\setminus Q_2. \tag{3\(_\omega\)} \]

Indeed, the inclusion (3\(_\omega\)) follows from the fact that all vertices of the complex \(g_\omega N_{\omega\Phi}\) are points of the form \(\varphi p_i\), where \(p_i\in O_i\cap \Phi\); that the simplices of this complex have diameter \(<2\varepsilon\), and that \(\varphi\Phi\subseteq \overline Q\setminus Q_1\).

Let us sum up: we have an essential mapping \(\psi=g_\omega \widetilde{\mathfrak d}_{\omega}^{\omega'}\) of the polyhedron \(Y\) (of dimension \(n\)) onto the \(n\)-dimensional complex \(\overline Q_0\), under which \(\psi^{-1}S_0\) is contained in the polyhedron \(\widetilde N_{\omega\Phi}\cap Y=\Pi_0\subseteq \psi(\overline Q_0\setminus Q_2)\). Denote now by \(N_{\omega'}^{(1)}\) such a subdivision of the triangulation \(\widetilde N_{\omega'}\) that the polyhedron \(Y\) is the body \(\widetilde K\) of some complex \(K\subseteq N_{\omega'}^{(1)}\), and the polyhedron \(\Pi_0\) is the body of some complex \(K_0\subseteq K\).

We are in the conditions of the already mentioned theorem of Hopf ((\(^{2}\)), p. 70), by virtue of which on the complex \(K\) there is a relative cycle \(z_{\omega'_1}^n \bmod K_0\), with respect to some modulus \(m=m_{\omega'}\), which covers under the mapping \(\psi:Y\to \overline Q_0\) the point \(c\) with degree \(\operatorname{gr}_c\psi=\gamma\ne0\). Under the canonical shift \(\sigma:N_{\omega'}^{(1)}\to N_{\omega'}\) we have \(\sigma K_0\subseteq N_{\omega'\Phi}\), and the relative cycle \(z_{\omega'_1}^n\bmod K_0\) passes into the relative cycle \(\sigma z_{\omega'_1}^n=z_{\omega'}^n\) of the complex \(N_{\omega'}\), \(\bmod N_{\omega'\Phi}\), while the boundary \(z_{\omega'_1}^{\,n-1}=\Delta z_{\omega'_1}^n\) passes (remaining throughout in the polyhedron \(\widetilde N_{\omega'\Phi}\subseteq \overline Q\setminus Q_2\)) into the cycle \(z_{\omega'}^{\,n-1}\) of the complex \(N_{\omega'}\). At the same time the cycle \(\psi z_{\omega'_1}^{\,n-1}\) undergoes a deformation in \(\overline Q\setminus Q_2\) which does not change the linking coefficient \(\nu(c,\psi z_{\omega'_1}^{\,n-1})=\operatorname{gr}_c\psi z_{\omega'_1}^n\), so that \(\operatorname{gr}_c\psi z_{\omega'}^n=\operatorname{gr}_c\psi z_{\omega'_1}^n=\gamma\ne0\). Obviously, \(\widetilde{\mathfrak d}_{\omega}^{\omega'}z_{\omega'}^n=z_\omega^n\) is a relative cycle of the complex \(N_\omega\), \(\bmod N_{\omega\Phi}\), with respect to the same modulus \(m=m_{\omega'}\), and \(\operatorname{gr}_c g_\omega z_\omega^n=\operatorname{gr}_c g_\omega \widetilde{\mathfrak d}_{\omega}^{\omega'}z_{\omega'}^n=\operatorname{gr}_c\psi z_{\omega'}^n=\gamma\ne0\). Consequently, \(z_\omega^n\) is not \(\sim0\) on \(N_\omega\bmod N_{\omega\Phi}\), as was required to prove.

6. Proof of assertion (6) of Theorem 1. We keep our notation. The projection \(\mathfrak d_{\omega}^{\omega'}z_{\omega'}^{\,n-1}\) is a cycle of the complex \(N_{\omega\Phi}\). Let us prove that it is not homologous to zero on \(N_{\omega\Phi}\). Otherwise, let \(\mathfrak d_{\omega}^{\omega'}z_{\omega'}^{\,n-1}\) be the boundary of a chain \(x_\omega^n\) of the complex \(N_{\omega\Phi}\). By formula (3\(_\omega\)), the chain \(g_\omega x_\omega^n\) lies in \(\overline Q\setminus Q_2\), so that \(\operatorname{gr}_c g_\omega x_\omega^n=0\); therefore for the cycle \(y_\omega^n=z_\omega^n-x_\omega^n\) we have \(\operatorname{gr}_c g_\omega y_\omega^n=\gamma\ne0\), which is obviously impossible, since the cycle \(g_\omega y_\omega^n\), like every \(n\)-dimensional cycle on the \(n\)-dimensional complex \(\overline Q_0\), is homologous in \(\overline Q_0\) to zero. Theorem 1 is proved.

In the metric case Theorem 1 assumes, as is easy to establish, the following form:

Theorem 2. For every compactum \(X\) of finite dimension \(n\ge1\) there exist a closed set \(\Phi\subset X\) and an \(\varepsilon>0\) such that for every (arbitrarily small) \(\varepsilon'>0\) there is, first, an \(n\)-dimensional relative \(\varepsilon'\)-cycle on \(X\bmod\Phi\), with respect to some (depending on \(\varepsilon'\)) modulus \(m_{\varepsilon'}\), not \(\varepsilon\)-homologous to zero on \(X\bmod\Phi\), and, secondly, on \(\Phi\) there is an \((n-1)\)-dimensional \(\varepsilon'\)-cycle \(z_{\varepsilon'}^{\,n-1}\) modulo \(m_{\varepsilon'}\), \(\varepsilon'\)-homologous to zero on \(X\), but not even \(\varepsilon\)-homologous to zero on \(\Phi\).

Here \(n\) is the greatest natural number for which at least one of these assertions holds.

The formulation of Theorem 2 (entirely in the spirit of Brouwer’s classical works) is in an obvious way equivalent to my main theorem in its original formulation (paper (\(^{1}\)), p. 195).

Mechanical-Mathematical Faculty
of Moscow State University
named after M. V. Lomonosov

Received
15 II 1968

REFERENCES

\(^{1}\) P. Alexandroff. Math. Ann., 106, 161 (1932).
\(^{2}\) P. S. Aleksandrov, UMN, 4, 6, 17 (1949).
\(^{3}\) P. S. Aleksandrov, Combinatorial Topology, Moscow, 1947.