PL Embeddings Research Articles

On PL embeddings of a 2-sphere in the 4-dimensional Euclidean space

On PL embeddings of a 2-sphere in the 4-dimensional Euclidean space

Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods*

For a given polyhedron K ⊂ M, the notation R M (K) denotes a regular neigh-borhood of K in M. The authors study the following problem: find all pairs (m, k) such that if K is a compact k-polyhedron and M a PL m-manifold, then R M (f(K)) ≅ R M (g(K)) for each two homotopic PL embeddings f, g : K → M. It is proved that R S k+2 (S k ) ≇ S k × D 2 for each k ≥ 2 and some PL sphere S k ⊂ S k +2 (even for any PL sphere S k ⊂ S k +2 having an isolated non-locally flat point with the singularity S k -1 ⊂ S k +1 such that π1(S k +1 – S k -1) ≇ ℤ).

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber on the deleted product criteria for embeddings and immersions. The isotopy and regular homotopy versions of the above theorems are also improved. We show by examples that they cannot be improved further. These results have many interesting corollaries, e.g. 1) Any closed homologically 2-connected smooth 7-manifold smoothly embeds in \mathbb{R}^11 . 2) If p \leq q and m \geq \frac{3q}2 + p + 2 then the set of PL embeddings S^{p} \times S^{q} \to \mathbb{R}^m up to PL isotopy is in 1-1 correspondence with \pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}) .

Spaces of embeddings of compact polyhedra into 2-manifolds

Let M be a PL 2-manifold and X be a compact subpolyhedron of M and let E(X,M) denote the space of embeddings of X into M with the compact-open topology. In this paper we study an extension property of embeddings of X into M and show that the restriction map from the homeomorphism group of M to E(X,M) is a principal bundle. As an application we show that if M is a Euclidean PL 2 -manifold and dimX≥1 then the triple (E(X,M) , ELIP(X,M) , EPL(X,M)) is an (s,Σ,σ) -manifold, where EKLIP(X,M) and EKPL(X,M) denote the subspaces of Lipschitz and PL embeddings.

On the deleted product criterion for embeddability of manifolds in $ \Bbb {R}^m $

For a space N let \( \tilde N = \{(x,y) \in N \times N|x \neq y \} \). Let \(\Bbb {Z}_2 \) act on N and on \( S^{m-1} \) by exchanging factors and antipodes, respectively. For an embedding \( f : N \to \Bbb {R}^m \) define the map \( \tilde f : \tilde N \to S^{m-1} \) by \( \tilde f(x,y) = {fx-fy \over|fx-fy|} \).¶Theorem. Let \( d = 3n - 2m + 2 \) and N be a closed PL n-manifold.¶a) If \( d \in \{0,1,2 \} \), N is d-connected and there exists an equivariant map \( F : \tilde N \to S^{m-1} \), then N is PL-embeddable in \( \Bbb {R}^m \).¶b) If \( d \in \{-1,0,1\} \), \( m-n \ge 3 \), N is (d + 1)-connected and \( f, g : N \to \Bbb {R}^m \) are PL-embeddings such that \( \tilde f,\tilde g \) are equivariantly homotopic, then f, g are PL-isotopic.¶Corollary. a) Every closed 6-manifold N such that H1(N) = 0 PL embeds in \( \Bbb {R}^{10} \);¶b) Every closed PL 2-connected 7-manifold PL embeds in \( \Bbb {R}^{11} \);¶c) There are exactly four PL embeddings \( S^{2l+1} \times S^{2l+1} \subset \Bbb {R}^{6l+4} \) up to PL isotopy (l > 0).

An embedding space triple of the unit interval into a graph and its bundle structure

Let l 2 denote a Hilbert space, and let l 2 Q ={(x i )∈l 2 |sup|i.x i |<∞} and l 2 f ={(x i )∈l 2 |x i =0 except for finitely many i}. We show that the triple (H(X), H LIP (X), H PL (X)) of spaces of homeomorphisms, of Lipschitz homeomorphisms, and of PL homeomorphisms of a finite graph X onto itself is an (l 2 ,L 2 Q , l 2 f )-manifold triple, and that the triple (E(I,X), E LIP (I,X), E PL 5I,X)) of spaces of embeddings, of Lipschitz embeddings, and of PL embeddings of I=[0,1] into a graph X is an (l 2 ,l 2 Q ,l 2 4 )-manifold triple

Triangulation and general position of PL diagrams

We give conditions under which a diagram R ← g P → g Q of polyhedra and PL maps is triangulable. Suppose there is an integer m such that if q 1,…, q r are distinct points of Q with gf -1( q i )∩ gf -1( q i+1 )≠ ∅, i = 1,…, r - 1, then r ⩽ m. Then there are triangulations K, H, and J of P, Q, and R, respectively, with respect to which f and g are simplicial. Moreover, if f and g are surjections, then the relation q ∼ q' if gf -1( q) ∩ gf -1( q') ≠ ∅ defines a quotient complex L of H and simplicial maps ψ : H → L and φ : J → L such that the diagram ▪ is commutative. Suppose R is a PL n-manifold, and for some triangulation T of P, dim σ + max{dim f -1( q) ∩ σ | q ∈ Q}< n for each σ ∈ T. Then g may be approximated by a PL map h : P → R such that the diagram R ← g P → g Q satisfies the hypothesis above with m = 4 n 2, and, hence, is triangulable. These are adaptations and generalizations of results of T. Homma, which we use to give a proof of R. Miller's codimension-three PL approximation theorem: If f : M m → N n , n − m ⩾ 3, is a topological embedding of a PL m-manifold M into a PL n-manifold N, then f can be approximated by PL embeddings.

PL embeddings ofPL manifolds into some Euclidean spaces

The main result in this paper is the following: Theorem.Assume that W is a k-connected compact PL n-manifold with boundary, BdW is (k−1)-connected, k⩾1(BdW is 1-connected for k=1), 0⩽h⩽2k, 2n−h>5and there exists a normal block (n-h-1)-bundle vover W, then Wheree denotes the trivial block 1-boundle,D 2n−h={x=(x 1,x 2, ⋯,x 2n− h) ∈R 2n− h¦ ¦x i¦⩽1} andS 2n−h}- 1=BdD 2n−h.

Linear isotopies in $E\sp{2}$

This paper deals with continuous families of linear embeddings (called linear isotopies) of finite complexes in the Euclidean plane ${E^2}$. Suppose f and g are two linear embeddings of a finite complex P with triangulation T into a simply connected open subset U of ${E^2}$ so that there is an orientation preserving homeomorphism H of ${E^2}$ to itself with $H \circ f = g$. It is shown that there is a continuous family of embeddings ${h_t}:P \to U(t \in [0,1])$ so that ${h_0} = f,{h_1} = g$, and for each t, ${h_t}$ is linear with respect to T. It is also shown that if P is a PL star-like disk in ${E^2}$ with a triangulation T which has no spanning edges and f is a homeomorphism of P which is the identity on Bd P and is linear with respect to T, then there is a continuous family of homeomorphisms ${h_t}:P \to P(t \in [0,1])$ such that ${h_0} = {\text {id}},{h_1} = f$, and for each t, ${h_t}$ is linear with respect to T. An example shows the necessity of the “star-like” requirement. A consequence of this last theorem is a linear isotopy version of the Alexander isotopy theorem-namely, if f and g are two PL embeddings of a disk P into ${E^2}$ so that $f|{\text {Bd}}\;P = g|{\text {Bd}}\;P$, then there is a linear isotopy with respect to some triangulation of P which starts at f, ends at g, and leaves the boundary fixed throughout.

Linear isotopies in 𝐸³

In this paper it is shown that if f and g are two PL embeddings of a finite complex K into E 3 {E^3} so that there is an orientation-preserving homeomorphism h of E 3 {E^3} with h ∘ f = g h \circ f = g , then there is a triangulation T of K and a linear isotopy h t : ( K , T ) → E 3 ( t ∈ [ 0 , 1 ] ) {h_t}:(K,T) \to {E^3}(t \in [0,1]) so that h 0 = f {h_0} = f and h 1 = g {h_1} = g .

The equivalence of close piecewise linear embeddings

The main theorem (2.1) says that if N is an abstract regular neighborhood of a polyhedron X with collapsible retraction r: N → X, dimX < dimN − 3, and if g: N → intN is an embedding such that rg is close to the inclusion map X↪ N, then g is isotopic to the inclusion X↪ N by an ambient isotopy which is limited by r. As a corollary two close PL embeddings of a polyhedron X into a PL manifold Q are equivalent by small PL ambient isotopy if dim X < dimQ − 3 and if the embeddings are sufficiently close to a given topological embedding of X into Q. Some related results using a slightly weaker dimension restriction are also discussed, and some other corollaries are presented.

Fiber preserving equivalence

We give a theory of fibered regular neighborhoods based on a remarkable property of simplicial fibered projections. All the usual properties of regular neighborhoods are retained. Using Millett’s fibered general position, together with the regular neighborhoods, we prove THEOREM. The simplicial space of codimension 4 PL embeddings of a complex into a PL manifold is locally contractible at each point of the space of topological embeddings.

Approximating Embeddings of Polyhedra In Codimension Three

Let $P$ be a $p$-dimensional polyhedron and let $Q$ be a PL $q$-manifold without boundary. (Neither is necessarily compact.) The purpose of this paper is to prove that, if $q - p \geqslant 3$, then any topological embedding of $P$ into $Q$ can be pointwise approximated by PL embeddings. The proof of this theorem uses the analogous result for embeddings of one PL manifold into another obtained by Černavskiĭ and Miller.

Approximating embeddings of polyhedra in codimension three

Let F be a p-dimensional polyhedron and let C be a PL q- manifold without boundary. (Neither is necessarily compact.) The purpose of this paper is to prove that, if q — p > 3, then any topological embedding of P into Q can be pointwise approximated by PL embeddin-gs. The proof of this theorem uses the analogous result for embeddings of one PL manifold into another obtained by Cernavskii and Miller. Introduction. Recently Cernavskii and Miller have shown that any topological embedding of a PL zzz-manifold M into a PL o-manifold Q can be approximated by PL embeddings provided q - m > 3- (See (5), (6), and (10).) In this paper we use this fact to prove that a topological embedding of an arbitrary p-dimensional poly- hedron into Q (fl - p > 3) can be approximated by PL embeddings. This general- izes the results of Berkowitz (2) and Weber (13)- Our theorem can be stated as follows. Theorem 1. Suppose that P is a (not necessarily compact) p-dimensional polyhedron, that 0 is a PL q-manifold without boundary (q — p > 3), and that f: P —* Q is a topological embedding. Then for each continuous function c P—» (0, 00) there exists a PL embedding g; P—»0 such that d(f(x),g(x)) < e(x) for each x £ P. Moreover, if R is a subpolyhedron of P on which f\R is PL, then we may choose g so that g\R=f\R.