UDC 513.821.838

MATHEMATICS

E. S. TIKHOMIROVA

THE SPECTRUM OF UNIFORM HOMOLOGIES

(Presented by Academician L. S. Pontryagin, 5 IX 1969)

In the paper \((^4)\) groups of uniform homologies were constructed. These groups, however, do not make it possible to distinguish even such nonhomeomorphic manifolds as \(w=u^2+v^2\) and \(w=u^4+v^4\) in \(E^3\). Let us note that between these manifolds there is the following distinction. In the first of them take a cycle \(Z\) with support \(w=c\) and an \(m\)-bounded chain \(X\) with support \(|X|\) of minimal diameter. It is easy to see that \(d(X)\), as a function of \(d(Z)\) (\(d(X)\) is the diameter of the set \(|X|\)), has order of growth equal to the order of growth of the function \(\varphi(t)=t^2\). An analogous construction for the second manifold gives the order of growth of the function \(\varphi(t)=t^4\).

In the present paper groups are constructed that reveal the indicated difference in orders of growth. To each order of growth there is assigned a group of uniform homologies of a metric space; moreover, with respect to the natural direction in the set of orders of growth, these groups form an inverse spectrum. In particular, for a function of the same order as \(\varphi(t)=t\), on manifolds with a uniform metric we obtain the groups \(Q_r\) constructed earlier \((^4)\).

1. We shall need the following definitions.

A. A Riemannian manifold \(R^n\) is called a manifold with a uniform metric \((^4)\) if there exist \(\gamma_1>0\) and \(\gamma_2>0\) such that for every point \(x\in R^n\) there exists a mapping \(f\) of some neighborhood \(U_x\) of this point onto the Euclidean \(n\)-dimensional ball of unit radius, satisfying the condition

\[ \gamma_1 \leq \rho(\bar{x},\bar{\bar{x}})/\rho(f(\bar{x}),f(\bar{\bar{x}})) \leq \gamma_2, \tag{1} \]

where \(\bar{x}\) and \(\bar{\bar{x}}\) are arbitrary points of \(U_x\).

B. We shall use the notion of the order of growth of a function only for continuous increasing functions satisfying the condition \(\varphi(t)\geq t>0\). The corresponding definition is given for this case in a form convenient for our purposes: the order of growth of the function \(\varphi(t)\) is not less than the order of growth of the function \(\psi(t)\) if there exists a constant \(\alpha>0\) such that \(\varphi(t)\geq \alpha\psi(\alpha t)\). The order of growth of the function \(\varphi(t)\) is denoted by \([\varphi]\). If \([\varphi]\geq[\psi]\) and \([\psi]\geq[\varphi]\), then \([\varphi]=[\psi]\).

1. Let \(R\) be a metric space, and let the function \(\varphi(t)\) satisfy the conditions formulated in 1.B. For each \(\beta>0\) construct in \(R\) the set \({}_{\varphi}P_\beta(R)\). By definition, \(x\in{}_{\varphi}P_\beta\) if there exists a singular cycle \(Z\), homologous to zero in \(R\), such that \(x\in |Z|\) and for every chain \(X\), \(\partial X=Z\), we have

\[ d(X)>\beta\varphi d(Z). \tag{2} \]

(Roughly speaking, it is required that the diameter of the chain relative to the diameter of the cycle have order not lower than the order of growth of the function \(\varphi(t)\); the condition \(\varphi(t)\geq t\) is a natural consequence of the fact that \(d(X)\geq d(Z)\).) It is clear that if \(\beta_1<\beta_2\), then \(\beta_2\varphi(\beta_2 d(Z))>\beta_1\varphi(\beta_1 d(Z))\), i.e.

\[ {}_{\varphi}P_{\beta_2}\subset{}_{\varphi}P_{\beta_1}. \tag{3} \]

Denote by \(H_r({}_{\varphi}P_\beta)\) the kernel of the natural homomorphism of the \(r\)-dimensional singular homology group of the set \({}_{\varphi}P_\beta\) into the \(r\)-dimensional singular homology group of the space \(R\). By virtue of (3), for \(\beta_1 < \beta_2\) there is a natural homomorphism
\[ h=h_{\beta_2\beta_1}: H_r({}_{\varphi}P_{\beta_2})\to H_r({}_{\varphi}P_{\beta_1}), \]
and for \(\beta_1<\beta_2<\beta_3\) we have \(h_{\beta_2\beta_1}h_{\beta_3\beta_2}=h_{\beta_3\beta_2}\), i.e., the system of groups \(H_r({}_{\varphi}P_\beta)\) and homomorphisms \(h\) forms an inverse spectrum \(\{H_r({}_{\varphi}P_\beta),h\}\). We shall denote the limiting group of this spectrum by \({}_{\varphi}Q_r(R)\).

Theorem 1. If \([\varphi]\ge [\psi]\), then there exists a canonical homomorphism of the group \({}_{\varphi}Q_r(R)\) into the group \({}_{\psi}Q_r(R)\).

Let \(x\in{}_{\varphi}P_\beta\); then there is a cycle \(Z\) from the definition of the set \({}_{\varphi}P_\beta\) such that, for any chain \(X\), \(\partial X=Z\), we have
\[ d(X)>\beta\varphi(\beta d(Z)). \]
Since \([\varphi]\ge[\psi]\), there exists an \(a>0\) such that \(\varphi(t)\ge a\psi(at)\). Hence
\[ d(X)>\beta\varphi(\beta d(Z))\ge a\beta\psi(a\beta d(Z)), \]
i.e.,
\[ {}_{\varphi}P_\beta\subset{}_{\psi}P_{a\beta}. \tag{4} \]

The identity mapping \(R\to R\), by virtue of (4), gives rise to natural homomorphisms
\[ f_\beta: H_r({}_{\varphi}P_\beta)\to H_r({}_{\psi}P_{a\beta}), \]
which together with the homomorphisms \(h\) form the commutative diagram
\[ \begin{array}{ccc} H_r({}_{\varphi}P_{\beta_2}) & \xrightarrow{\ h\ } & H_r({}_{\varphi}P_{\beta_1})\\ {\scriptstyle f_{\beta_2}}\downarrow & & \downarrow{\scriptstyle f_{\beta_1}}\\ H_r({}_{\psi}P_{a\beta_2}) & \xrightarrow{\ h\ } & H_r({}_{\psi}P_{\beta_1}) \end{array} \]

Therefore the system of homomorphisms \(f_\beta\) may be regarded as a homomorphism of the spectrum \(\{H_r({}_{\varphi}P_\beta),h\}\) into the spectrum \(\{H_r({}_{\psi}P_{a\beta}),h\}\). This homomorphism gives rise to a canonical homomorphism \(f=f_{\varphi\psi}\) of the limiting groups of these spectra. It remains to note that the limiting groups of the spectra \(\{H_r({}_{\psi}P_{a\beta}),h\}\) and \(\{H_r({}_{\psi}P_\beta),h\}\) may be identified.

Remark. It is easy to see that from \([\varphi_1]\le[\varphi_2]\le[\varphi_3]\) it follows that
\[ f_{\varphi_3\varphi_1}=f_{\varphi_2\varphi_1}f_{\varphi_3\varphi_2}. \]

Theorem 2. If \([\varphi]=[\psi]\), then the groups \({}_{\varphi}Q_r(R)\) and \({}_{\psi}Q_r(R)\) are canonically isomorphic.

By the preceding theorem we have canonical homomorphisms
\[ f:{}_{\varphi}Q_r(R)\to{}_{\psi}Q_r(R) \]
and
\[ g:{}_{\psi}Q_r(R)\to{}_{\varphi}Q_r(R). \]
It remains to verify that \(gf\) and \(fg\) are the identity mappings. It suffices to do this for \(gf\). The mapping \(gf\) is obtained from a homomorphism of the inverse spectrum \(\{H_r({}_{\varphi}P_\beta),h\}\) into the spectrum
\[ \{H_r({}_{\varphi}P_{m\beta}),h\} \]
(\(m\) is some constant independent of \(\beta\)), generated by the inclusions
\[ {}_{\varphi}P_\beta\to{}_{\psi}P_{a\beta}\to{}_{\varphi}P_{m\beta}. \]
This homomorphism of spectra corresponds to the identity mapping of the group \({}_{\varphi}Q_r(R)\) into itself.

By Theorem 2, to each order of growth \([\varphi]\) there corresponds a group, which we denote by \({}_{[\varphi]}Q_r(R)\). It is easy to see that in Theorem 1, instead of the groups \({}_{\varphi}Q_r(R)\) and \({}_{\psi}Q_r(R)\), one may speak respectively of the groups \({}_{[\varphi]}Q_r(R)\) and \({}_{[\psi]}Q_r(R)\). The system of groups \({}_{[\varphi]}Q_r(R)\), together with the homomorphisms
\[ f_{\varphi\psi}:{}_{[\varphi]}Q_r(R)\to{}_{[\psi]}Q_r(R), \]
forms, in the partially ordered directed set of orders of growth, an inverse spectrum
\[ \{{}_{[\varphi]}Q_r(R),f\}. \]
We shall call it the spectrum of uniform homologies.

Example 1. Consider the surfaces \(\Gamma_1\) and \(\Gamma_2\), given respectively by the equations
\[ z=x^2+y^2 \]
and
\[ z=x^4+y^4 \]
in Euclidean space \(E^3\), and the functions \(\varphi_1(t)=t\), \(\varphi_2(t)=t^2\), \(\varphi_3(t)=t^4\). It can be shown that for \(\Gamma_1\) the group \({}_{[\varphi_1]}Q_1\) is free cyclic, \({}_{[\varphi_2]}Q_1\), \({}_{[\varphi_3]}Q_1\) are trivial, while for \(\Gamma_2\) the groups \({}_{[\varphi_1]}Q_1\), \({}_{[\varphi_2]}Q_1\) are free cyclic, and \({}_{[\varphi_3]}Q_1\) is trivial.

Theorem 3. An equimorphism \(g:R\to R'\) of spaces with uniform metric gives rise to an isomorphism
\[ g:{}_{[\varphi]}Q_r(R)\to{}_{[\varphi]}Q_r(R'). \]

It is easy to see (cf. (4)) that if \(R\) is a space with a uniform metric, then there exists a \(q>0\) such that for \(\beta>q\), in constructing the sets \({}_{\varphi}P_\beta\), one may restrict oneself to considering only those cycles for which
\[ d(Z)\ge \gamma_2 \]
(for the notation see 1.A). We note that \(\gamma_1\) and \(\gamma_2\) may be taken to be ob—

for both manifolds. Since \(g\) is an equimorphism, from \(\rho(x,y)\geq \gamma_2\), \(x,y\in R\), it follows that \(\rho(g(x),g(y))\geq l\), where \(l\) is a constant independent of the choice of the points \(x,y\in R\). Let \(c\) denote the smaller of the numbers \(\gamma_2\) and \(l\). Then \(\left({}^{2}\right)\) there exist such positive constants \(C_1\) and \(C_2\) that

\[ C_1 \leq \rho(x,y)/\rho(g(x),g(y))\leq C_2 \tag{5} \]

as soon as \(\rho(x,y)>c\) or \(\rho(g(x),g(y))>c\). Let now \(x\in {}_{\varphi}P_\beta(R)\), \(\beta>q\). This means that there is a cycle \(Z\), \(x\in |Z|\), \(d(Z)\geq \gamma_2\), such that for any chain \(X\), \(\partial X=Z\), the inequality (2) holds. Denote by \(Z'\) and \(X'\) the images of the cycle \(Z\) and the chain \(X\), respectively, under the mapping \(g\). Then, as is easily obtained, \(C_1\leq d(Z)/d(Z')\leq C_2\) and \(C_1\leq d(X)/d(X')\leq C_2\); hence

\[ d(X')\geq \frac{1}{C_2}d(X)\geq \frac{1}{C_2}\,\beta\varphi(\beta d(Z))\geq \frac{1}{C_2}\,\beta\varphi(\beta C_1 d(Z'))\geq a\beta\varphi(a\beta d(Z')),\, a= \]

\[ =\min\left(C_1,\frac{1}{C_2}\right). \]

Thus, \(g(x)\in {}_{\varphi}P'\), i.e.

\[ g({}_{\varphi}P_\beta)\subset {}_{\varphi}P'_{a\beta} \tag{6} \]

(here \({}_{\varphi}P'_{\alpha\beta}={}_{\varphi}P_{\alpha\beta}(R')\)). Similarly, the existence of such a constant \(b\) is proved that for \(\beta>q\) we shall have

\[ g^{-1}({}_{\varphi}P'_\beta)\subset {}_{\varphi}P_{b\beta}. \tag{7} \]

Using the inclusions (6) and (7), by arguments analogous to those given in the proofs of Theorems 1 and 2, we obtain that the mapping \(g\) induces a canonical isomorphism.

Remark. In the case of geodesic spaces the groups \({}_{[\varphi]}Q_r\) are invariants of strong homeomorphisms.

Example 2. Returning to the surfaces \(\Gamma_1\) and \(\Gamma_2\) considered in Example 1, we obtain, by Theorem 3, that they are not equimorphic, although the groups \(Q_r={}_{\varphi_1}Q_r\) (see (4)) are isomorphic for them. Note, however, that the non-equimorphism of these surfaces can also be established by comparing their volume invariants \(\left({}^{1}\right)\). For the non-equimorphic manifolds considered in the following example, the volume invariants are the same and the groups \(Q_r\) are isomorphic.

Example 3. Consider the manifolds \(\Pi_1\) and \(\Pi_2\), given in Euclidean space \(E^4\) by the equations \(u=x^2+y^2-z^2\) and \(x^4+y^4-z^2\), respectively. Let \(\psi(t)=t^2\). It can be proved that \({}_{[\psi]}Q_1(\Pi_2)\) is a free cyclic group, while \({}_{[\psi]}Q_1(\Pi_1)\) is trivial. It follows that \(\Pi_1\) and \(\Pi_2\) are not equimorphic.

A strengthening of Theorem 3 is

Theorem 4. An equimorphism \(g:R\to R'\) of spaces with a uniform metric induces an isomorphism of spectra
\[ \hat g:\{[{}_{\varphi}]Q_r(R),f\}\to \{[{}_{\varphi}]Q_r(R'),f\}. \]

The assertion of the theorem follows from Theorem 3 and the commutativity of the diagrams

\[ \begin{array}{ccc} {}_{[\varphi]}Q_r(R) & \to & {}_{[\psi]}Q_r(R)\\ \downarrow & & \downarrow\\ {}_{[\varphi]}Q_r(R') & \to & {}_{[\psi]}Q_r(R') \end{array} \qquad \begin{array}{ccc} {}_{[\varphi]}Q_r(R') & \to & {}_{[\psi]}Q_r(R')\\ \downarrow & & \downarrow\\ {}_{[\varphi]}Q_r(R) & \to & {}_{[\psi]}Q_r(R) \end{array} \]

I take this opportunity to thank V. A. Efremovich and especially V. Yu. Sandberg for valuable suggestions and attention.

Voronezh State University

Received
4 IX 1969

REFERENCES

1. V. A. Efremovich, UMN, 8, 5 (1953).
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3. V. Gurevich, G. Wallman, Dimension Theory, IL, 1948.
4. E. S. Tikhomirova, Izv. AN SSSR, ser. matem., 26, No. 6, 865 (1962).