MATHEMATICS

S. S. RYSHKOV

ON ONE CLASS OF CONTINUOUS MAPPINGS OF CERTAIN INFINITE-DIMENSIONAL SETS

(Presented by Academician P. S. Aleksandrov on 28 XII 1956)

In this note an (\Lambda)-class of mappings of subsets of Hilbert space (H) is defined. This class is of interest, for example, in that for (H)-polyhedra homeomorphic in this class, the infinite homology groups (\left(^{1}\right)) are isomorphic. Moreover, every mapping (\varphi: M \to H) having the form (\lambda e + a) (where (\lambda) is a real number; (a) is a continuous mapping of the set (M \subset H) into a compact set; (e) is the identity mapping) belongs to the class (\Lambda).

I. Definition of the class (\Lambda_{\mathfrak B,\widetilde{\mathfrak B}}). Let (\mathfrak B={f_1,f_2,\ldots,f_n,\ldots}) and (\widetilde{\mathfrak B}={\tilde f_1,\tilde f_2,\ldots,\tilde f_n,\ldots}) be arbitrary orthonormal bases of Hilbert space (H); let (M) be a subset of this space. We shall say that a continuous mapping (\varphi: M \to H) belongs to the class (\Lambda_{\mathfrak B,\widetilde{\mathfrak B}}) if it has the following properties:

1) For every point (x \in M) and numbers (r>0) and (\varepsilon>0), there exist numbers
[
\lambda_2=\lambda_2(x)\geq \lambda_1=\lambda_1(x)>0
]
and
[
N=N(x,r,\varepsilon,\mathfrak B,\widetilde{\mathfrak B}),
]
such that for (n>N) and (y\in M\cap O_r x) the inequalities
[
\lambda_1\rho_n(x,y)-\varepsilon \leq \widetilde{\rho}_n(\varphi(x),
]
[
\varphi(y)) \leq \lambda_2\rho_n(x,y)+\varepsilon,
]
are satisfied, where (\rho_n(x,y)) is the distance between the projections of the points (x) and (y) onto the linear span ({}^{n}H) of all vectors of the basis (\mathfrak B), with the exception of the first (n); (\widetilde{\rho}_n(x,y)) is defined similarly, only with (\mathfrak B) replaced by (\widetilde{\mathfrak B}).

2) For every point (x\in M) and numbers (r>0) and (\varepsilon>0), there exists a number (N=N(x,r,\varepsilon,\mathfrak B,\widetilde{\mathfrak B})) such that for any (n>N) and
[
y\in M\cap(H_+^n+x)\cap O_r x,
]
where (O_r x) is the open ball of radius (r) with center at the point (x), the inequality
[
\rho(\varphi(y),\varphi(x)+\widetilde H_+^n)\leq \varepsilon
]
is satisfied

(here (H_+^n) denotes the half-space formed by the vectors
[
b_1f_1+b_2f_2+\cdots+b_nf_n,
]
where (b_1,b_2,\ldots,b_{n-1}) are arbitrary and (b_n>0); the half-space (\widetilde H_+^n) is defined analogously).

3) The preimage of a bounded set is bounded.

II. Definition of the class (\Lambda). We shall say that a continuous mapping (\varphi: M\to H) belongs to the class (\Lambda) if it satisfies conditions 2) and 3), as well as the following condition:

1′) There exist two bases
[
\mathfrak B={f_1,f_2,\ldots,f_n,\ldots}
]
and
[
\widetilde{\mathfrak B}={\tilde f_1,\tilde f_2,\ldots,\tilde f_n,\ldots},
]
such that for any (x\in H), (r>0), and (\varepsilon>0) one can

choose the numbers (\lambda_2=\lambda_2(x)), (\lambda_1=\lambda_1(x)) and (N=N(x,r,\varepsilon,\mathfrak B,\tilde{\mathfrak B})), having the property that, for the plane (R) spanned by the vectors

[
f_1,f_2,\ldots,f_N,\quad \sum_{i=N+1}^{\infty} a_1^i f_i,\ \ldots,\ \sum_{i=N+1}^{\infty} a_k^i f_i,
]

where

[
\sum_{i=N+1}^{\infty} (a_l^i)^2<\infty,
]

the inequality

[
\lambda_1\rho(R+x,R+y)-\varepsilon
\leq
\rho(\tilde R+\varphi(x),\tilde R+\varphi(y))
<
\lambda_2\rho(R+x,R+y)+\varepsilon;
]

holds; here (y\in M\cap O_r x); (\tilde R) is the plane spanned by the vectors

[
\tilde f_1,\tilde f_2,\ldots,\tilde f_N,\quad
\sum_{i=N+1}^{\infty} a_1^i\tilde f_i,\ldots,\quad
\sum_{i=N+1}^{\infty} a_k^i\tilde f_i .
]

Theorem. If a mapping (\varphi:M\to H) belongs to the class (\Lambda), then for every basis (\mathfrak B) there exists a basis (\tilde{\mathfrak B}) such that (\varphi\in\Lambda_{\mathfrak B,\tilde{\mathfrak B}}).

II. Lemmas, examples.
A. Let (\varphi\in\Lambda_{\mathfrak A,\mathfrak A'}) and (\psi\in\Lambda_{\mathfrak A,\mathfrak A'}); then also (\varphi+\psi\in\Lambda_{\mathfrak A,\mathfrak A'}).

B. Let (\varphi\in\Lambda_{\mathfrak A',\mathfrak A'}) and (\psi\in\Lambda_{\mathfrak A',\mathfrak A''}); then (\psi\varphi\in\Lambda_{\mathfrak A,\mathfrak A''}).

C. If (\varphi\in\Lambda_{\mathfrak A,\mathfrak B}) and the basis (\mathfrak A) is comparable with the basis (\mathfrak A'), while the basis (\mathfrak B) is comparable with (\mathfrak B') ((^1)), then (\varphi\in\Lambda_{\mathfrak A',\mathfrak B'}).

By virtue of the last assertion, instead of (\Lambda_{\mathfrak A,\mathfrak B}) we may write (\Lambda_{\alpha,\beta}), where (\alpha) and (\beta) are the corresponding senses of orientation.

D. Any mapping (\varphi:M\to H) of the form (\lambda e+a) (where (\lambda) is a real number; (a) is a continuous mapping of the set (M) into a compact set; (e) is the identity mapping) belongs to the class (\Lambda_{\alpha,\alpha}) for any sense of orientation (\alpha), and also to the class (\Lambda).

E. Let (\tilde\varphi) be some continuous positive function of a real variable; then the mapping (\varphi:H\to H), given by the formula (\varphi(x)=x\tilde\varphi(\rho(0,x))), belongs to the class (\Lambda_{\alpha,\alpha}) for any sense of orientation (\alpha), and also belongs to the class (\Lambda).

IV. Invariance theorems. Let (K) be a subcomplex of some triangulation of the Hilbert space (H). The closure (P) of the body of the subcomplex (K) will be called an (H)-polyhedron. Note that, on the basis of the invariance theorem formulated in ((^2)), one may speak of the homotopy groups of the (H)-polyhedron (P), and not of the subcomplex defining it; we shall denote the homology groups of an (H)-polyhedron by ({}_r H(P)).

Theorem 1. Let (P_1) and (P_2) be two (H)-polyhedra; let (f:P_1\to H) and (g:P_2\to H) be mappings belonging, respectively, to the classes (\Lambda_{\alpha,\beta}) and (\Lambda_{\beta,\alpha}), such that (fg=e) and (gf=e), where (e) denotes the identity mappings. Then ({}n H(P_1)\cong {}_n H(P_2)) for any (n). If, moreover, (f\in\Lambda) and (g\in\Lambda), then the isomorphism ({}_n H(P_1)\cong {}_n H(P_2)) constructed from these mappings does not depend on the choice of the classes (\Lambda).}) and (\Lambda_{\beta,\alpha

Theorem 2. Let (F) and (F') be (closed) subsets of the space (H); suppose there exist two systems of open sets (O_i) and (O'_i), for which the inclusions

[
F\subset \ldots \subset O_{\eta_i}F\subset O_i\subset O_{\eta_{i-1}}F\subset O_{i-1}\subset \ldots \subset O_1\subset H,
]

[
F'\subset \ldots \subset O_{\xi_i}F'\subset O'i\subset O\subset \ldots \subset O'_1\subset H,}}F'\subset O'_{i-1
]

hold, where (\xi_i) and (\eta_i) are numbers tending to zero as the index (i) increases, and

by (O_\delta M) is denoted the (\delta)-neighborhood of the set (M). Let also
(f: O_1 \to H) and (g: O'1 \to H) be mappings belonging, respectively, to the classes (\Lambda), and having the property that for each pair of sets (F) and (F'), (O_1) and (O'_1), (O_2) and (O'_2), etc., the mappings (f) and (g) are mutually inverse homeomorphisms.}) and (\Lambda_{\beta,\alpha

Then
[
{}n H}}(F) \cong {n H(F'),}
]
and, by means of the mappings (f) and (g), a quite definite isomorphism of these groups is constructed. If, moreover, (f \in \Lambda) and (g \in \Lambda), then the constructed isomorphism does not depend on the choice of the classes (\Lambda_{\alpha,\beta}) and (\Lambda_{\beta,\alpha}). (For the definition of ({}n H(F)), see ((^2)).)}

The following lemmas are central in the proof of these assertions.

Lemma 1. Let (\varphi: K \to H) be a mapping of the complex (K) into the space (H) such that, on the preimage of some neighborhood (O_\rho y) of some point (y \in H), the mapping (\varphi) belongs to the class (\Lambda_{\alpha,\beta}). Then, for any (n)-plane ({}^n R) of defect (n) passing through the point (y), there exists a completely continuous mapping (\varphi^*: K \to H) such that:

a) (\varphi^*(K \setminus \varphi^{-1}(O_\rho y)) = 0);

b) the complex (K) is mapped into a plane parallel to ({}^n R);

c) (\operatorname{diam}\varphi^*(K) < 2\rho);

d) (y \notin (\varphi + \varphi^*)({}^{r+1}K)), where ({}^{r+1}K) denotes the ((r+1))-defective skeleton of the complex (K).

Lemma 2. Let (\varphi: F \to H) be a mapping of a closed subset (F \subset H) into (H). If some point (y \in H) does not belong to the image of the set (F), and (\varphi \in \Lambda_{\alpha,\beta}), then there exists (\rho) such that the intersection (\varphi(F)\cap O_\rho y) is empty.

I take this opportunity to express my gratitude to my supervisor, P. S. Aleksandrov.

Moscow State University
named after M. V. Lomonosov

Received
26 XII 1956

References Cited

(^{1}) V. G. Boltyanskii, DAN, 105, No. 6 (1955).
(^{2}) S. S. Ryshkov, DAN, 114, No. 3 (1957).