Neighborhood Theorem Research Articles

A generalization of the punctured neighborhood theorem

If T ∈ L ( X ) T \in \mathcal {L}(X) is regular on a Banach space X X , with finite- dimensional intersection T − 1 ( 0 ) ∩ T ( X ) {T^{ - 1}}(0) \cap T(X) , and if S , S ′ S,\;S’ are invertible, commute with T T and have sufficiently small norm, then dim ⁡ ( T − S ′ ) − 1 ( 0 ) = dim ⁡ ( T − S ) − 1 ( 0 ) \dim {(T - S’)^{ - 1}}(0) = \dim {(T - S)^{ - 1}}(0) and dim ⁡ X / ( T − S ′ ) X = dim ⁡ X / ( T − S ) X \dim X/(T - S’)X = \dim X/(T - S)X .

The stable neighborhood theorem and lengths of closed geodesics

We derive a generalized collar lemma, called the stable neighborhood theorem, for nonsimple closed geodesics. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number k k on a hyperbolic surface. This lower bound only depends on k k and tends to infinity as k k goes to infinity. Also, we show that the shortest nonsimple closed geodesic on a closed hyperbolic surface has (geometric) crossing number bounded above by a constant which only depends on the genus.

Generalizing the hyperbolic collar lemma

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a "stable neighborhood" whose width only depends on the length of the geodesic. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number k on a hyperbolic surface. This lower bound tends to infinity with k. Our second generalization is to totally geodesic hypersurfaces of hyperbolic manifolds. Namely, we construct a tubular neighborhood function and show that an embedded closed totally geodesic hypersurface in a hyperbolic manifold has a tubular neighborhood whose width only depends on the area of the hypersurface (and hence not on the geometry of the ambient manifold). The implications of this result for volumes of hyperbolic manifolds is discussed. We also derive a (hyperbolic) quantitative version of the Klein-Maskit combination theorem (in all dimensions) for free products of fuchsian groups. Using this last theorem, we construct examples to illustrate the qualitative sharpness of the tubular neighborhood function.

Decompositions and the relative tubular neighbourhood conjecture

THIS is a sequel to “ Block bundles: IT, transversality ” [S]. We expand some of the statements made in $6 of [S] and prove, as announced there, that BE”+,. “. 4 ‘v G/PL for n, q 2 3. This makes it easy to construct decompositions which are not block decompositions and hence counterexamples to the various conjectures we listed (see also Hudson [4] and Lickorish-Rourke [5]). As an interesting corollary of the existence of such examples we prove that there is no strong relative tubular neighbourhood theorem in the PL case; precisely, we give a counterexample (for all types of bundle) to the following:

Embedding spheres and balls in codimension $\leqq 2$

regular neighborhood of (jV=lS~-l ) v (k~lST,-2 ) with codimension one as an addendum to Theorem 2.1. Addendum to Theorem 2.1. Let N be a compact q-manifold. Suppose that q>=6. 7hen N is an abstract regular neighborhood of jV v kVISZ, i f N is ( q 3)-connected and ON consists o f r + l connected components Mo, M1 . . . . . M, whose fundamental groups rq(Ml) are of weights r~ equal to the Betti numbers of HI(M.t ). (For the weight of a group, see [18].) Remark. One more effort shows that the converse of the above is true for q_~ 4. Furthermore, if the 4-dimensional Poincar6 conjecture is Embedding Spheres and Balls in Codimension ~ 2 97 true, then the above is true for q= 5. Notice that the images of S~ -~ and S~-2 represent a basis of/2/. (N). This fact was the starting point of our investigation of the paper. (Refer to [16], Theorem 3.12.) Proof. For each j, let ;tjk, k= 1, ..., rj, be elements of ~l(Mi) whose normal closure equals ~I(Mj). Since r~ equals the Betti number of HI(Mj), it follows that the Hurewicz images of ,;tjk form a basis of HI(M~). Thus in the proof of the structure theorem we may take f2 k SO that f2 k] O q2 X S ~ represent ~-jk. Therefore, in the handle decomposition of N, handles (ffl-2) kill the fundamental groups ~(Mj) and hence C can be taken so that OC is simply connected. Since q>6, it follows from Poincar6Smale theorem that C is a q-ball. Consequently, N is obtained from the q-ball D ~= C by attaching to S~-~cD ~ handles of indices q 1 and q 2 via a framed link f : r S q 2 x D w t S ~3x D2--* S q-l. Now a subpolyhedron 0 * f (r S q2 x O w t S q3 x 02) w ((r D ~-1 x O) w (t D q2 x 02)) of N is the required spine of N, where 0 9 X denotes the cone of X S q~ from the origin 0, completing the proof. w 3. Nuli-Cobordism of Links in Homology Spheres Let C be a contractible (m+ 1)-manifold. Definition of Null-Cobordism. A link f : S (n, rn) ~ ~C is null-cobordant, if it can be extended to a locally fiat embedding F: D(n + 1, r , ) ~ O C, called a null-cobordism off. A framed link f : ~ S ( n , rn)--.aC is nullcobordant, if it can be extended to an embedding F: ~m+ 1D(n + 1, r,)--* C, called a null-cobordism of f, such that F(~ '~+ID(n+ 1, r,))c~ t~C= F ( ~ S (n, rn) ). Notice that in case m => 4, since by Lemma 2.2 a homology m-sphere bounds a unique contractible (m+ 1)-manifold, for a (framed) link the notion of null-cobordism makes sense without mentioning a contractible manifold bounding the homology sphere. It follows again from Lemma 2.2, (1) that if f is null-cobordant, then a link equivalent t o f i s also nullcobordant, provided at least m~ 3 or C= D m+~. We shall be concerned mainly with the case m n < 2. In that case, by the product neighborhood theorem of locally flat spheres in a homology sphere with codimension < 2 (for example refer to [31]), a link f : r S ~l w t S m2 ~ M is always framed and two framed links are equivalent if and only if their underlying links are equivalent, provided m 2 ~ : 1. Thus a framed link of codimension < 2 is null-cobordant if and only if its underlying link is null-cobordant. Moreover, the cone extension argument of a homeomorphism of S n shows that a link is null-cobordant if the image spheres in 0C bound mutually disjoint locally fiat balls in C.

A general theory of relative regular neighborhoods

Regular neighborhoods of polyhedra in manifolds have been discussed in many papers, most notably, in the present context, in [W], [H-Z], [H] and [S]. We study regular neighborhoods of X mod Y in Z where (Z, X, Y) is an arbitrary triad of finite dimensional, locally compact polyhedra. When applied to manifolds our theory generalizes previous results in that we make no compactness assumptions and place no restrictions on the polyhedra X and Y to be considered. The paper is organized as follows: 1. Definitions and notation 2. Simplicial preliminaries 3. The uniqueness theorem 4. The characterization of collared subpolyhedra 5. Cone-retracting and collaring regular neighborhoods 6. The stellar neighborhood theorem 7. Relationships between regular neighborhoods 8. Collapsibility and collapsible retractions 9. Regular neighborhoods via collapsibility. V is defined to be a regular neighborhood of X mod Y in Z if there exists a full triangulation (J, K, L; h) of (Z, X, Y) in which V underlies a relative first derived neighborhood, V=hN(K-L, J'). Thus, on the one hand, a given regular neighborhood can be presented in an explicit manner with a fixed triangulation. On the other hand, regular neighborhoods are piecewise linear rather than simplicial objects (there exists a triangulation) and there are uncountably many regular neighborhoods of X mod Y in Z implicit in the choice of triangulating complex and triangulating homeomorphism. We develop the theory by first exploiting the seeming concreteness (??2-5) and then developing the implicit generality (??6-9). After presenting the basic simplicial data we turn immediately to the uniqueness theorem (3.1). (Existence is automatic.) This asserts that, given two regular neighborhoods of X mod Y in Z, there is an ambient piecewise linear isotopy taking one onto the other and keeping X u Y fixed. The proof is quite direct and carries a great deal of extra information. For example (3.4) if Z is an n-manifold the isotopy can be realized by a sequence of 2n moves relative to X u Y.

Embeddings and compressions of polyhedra and smooth manifolds

Topology Vol. 4 pp. 361-369. EMBEDDINGS Pergamon Press, 1966. Printed in Great Britain AND COMPRESSIONS OF POLYHEDRA SMOOTH MANIFOLDS? AND MORRIS W. HIRSZH (Receiued 23 July 1965) $1. INTRODUCTION IT IS OFTEN desirable to compress a subset X of a manifold V into a submanifold Y’ by an isotopy of V. For example, V might be a Euclidean space R” and V’ might be p, with k much smaller than n. If X has some special properties ensuring that such a compression is always possible, then we conclude that X embeds in ti. In this article the problem of compressing X into the boundary of an s-cell is studied, where s = dim V. As applications several theorems are proved about embedding polyhedra and smooth manifolds in Eucli- dean space. The main theorems say that a compact polyhedron or smooth submanifold XC V compresses into the boundary of an s-cell provided (1) there exists a “Dehn cone” on X, and (2) V-X is highly connected. A Dehn cone on XC V is an embedding X x I c V with X x 0 = X together with a null homotopy of X x 1 in Y - X x 0. If V-X is sufhci- ently connected, a regular neighborhood theorem of Hudson and Zeeman, together with an engulfing theorem due to Zeeman and the author, provides an s-cell E c V with X c i?E. If V = R’, this implies the compressibility of X into R* -I or Ss -‘. First the piecewise linear (=PL) theory is developed, then the smooth case is reduced to the PL case. In the applications we take X c R4+’ and try to compress X into R4. There are three steps : (1) extend X to X x I c RQ+’ ; (2) choose the extension so that X x 1 3: 0 in RQ+’ - X; (3) prove P+’ - X is sufficiently connected for the Enguhing Theorem to apply. Step (1) is difficult in the PL case, so in the applications it is assumed as part of the hypothesis. In the smooth case, however, (1) is equivalent to the existence of a normal vector field on the smooth submanifold X. Step (2) is accomplished through algebraic topology, sometimes by luck-as when X is a smooth homology sphere-more usually by just assuming that the obstructions to a t This work was supported by the National Science Foundation grant GP-4035

The nonstable neighborhood theorem

The nonstable neighborhood theorem