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^N\cap M) are convex (N)-dimensional bodies and, consequently, cells, the degree of the mapping ([\varphi'N{}^{r}t_x^N:{}^{r}\tau^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\tau), and for sufficiently large (N), depending on (x), the number ([\varphi'}^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{}^{rN{}^{r}t_x^N:{}^{r}\tau\tau]).}^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

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)+p, & \text{for } x\in\overline{O p},\
\nu\gamma^{-1}+(1-\nu)+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)+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(M)}
]
and will be called the exterior cohomology group of the set (M).

The groups ({}r H(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).