MATHEMATICS

A. B. SOSINSKII

ON THE EMBEDDING OF A (k)-DIMENSIONAL ELEMENT IN (E^n)

(Presented by Academician P. S. Aleksandrov on 3 IV 1961)

The paper gives necessary and sufficient conditions for a tame embedding of a (k)-dimensional element in (n)-dimensional Euclidean space (E^n) (for arbitrary (k<n)) and investigates the connection between the notions of tame and normal embedding. An embedding of a (k)-dimensional triangulable manifold (M^k \subset E^n) is called tame if there exists a homeomorphism (g) of the space (E^n) onto itself such that (g(M^k)) is a polyhedron; an embedding (M^k \subset E^n) is called normal if there exists a homeomorphism
[
h:(M^k \times I^{\,n-k}) \to E^n
]
(where (I^{\,n-k}) is an ((n-k))-dimensional cube with center (O)) such that
[
h(M^k \times O)=M^k.
]
The first examples of “wild” (i.e. non-tame) embeddings were constructed by Antoine ((^1)), Urysohn ((^2)), and Aleksandrov ((^3)). Criteria for tame embedding in (E^3) were given by Harrold ((^4)) and Bing ((^5)). Brown proved that the closures of domains bounded by an ((n-1))-dimensional sphere (S^{n-1}), normally embedded in the (n)-dimensional sphere (S^n), are (n)-dimensional elements ((^6)).

§ 1. Some consequences of Brown’s work. An embedding of a continuum (F) in (E^n) (or in (S^n)) is called cellular ((^6)) if there exists a system of (n)-dimensional elements ({Q_i^n}) such that (Q_{i+1}^n) is contained strictly inside (Q_i^n) and
[
\bigcap_{i=1}^{\infty} Q_i=F.
]

Corollary 1. In order that an embedding (F \subset S^n) be cellular, it is necessary and sufficient that (S^n \setminus F) be homeomorphic to (S^n \setminus x), where (x) is a point.

Let the embedding (F \subset S^n) be cellular. Take in (S^n \setminus F) a small geometric sphere (S^{n-1}) and denote by (Q) the closure of that component of (S^n \setminus S^{n-1}) which contains (F). Then, by Brown’s Theorem 1 ((^6)), there exists a continuous mapping (f) of the element (Q) onto itself, identical on the boundary of (Q), for which (F) is the unique nondegenerate inverse image of a point. Let (h) be a mapping identical on (S^n \setminus Q) and coinciding with (f) on (Q \setminus F). Then (h) is a homeomorphism of (S^n \setminus F) onto (S^n \setminus f(F)), and since (f(F)) is a point, the sufficiency is proved.

Let the embedding (F \subset S^n) be such that (S^n \setminus F) is homeomorphic to (S^n \setminus x), where (x) is a point. Then the mapping (f') of the sphere (S^n) (under which (F) is the unique nondegenerate inverse image of a point) is a mapping of this sphere onto itself; but then the embedding is cellular by Brown’s Theorem 3, and the corollary is proved.

We note that a cellular set may be wild ((^7)).

Lemma 1. In order that an embedding of the ((n-1))-dimensional sphere (S^{n-1}) in the (n)-dimensional sphere (S^n) be tame, it is necessary and sufficient that the closures of both components of the complement of (S^{n-1}) in (S^n) be (n)-dimensional elements.

Necessity is obvious. Sufficiency is not hard to prove from the fact that a homeomorphism of the sphere onto the boundary of an (n)-element can always be extended along radii to a homeomorphism of the ball onto the entire (n)-element.

In view of this same remark it is easy to prove:

Lemma 2. If the embeddings (S^{n-1}\subset S^n) and (S_^{\,n-1}\subset S^n) are tame and a homeomorphism (g:S^{n-1}\to S_^{\,n-1}) is given, then (g) can be extended to all of (S^n).

Corollary 2. In order that an embedding (S^{n-1}\subset S^n) be tame, it is necessary and sufficient that it be normal.*

Necessity is obvious. Sufficiency follows at once from Lemma 1 and the theorem of Brown mentioned above (\bigl({}^{(6)}), Theorem 5()).

§ 2. Criterion for a tame embedding of a (k)-dimensional element in (E^n).

In connection with Corollary 2 it is natural to pose the question: will a normally embedded (k)-dimensional element in (E^n) be tame? It turns out, not always: the simple arc (with one infinitely knotted end) of Artin–Fox (\bigl({}^{(7)}), Example 1.2()) is wildly embedded in (E^3), although it is not hard to show that it is normally embedded in (E^3). In this example, as in any normal embedding of a (k)-element in (E^n), wildness can arise only at boundary points; therefore it is natural to impose additional conditions on the embedding of the boundary.

Definition. A (k)-dimensional element (Q^k) is embedded in (E^n) with a collar if there exists a homeomorphism of a (k)-dimensional ball into (E^n) under which a smaller concentric ball is mapped onto (Q^k).

Theorem 1. In order that an embedding of a (k)-dimensional element (Q^k) in (E^n) be tame, it is necessary and sufficient that (Q^k) be embedded in (E^n) with a collar (L) and that the element (Q^k\cup L) be embedded normally in (E^n).

Necessity is obvious. To prove sufficiency, consider a system of closed concentric (k)-dimensional balls (\Delta_r^k) of radius (r), where (0\le r\le 2); take an analogous system of balls (\Delta_s^{\,n-k}) of dimension (n-k) with center (O) and radii (0\le s\le 1), and consider the topological product (\Delta_2^k\times \Delta_1^{\,n-k}). By the hypotheses of the theorem there exists a homeomorphism
[
h:\bigl(\Delta_2^k\times \Delta_1^{\,n-k}\bigr)\to E^n
]
such that
[
h(\Delta_1^k\times 0)=Q^k.
]
Consider the set
[
\partial\bigl(\Delta_{3/2}^k\times \Delta_{1/2}^{\,n-k}\bigr)=W_0,
]
where (\partial) denotes the boundary; (W_0) is an ((n-1))-dimensional sphere bounding the topological ball
[
\bigl(\Delta_{3/2}^k\times \Delta_{1/2}^{\,n-k}\bigr)=Z,
]
strictly inside which lies ((\Delta_1^k\times 0)). We show that (h(W_0)) is normally embedded in (E^n). Consider the system of topological spheres
[
W_t=\partial\bigl(\Delta_{3/2+t}^k\times \Delta_{1/2+t}^{\,n-k}\bigr),
]
where (-1/4\le t\le 1/4); (h'=h|Z) realizes an embedding of (h(W_0)) in (E^n) with topological product by an interval, i.e. (h(W_0)\subset E^n) is normal and, by Corollary 2, also tame. Embed
[
\bigl(\Delta\bigr)}^k\times \Delta_{1/2}^{\,n-k
]
identically in (E^n) and extend (h') to a homeomorphism (h_) of all (E^n) onto itself. This can be done, according to Lemma 2, because the spheres (W_0) and (h(W_0)) are tame. But (h_^{-1}) is a homeomorphism of (E^n) onto itself, and
[
h_*^{-1}(Q^k)=h^{-1}(Q^k)=(\Delta_1^k\times 0),
]
a polyhedron. The theorem is proved.

It is not hard to construct an example of a wild (2)-element (Q^2) which is embedded with a collar (L), is embedded normally, but (Q^2\cup L) is not embedded normally. Let us also note that every simple arc is embedded with a collar in (E^3), whereas a topological square may have no collar.

§ 3. On normally embedded (k)-dimensional manifolds in (E^n).

It is natural to pose the question: will a normal embedding of a closed triangulable (k)-dimensional manifold (M^k) in (E^n) be tame? We shall show only that a normal embedding of (M^k) in (E^n) is locally tame

* This result was recently announced by Brown himself ({}^{(8)}).

at every point (i.e., there exists a neighborhood (Ox) of every point (x\in M^k) and a homeomorphism (g_x) such that (g_x(\overline{Ox})) and (g_x(\overline{Ox}\cap M^k)) are polyhedra); in the case (n=3) this also means that it is tame ((^{9,10})).

Theorem 2. If the embedding in (E^n) of a closed triangulated (k)-dimensional manifold (M^k) is normal (or locally normal), then it is locally tame at every point.

By hypothesis, for (x\in M^k) there exists a (k)-dimensional element (Q^k), (x\in Q^k\setminus \partial Q^k), lying with its rim (L) in (M^k), and moreover (Q^k\cup L) is normally embedded in (E^n). By Theorem 1 the embedding (Q^k\subset E^n) will be tame; therefore there exists a homeomorphism (h(E^n)=E^n) such that (h(Q^k)=\sigma^k), a closed simplex. Choosing in it a smaller concentric simplex (\sigma_0^k\ni h(x)), it is not hard to construct two open (n)-dimensional simplexes (\sigma_1^n) and (\sigma_2^n), not intersecting each other and with (\sigma_0^k), such that the open kernel (V) of the set (\overline{\sigma_1^n\cup \sigma_2^n}) contains (\sigma_0^k), and the intersection (\overline V\cap h(M^k)) is equal to (\sigma_0^k). Then (h^{-1}(V)) will be a neighborhood of (x) satisfying the condition of a locally tame embedding. The theorem is proved.

For (E^3) there is the Bing–Moise theorem ((^{9,10})), according to which an embedding in this space is tame if and only if it is locally tame at every point. Therefore Theorem 2 implies:

Theorem 3. A normal embedding of a simple closed contour and of a two-dimensional closed manifold in (E^3) is tame.*

For a two-dimensional manifold this assertion follows at once from the well-known Bing criterion ((^5)).

We note that Theorem 2 is not true for manifolds with boundary, since boundary points may turn out to be locally wild. It is natural to impose additional conditions on the embedding of the boundary.

Definition. A (k)-dimensional manifold (with boundary) (M^k) is embedded in (E^n) with rim if there exists a ((k-1))-dimensional polyhedron (\partial\mu^k) (possibly disconnected) and a homeomorphism (h:(\partial\mu^k\times[-1,1])\to E^n) such that (h(\partial\mu^k\times 0)=\partial M^k), (h(\partial\mu^k\times(0,1])\cap M^k=\Lambda), and (h(\partial\mu^k\times[-1,0])\subset M^k).

By a simple modification of the proof of Theorem 2 one can prove:

Theorem 4. If a triangulated (k)-dimensional manifold (M^k) is embedded in (E^n) with rim (L) and (M^k\cup L) is embedded normally, then the embedding (M^k\subset E^n) is locally tame at every point.

The author expresses gratitude to L. V. Keldysh for the attention given to this work.

Moscow State University
named after M. V. Lomonosov

Received
29 III 1961

REFERENCES

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3. J. W. Alexander, Proc. Nat. Acad. USA, 10, 8 (1924).
4. O. G. Harrold, Ann. of Math., 69, No. 2 (1959).
5. R. H. Bing, Fund. Math., 47, 105 (1959).
6. M. Brown, Bull. Am. Math. Soc., 66, No. 2, 74 (1960).
7. E. Artin, R. Fox, Ann. of Math., 49, 979 (1948).
8. M. Brown, Not. Am. Math. Soc., 7, No. 7, 50, 576 (1960).
9. R. H. Bing, Ann. of Math., 59, No. 1, 145 (1954).
10. E. E. Moise, Ann. of Math., 59, No. 1, 159 (1954).

* Evidently, the converse assertion is also true.