S. S. RYSHKOV

ON THE COMBINATORIAL TOPOLOGY OF HILBERT SPACE

(Presented by Academician P. S. Aleksandrov, 12 XII 1955)

This note briefly sets forth a proof of the invariance (under different cell decompositions) of the cohomology groups of infinite-dimensional sets \((^1)\). Also defined are the exterior homology groups of an arbitrary closed set \(F\) lying in a Hilbert space \(H\).

Theorem. Let some set \(P \subset H\) be representable as the closure of the bodies of closed subcomplexes \(K_1\) and \(K_2\) of certain cell decompositions of the Hilbert space \(H\). Then for every \(r\) the relation \({}_r H(K_1) \approx {}_r H(K_2)\) holds, where \({}_r H(K)\) is the \(r\)-defective cohomology group* of the complex \(K\), defined in note \((^1)\), taken with an arbitrary coefficient group.

For the proof we introduce below the notions of a cellular mapping and of the degree of a mapping.

I. Definition of a cellular mapping. Let \({}^r t\) and \(r_\tau\) be cells of defect \(r\) of certain cell decompositions of the Hilbert space \(H\). A continuous mapping \(\varphi : {}^r t \to H\) will be called cellular with respect to the pair of cells \({}^r t\) and \(r_\tau\), if the following two conditions are satisfied:

a) For every number \(k\) between zero and one, and every polyhedron \(M\), homothetic to \(r_\tau\) with homothety coefficient \(k\) and lying together with its closure in \(r_\tau\), there exists a number \(\xi > 0\) such that the set \(\varphi({}^r t) \cap (O_\xi M \setminus r_\tau)\) is empty.

b) \(\varphi(\dot r^{\,t}) \subset H \setminus r_\tau\), where by \(\dot r^{\,t}\) is denoted the boundary of the cell \(rt\).

We now fix some orientation sense \(\alpha\) of the Hilbert space \(H\) \((^1)\) and choose two bases of this space, compatible with the orientation sense \(\alpha\): the basis \(\{f_i\} = \{f_1, \ldots, f_r, f_{r+1}, \ldots\}\) and the basis \(\{\theta_i\} = \{\theta_1, \theta_2, \ldots, \theta_r, \theta_{r+1}, \ldots\}\), and we choose these bases so that the vectors \(f_{r+1}, f_{r+2}, \ldots\) lie in the carrier plane of the cell \({}^r t\), and the vectors \(\theta_{r+1}, \theta_{r+2}, \ldots\) in the carrier plane of the cell \(r_\tau\); from this moment on we shall consider the cells \({}^r t\) and \(r_\tau\) oriented in the sense \(\alpha\).

We now single out a class of cellular mappings admissible in the sense \(\alpha\), narrow enough so that within it the notion of the degree of a mapping can be meaningfully and correctly defined.

A mapping \(\varphi : {}^r t \to H\), cellular with respect to the pair of cells \({}^r t\) and \(r_\tau\), will be called admissible in the sense \(\alpha\) if the following conditions are satisfied.

\(1^\circ\). For any point \(x \in rt\), whose image belongs to the cell \(r_\tau\), and any positive number \(\varepsilon'\), there exist a number \(N'\) and numbers \(\lambda_2 > \lambda_1 > 0\), depending only on \(x\), such that for \(n > N'\) the inequalities

\[ \lambda_1 \rho\bigl({}^r t_x^{\,n}, {}^r t_y^{\,n}\bigr) - \varepsilon' \leq \rho\bigl({}^r \tau_{\varphi(x)}^{\,n}, {}^r \tau_{\varphi(y)}^{\,n}\bigr) \leq \lambda_2 \rho\bigl({}^r t_x^{\,n}, {}^r t_y^{\,n}\bigr) + \varepsilon', \]

hold.

* The terms “cohomology” and “contrahomology” are used in the sense of the previously used terms “lower homology” and “upper homology,” respectively.

as soon as \(y\in{}^{r}t\), and \(\varphi(y)\in{}^{r}\tau\). (By \({}^{r}t_x^n\) is denoted the intersection of the cell \({}^{r}t\) with the Euclidean space \(x+H_{\{f_i\}}^{\,n+r}\), where \(H_{\{f_i\}}^{\,n+r}\) is the linear span of the vectors \(f_1,f_2,\ldots,f_{n+r}\).)

\(2^\circ\). For each \(x\in{}^{r}t\), \(\varphi(x)\in{}^{r}\tau\), and each \(\varepsilon''\) there exists an \(N''\) such that for \(n>N''\) the inequality

\[ \rho\bigl({}^{r}\tau_{\varphi(x)}^{\,n},\varphi(y)\bigr)<\varepsilon'' \]

is satisfied as soon as \(y\in{}^{r}t_x^n\), and \(\varphi(y)\in{}^{r}\tau\); here \({}^{r}\tau_{\varphi(x)}^{\,n}\) is understood as the intersection of the cell \({}^{r}\tau\) with the half-space \(\varphi(x)+\bigl(H_{\{v_i\}}^{\,r+n-1}\times l\bigr)\), where \(l\) is the positive ray directed along the vector \(v_{r+n}\).

II. The degree of a cellular mapping. Let \(x\in{}^{r}t\), \(\varphi(x)\in{}^{r}\tau\), and let the distance from the point \(\varphi(x)\) to the boundary \(\dot{{}^{r}\tau}\) of the cell \({}^{r}\tau\) be equal to \(10\rho\); then choose \(k\) so that any polyhedron \(M\) defined in condition a) contains within itself the \(4\rho\)-neighborhood of the point \(\varphi(x)\) relative to the cell \({}^{r}\tau\). By the number \(k\), from condition a) one finds a number \(\xi\), which may be assumed smaller than \(5\rho\). We also choose the numbers \(\varepsilon'\) and \(\varepsilon''\) smaller than \(\xi/10(\lambda_1+\lambda_2)\), if \(\lambda_1+\lambda_2>1\), and smaller than \(\xi/10\) otherwise. From \(\varepsilon'\) and \(\varepsilon''\), by conditions \(1^\circ\) and \(2^\circ\), we find \(N'\) and \(N''\) and put \(N=\max(N',N'')\). We orthogonally project the set \((O_{\xi/2}M)\cap\varphi({}^{r}t_x^N)\) onto \({}^{r}\tau_{\varphi(x)}^N\) and extend this projection in such a way to the intersection of the set \(\varphi({}^{r}t_x^N)\) with the set \(O_{\xi}M\setminus O_{\xi/2}M\) that a continuous mapping of the set \({}^{r}t_x^N\) into the space \(H\) is obtained; denote the resulting mapping by \(\varphi'_N\). Since the sets \({}^{r}t_x^N\) and \({}^{r}\tau_{\varphi(x)}^N\cap M\) are convex \(N\)-dimensional bodies and, consequently, cells, the degree of the mapping \([\varphi'_N{}^{r}t_x^N:{}^{r}\tau_{\varphi(x)}^N\cap M]\) is defined by virtue of the choice of the mapping \(\varphi'_N\).

It is proved that, for sufficiently large \(N\), the degree \([\varphi'_N{}^{r}t_x^N:{}^{r}\tau_{\varphi(x)}^N\cap M]\) depends neither on the choice of \(N\) nor on the choice of the polyhedron \(M\). It is also proved that, for all points \(x\in{}^{r}t\) for which \(\varphi(x)\in{}^{r}\tau\), and for sufficiently large \(N\), depending on \(x\), the number \([\varphi'_N{}^{r}t_x^N:{}^{r}\tau_{\varphi(x)}^N\cap M]\) is one and the same. This number we shall call the degree of the mapping \(\varphi\) of the cell \({}^{r}t\) onto the cell \({}^{r}\tau\) and shall denote by \([\varphi{}^{r}t:{}^{r}\tau]\).

III. Standard extensions. Choose in the cell \(\tau\) an arbitrary point \(p\), take its neighborhood \(O_p\) relative to the cell \(\tau\), having the form \(\mu(\tau-p)+p\), where \(\mu\) is a number between zero and one. Define a deformation \(\Pi_\nu:\tau\to\tau\) by the formula:

\[ \Pi_\nu= \begin{cases} [\nu\mu^{-1}+(1-\nu)](x-p)+p, & \text{for } x\in\overline{O p},\\ [\nu\gamma^{-1}+(1-\nu)](x-p)+p, & \text{for } x\in\gamma(\dot{\tau}-p)+p, \end{cases} \]

where \(1\ge\gamma\ge\mu\).

We shall denote the mapping \(\Pi_1\) by \(\Pi_p\), or, if this mapping is carried out in several cells, none of which is contained in the geometric boundary of another, by \(\Pi_{\{p\}}\), where the index is the set of the corresponding points.

Suppose further that there are given: a number \(\mu\), lying between zero and one, a point \(p\in\tau\), and a deformation \(f_\nu:\tau\to\tau\); the formula

\[ f_\nu= \begin{cases} [\nu\mu^{-1}+(1-\nu)](x-p)+p, & \text{for } x\in\overline{O p},\\ [\nu+(1-\nu)\gamma]\left[f_{\frac{\nu-\mu}{1-\mu}}\left(\frac{x-p}{\gamma}+p\right)-p\right]+p, & \text{for } x\in\gamma(\dot{\tau}-p)+p, \end{cases} \]

where \(1 \ge \gamma > \mu\), defines the extension of the deformation \(f_\nu\) to the whole cell \(\tau\); we shall call this extension standard. The mapping \(f_1\) will be called a standard extension of the mapping \(f_1\).

IV. Approximation of the identity mapping.
Let us now consider two complexes \(K_1\) and \(K_2\), which are subcomplexes of some cell decompositions of the space \(H\). The closures of the point sets—the bodies of these subcomplexes—will likewise be denoted by \(K_1\) and \(K_2\). Suppose also that \(K_1 \subset K_2\), and that \(i: K_1 \to K_2\) is the inclusion mapping. We shall construct an \(r\)-defective cellular approximation \(i_r\) of the mapping \(i\).

For each \(r = 0, 1, 2, \ldots\) let us number in some way all cells of defect \(r\) of the complex \(K_2\), and denote them by \({}^r\tau_j\), where \(r\) is the defect and \(j\) the number. We shall denote the totality of cells of the complex \(K\) having the given defect \(r\) by \({}^rK\), and shall call this totality the \(r\)-defective skeleton of the complex \(K\).

Construction of the approximation \(i_0\). Choose in each cell \({}^0\tau_j\) a point \({}^0p_j\) so that \({}^0p_j \notin i({}^1K_1)\) for any \(j\); next choose, for each point \({}^0p_j\), a number \({}^0\mu_j\) such that the neighborhood \({}^0\mu_j({}^0\tau_j - {}^0p_j) + {}^0p\) of the point \({}^0p_j\) does not intersect the set \(i({}^1K_1)\). After this we perform the mapping \(\Pi_{\{{}^0p_j\}}: K_2 \to K_2\), and denote the mapping \(\Pi_{\{{}^0p_j\}} i\) by \(i_0\).

Construction of the approximation \(i_r\). Suppose that all the preceding approximations have already been constructed. In each cell \({}^r\tau_j\) choose a point \({}^rp_j\) so that \({}^rp_j \notin i_{r-1}({}^{r+1}K_1)\) for all \(j\). Also choose, for each point \({}^rp_j\), a number \({}^r\mu_j\) such that the set \({}^r\mu_j({}^r\tau_j - {}^rp_j) + {}^rp_j\) does not intersect the set \(i_{r-1}({}^{r+1}K_1)\). After this choice we perform the mapping \(\Pi_{\{{}^rp_j\}}: {}^rK_2 \to {}^rK_2\), extend it standardly to the skeleton \({}^{r-1}K_2\), then to \({}^{r-2}K_2\), and so on down to the skeleton \({}^0K_2\), using the fact that for every cell of each of these skeletons a point \(p\) and a number \(\mu\) have already been chosen. The superposition \(\Pi i_{r-1}\) of the resulting mapping \(\Pi\) and the mapping \(i_{r-1}\) we shall call an \(r\)-defective cellular approximation of the mapping \(i\) and denote it by \(i_r\).

The following assertion is essential: The mapping \(i_r\) is an admissible cellular mapping with respect to any pair of cells \({}^r t\) and \(\tau\), where \({}^r t \in {}^rK_1\).

V. Invariance of infinite-dimensional homologies.
The constructions carried out above make it possible, in the usual way, to define a homomorphism
\[ {}_* i : {}_r L(K_1) \to {}_r L(K_2) \]
of cochain groups, then to prove commutativity of the coboundary homomorphism with the homomorphism \({}_* i\), and thereby to construct a homomorphism
\[ {}_* i : {}_r H(K_1) \to {}_r H(K_2) \]
of cohomology groups. For two homomorphisms
\[ {}_* i : {}_r H(K_1) \to {}_r H(K_2) \]
and
\[ {}_* j : {}_r H(K_2) \to {}_r H(K_3) \]
of this kind, generated by the inclusion mappings \(i: K_1 \to K_2\) and \(j: K_2 \to K_3\), the relation
\[ {}_*(j \circ i) = {}_* j \circ {}_* i \]
holds, expressing the naturality of these homomorphisms; this proves the invariance of the cohomology groups.

VI. Exterior cohomology groups.
Let \(M\) be a (closed) set lying in the space \(H\). Choose a system of numbers \(\varepsilon_i, i = 1, 2, \ldots\), decreasing monotonically to zero. Let, further, \(K_i\) \((i = 1, 2, \ldots)\) be a subcomplex of some cell decomposition of the space \(H\), with cell diameter less than \(\varepsilon_i/4\), such that
\[ O_{\varepsilon_{i+1}}M \subset K_i \subset O_{\varepsilon_i}M, \]
where \(O_{\varepsilon_i}M\) is the \(\varepsilon_i\)-neighborhood of the set \(M\). It is clear that the cohomology groups of defect \(r\) of the complexes \(K_i\) form an inverse spectrum; the limit of this spectrum will be denoted by
\[ {}_r H_{\mathrm{вн}}(M) \]
and will be called the exterior cohomology group of the set \(M\).

The groups \({}_r H_{\mathrm{вн}}(M)\) do not depend on the arbitrary elements of the construction.

VII. Analogous results (proved analogously) hold for cohomology groups.

In conclusion, I consider it my duty to express my gratitude to V. G. Boltyanskii for valuable advice and comments.

Moscow State University
named after M. V. Lomonosov

Received
12 XII 1956

References

¹ V. G. Boltyanskii, DAN, 105, No. 6 (1955).