Regular Neighborhood Research Articles

Construction of Manifolds of Positive Scalar Curvature

We prove that a regular neighborhood of a codimension $\geqslant 3$ subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact $4$-manifold with such a metric.

Construction of manifolds of positive scalar curvature

We prove that a regular neighborhood of a codimension ⩾ 3 \geqslant 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact 4 4 -manifold with such a metric.

Peripheral acyclicity in 3-manifolds

AbstractWe study weak and strong peripheral 1-acyclicity, a homology version of D. R. McMillan, Jr.'s weak cellularity criterion and cellularity criterion, for embeddings of compacta in 3-manifolds. In contrast with the two cellularity criteria we prove that the two peripheral acyclicities are equivalent and moreover, for compacta of dimension at most 1, independent of the embedding. We also give some results concerning regular neighborhoods of compact polyhedra in 3-manifolds.

Non-simply connected copes

Let ( W 4,∂ W 4) be a 4-manifold. Let f 1, f 2,…, f k :( D 2, ∂D 2)→ ( W 4, ∂W 4) be transverse immersions that have spherical duals α 1,α 2,…,α k:S 2→ W ̊ . Then there are open disjoint subsets V 1, V 2,…, V k of W, such that for each 1⩽ i⩽ k, ( a) ∂V i = V 1∩ ∂W and ∂V i is an open regular neighborhood of f i ( ∂D 2) in ∂ W, and (b) ( V i , ∂V i , f i ( ∂D 2)) is proper homotopy equivalent to ( M, ∂ M, d)—a standard object in which d bounds an embedded flat disk. If we could get a homeomorphism instead of a proper homotopy equivalence, then we would be able to prove a 5-dimensional s-cobordism theorem.

Primitive links of non-singular Morse-Smale flows on the special Seifert fibered manifolds

In this paper we consider a non-singular Morse-Smale flow Φ t on an irreducible, simple, closed, orientable 3-manifold M. We define a primitive flow ψ t from Φ t , and call the link type of the closed orbits of ψ t a primitive link of Φ t . We show that the link types of the primitive links are finite and every non-singular Morse-Smale flow on M is obtained from a primitive flow by exchanging the flow in a regular neighborhood of attracting or repelling closed orbits.

On properly embedding planes in arbitrary $3$-manifolds

We prove an analog of the loop theorem for an arbitrary noncompact 3-manifold. In particular, we show that the existence of a proper map of a plane into a 3-manifold implies the existence of a nearby nontrivial embedding of a plane into the 3-manifold. Introduction. In (2) we showed that if M is an eventually end-irreducible 3-mani- fold and f: R2 -- M is a proper essential map, then there is a proper essential embedding g: R2 -_ M. In this paper we remove the restriction to manifolds which are eventually end-irreducible. This permits us to choose the image of g to lie in a preassigned neighborhood of the image of f. This could not be done with the weaker theorem, for even when M is eventually end-irreducible, a regular neighborhood of f(R2) need not be. We also extend the theorem in another direction; roughly, we show that if a normal subgroup of the fundamental group of the complement of a compact set in M is given and large loops around the origin in R2 are mapped by f to homotopy classes not in the subgroup, then g can be chosen to have the same property. The proof of the main theorem, (2.2), parallels that of (2). At one point, (2) makes major use of the eventually end-irreducible hypothesis. In this paper we prove Lemma (2.1) to substitute where eventually end-irreducible was used before. We believe the technique used in Lemma (2.1) is new. We expect it to be generally applicable to 3-manifolds which are not eventually end-irreducible. 1. Notational conventions and a preliminary lemma. We work in the category of simplicial complexes and piecewise linear maps. A map f: X -* Y is proper if f -1(C) is compact for every compact C C Y. If X C Y we use Fr( X) to mean the frontier of X in Y and Cl(X) to mean the closure of X in Y. If M is a manifold, we use aM for the boundary of M. We follow Waldhausen's convention (6) on regular neighbor- hoods; specifically, if X c Y, choose a triangulation of Y in which all previously mentioned subspaces are subcomplexes, and let U(X) be the simplicial neighbor- hood of X in the second barycentric subdivision. If M is a 3-manifold, an exhausting sequence for M will be a sequence { M, } of compact 3-manifolds in M with Mn C Mn+1 - Fr(Mn+1), Mn n aM a 2-manifold,

Neighborhoods of algebraic sets

In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighborhoods will be defined as level sets of polynomial or semialgebraic functions.

ON TWINS IN THE FOUR-SPHERE I

E. C. Zeeman [Trans. Amer. Math. Soc. 115 (1965), 471–495; MR0195085 (33 #3290)] introduced the process of twist spinning a 1-knot to obtain a 2-knot (in S4), and proved that a twist-spun knot is fibered with finite cyclic structure group. R. A. Litherland [ibid. 250 (1979), 311–331; MR0530058 (80i:57015)] generalized twist-spinning by performing during the spinning process rolling operations and other motions of the knot in three-space. The first paper generalizes those results by introducing the concept of a twin. A twin W is a subset of S4 made up of two 2-knots R and S that intersect transversally in two points. The prototype of a twin is the n-twist spun of K (that is, the union of the n-twist spun knot of K and the boundary of the 3-ball in which the original knot lies). The exterior of a twin, X(W), is the closure of S4−N(W), where N(W) is a regular neighborhood of W in S4. The first paper considers properties of X(W), and uses these to characterize the automorphisms of a 2-torus standardly embedded in S4, which extend to S4, and also to prove that any homotopy sphere obtained by Dehn surgery on such a 2-torus is the real S4. The second paper is devoted to the fibration problem, i.e. given a twin in S4, try to understand what surgeries in W give a twin W′ which has a component that is a fibered knot (as in the Zeeman theorem). This approach yields alternative proofs of the twist-spinning theorem of Zeeman, and of the roll-twist spinning results of Litherland. New fibered 2-knots are produced through these methods.

Non-trivial ΓΔ-regular graphs

Using the results of C.D. Godsil and B.D. McKay in “Graphs with regular neighbourhoods” we prove that there are only two non-trivial ΓΔ-regular graphs with diameter 3, and that all the other non-trivial ΓΔ-regular graphs have diameter 2. We also prove that there are no non-trivial ΓΔ-regular graphs with η> λ+1. Next we provethat if a (0,1)-geometric graph is a non-trivial ΓΔ-regular graph, it is one of the two non-trivial ΓΔ-regular graphs with diameter 3. At last we construct two new non-trivial ΓΔ-regular graphs on 28 vertices using the Chang graphs.

Simple knots in compact, orientable 3-manifolds

A simple closed curve J J in the interior of a compact, orientable 3 3 -manifold M M is called a simple knot if the closure of the complement of a regular neighborhood of J J in M M is irreducible and boundary-irreducible and contains no non-boundary-parallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable 3 3 -manifold M M such that ∂ M \partial M contains no 2 2 -spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a 3 3 -manifold is completely determined by its set K ( M ) \mathcal {K}(M) of knot groups, i.e, the set of groups π 1 ( M − J ) {\pi _1}(M - J) as J J ranges over all the simple closed curves in M M . In addition, it is proven that a closed 3 3 -manifold M M is homeomorphic to S 3 {S^3} if and only if every simple closed curve in M M shrinks to a point inside a connected sum of graph manifolds and 3 3 -cells.

Knots with Heegaard genus $2$ complements are invertible

Let K be a polyhedral oriented knot in S3 and N(K) be a regular neighborhood of K. If S3 ,N(K) can be constructed by attaching a single 2-handle to a genus two handlebody, then there is a homeomorphism of S3 onto itself mapping K onto itself and reversing the orientation of K. We prove the title. A somewhat more careful statement is the following. THEOREM. Let K c S3 be a polyhedral knot and let N(K) be a regular neighborhood of K. IfS3 N(K) can be constructed by attaching a single 2-handle to a genus two handlebody, then K is invertible. PROOF. By a meridian of the knot K we mean a polyhedral disk D in N(K) with aD c aN(K) and N(K) N(D) is homeomorphic with a ball. (N(D) is a regular neighborhood of D in N(K).) We show how to construct a self-homeomorphism of S3 N(K) which maps the boundary of a meridian of the knot onto its inverse. It is easy to see that such a homeomorphism can be extended to an involution of S3 taking the (oriented) knot to its inverse. Let H2 be a genus two solid handlebody, let D, and D2 be meridian disks for H2 and let y be the simple closed curve on H2 to which a 2-handle B is attached to get S3 N(K). Let m be the boundary of a meridian of K. Without loss of generality we may assume that m c aH2 y. Let h: H2 -* H2 be a rotation of H2 through 180' about its axis. (Think of the standard picture of H2. The axis passes through both holes of the handlebody.) Now h induces the symmetry q as defined in [O & S, p. 248]. Thus h can be assumed to map y and m onto themselves while reversing their orientations (see also [B & H, ?5]). The underlying reason for this is that the rotation inverts the Lickorish twists that generate the homeotopy group of the surface. Clearly then h may be extended to a homeomorphism h which maps B onto itself while reversing orientation. This completes our proof. Note. This result shows that knots such as 820 and 10132 from the table of knots in [Rolf] are invertible. This does not appear evident from the presentations given. These knots are not torus knots or 2-bridge knots. The knot 810 does not have a complement with Heegaard genus 2 but it is invertible. That it does not have such a complement follows from the fact that its second elementary ideal is proper [Fox]. Of course, the result above means that the noninvertible pretzel knots of Trotter Received by the editors February 18, 1980 and, in revised form, April 21,1980. 1980 Mathematics Subject Classification. Primary 57M25. ( 1981 American Mathematical Society 0002-9939/81/0000-01 34/$01.50 501 This content downloaded from 157.55.39.35 on Tue, 30 Aug 2016 04:18:33 UTC All use subject to http://about.jstor.org/terms

A Simple Proof of the Algebraic Unknotting of Spheres in Codimension Two

By a knot we mean a locally flat PL embedding f:Sn Sn+2 of the nsphere in the (n + 2)-sphere. By Alexander duality the complement of the knot C = Sn+2 f(Sn) has the homology groups of the circle S1, but C is not, in general, homotopy equivalent to S1. The purpose of this note is to give a simple proof of the following theorem. THEOREM 1. Let f:Sn _ Sn+2 be a knot with complement C. If ri(C) =ri (SDfor i c (n + 1)/2, then C is homotopy equivalent to S'. This theorem is known from previous work (for n = 1 by [1], for n = 2 by [2, 8], for n = 3 by [4], and for n 2 4 by [3]), but the available proofs are all rather long and technical. One can also combine our proof of Theorem 1 with the s-cobordism theorem (as done in [7, p. 92]) to show that for n 2 4 the knot is actually trivial, thus avoiding the ambient surgery techniques of [3]. That Theorem 1 is the best possible result can be seen by considering [5, p. 25]. Proof of Theorem 1. First we replace C by a finite CW-complex. Let X be the complement in Sn+2 of an open regular neighborhood of f(Sn). Thus X is a compact orientable (n + 2)-manifold with boundary S1 X sn, and is homotopy equivalent to C. We will show that X is homotopy equivalent to a CW-complex of dimension (n + 1)/2 or (n + 2)/2, depending on whether n is odd or even. If n is odd, this implies that the universal cover of X is contractible, and so X must be homotopy equivalent to S1. If n is even, we can then show (following the argument of Wall [9, Prop. 3.3] in the 2dimensional case) that X is homotopy equivalent to a wedge of S1 with some (n + 2)/2-spheres. Since X has the homology of S1, this implies that X is homotopy equivalent to S1. We use the following special case of a basic result of Wall which relates the cohomology with local coefficients in the group ring of a finite CW-complex to its homotopy dimension.

Homeomorphisms of 𝑆³ leaving a Heegaard surface invariant

We find a finite set of generators for the group H g {\mathcal {H} _g} of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere S 3 {S^3} which leave a Heegaard surface T of genus g in S 3 {S^3} invariant. We also show that every element of the group H g {\mathcal {H} _g} can be represented by a deformation of the surface T in S 3 {S^3} of a very special type: during the deformation the surface T is the boundary of the regular neighborhood of a graph embedded in a fixed 2-sphere. The only exception occurs when a subset of the graph contained in a disc on the 2-sphere is “flipped over."

Irrational connected sums and the topology of algebraic surfaces

Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by V m {V_m} a nonsingular element of | m V | \left | {mV} \right | . Let V 1 {V_1} , V m {V_m} , V m + 1 {V_{m\, + \,1}} be generic elements of | V | \left | V \right | , | m V | \left | {mV} \right | , | ( m + 1 ) V | \left | {(m\, + \,1)V} \right | respectively such that they have normal crossing in W. Let S 1 m = V 1 ∩ V m {S_{1m}}\, = \,{V_1}\, \cap \,{V_m} and C = V 1 ∩ V m ∩ V m + 1 C\, = \,{V_1}\, \cap \,{V_m}\, \cap \,{V_{m + 1}} . Then S 1 m {S_{1m}} is a nonsingular curve of genus g m {g_m} and C is a collection of N = m ( m + 1 ) V 1 3 N\, = \,m\left ( {m + 1} \right )V_1^3 points on S 1 m {S_{1m}} . By [MM2] we find that ( ∗ ) V m + 1 ( \ast )\,{V_{m\, + \,1}} is diffeomorphic to V m − T ( S 1 m ) ¯ ∪ η V 1 ′ − T ( S 1 m ′ ) ¯ \overline {{V_m}\, - \,T({S_{1m}})} \,{ \cup _\eta }\,\overline {{V_1}’\, - \,T({S_{1m}}’)} , where T ( S 1 m ) T\left ( {{S_{1m}}} \right ) is a tubular neighborhood of S 1 m {S_{1m}} in V m {V_m} , V 1 ′ {V_1}’ is V 1 {V_1} blown up along C, S 1 m ′ {S_{1m}}’ is the strict image of S 1 m {S_{1m}} in V 1 ′ {V_1}’ , T ( S 1 m ′ ) T({S_{1m}}’) is a tubular neighborhood of S 1 m ′ {S_{1m}}’ in V 1 ′ {V_1}’ and η : ∂ T ( S 1 m ) → ∂ T ( S m ′ ) \eta :\,\partial T\left ( {{S_{1m}}} \right ) \to \partial T({S_m}’) is a bundle diffeomorphism. Now V 1 ′ {V_1}’ is well known to be diffeomorphic to V 1 # N ( − C P 2 ) {V_1}\, \# \,N\left ( { - C{P^2}} \right ) (the connected sum of V 1 {V_1} and N copies of C P 2 C{P^2} with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of V m {V_m} to ones about V 1 {V_1} we must simplify the “irrational sum” ( ∗ ) ( \ast ) above. The general question we can ask is then the following: Suppose M 1 {M_1} and M 2 {M_2} are compact smooth 4-manifolds and K is a connected q-complex embedded in M i {M_i} . Let T i {T_i} be a regular neighborhood of K in M i {M_i} and let η : ∂ T 1 → ∂ T 2 \eta :\,\partial {T_1}\, \to \,\partial {T_2} be a diffeomorphism: Set V = M 1 − T 1 ¯ ∪ M 2 − T 2 ¯ V\, = \,\overline {{M_1}\, - \,{T_1}} \, \cup \,\overline {{M_2}\, - \,{T_2}} . How can the topology of V be described more simply in terms of those of M 1 {M_1} and M 2 {M_2} . In this paper we show how surgery can be used to simplify the structure of V in the case q = 1 , 2 q\, = \,1,\,2 and indicate some applications to the topology of algebraic surfaces.

Irrational Connected Sums and the Topology of Algebraic Surfaces

Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by ${V_m}$ a nonsingular element of $\left | {mV} \right |$. Let ${V_1}$, ${V_m}$, ${V_{m + 1}}$ be generic elements of $\left | V \right |$, $\left | {mV} \right |$, $\left | {(m + 1)V} \right |$ respectively such that they have normal crossing in W. Let ${S_{1m}} = {V_1} \cap {V_m}$ and $C = {V_1} \cap {V_m} \cap {V_{m + 1}}$. Then ${S_{1m}}$ is a nonsingular curve of genus ${g_m}$ and C is a collection of $N = m\left ( {m + 1} \right )V_1^3$ points on ${S_{1m}}$. By [MM2] we find that $( \ast ) {V_{m + 1}}$ is diffeomorphic to $\overline {{V_m} - T({S_{1m}})} { \cup _\eta } \overline {{V_1}’ - T({S_{1m}}’)}$, where $T\left ( {{S_{1m}}} \right )$ is a tubular neighborhood of ${S_{1m}}$ in ${V_m}$, ${V_1}’$ is ${V_1}$ blown up along C, ${S_{1m}}’$ is the strict image of ${S_{1m}}$ in ${V_1}’$, $T({S_{1m}}’)$ is a tubular neighborhood of ${S_{1m}}’$ in ${V_1}’$ and $\eta : \partial T\left ( {{S_{1m}}} \right ) \to \partial T({S_m}’)$ is a bundle diffeomorphism. Now ${V_1}’$ is well known to be diffeomorphic to ${V_1} \# N\left ( { - C{P^2}} \right )$ (the connected sum of ${V_1}$ and N copies of $C{P^2}$ with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of ${V_m}$ to ones about ${V_1}$ we must simplify the “irrational sum” $( \ast )$ above. The general question we can ask is then the following: Suppose ${M_1}$ and ${M_2}$ are compact smooth 4-manifolds and K is a connected q-complex embedded in ${M_i}$. Let ${T_i}$ be a regular neighborhood of K in ${M_i}$ and let $\eta : \partial {T_1} \to \partial {T_2}$ be a diffeomorphism: Set $V = \overline {{M_1} - {T_1}} \cup \overline {{M_2} - {T_2}}$. How can the topology of V be described more simply in terms of those of ${M_1}$ and ${M_2}$. In this paper we show how surgery can be used to simplify the structure of V in the case $q = 1, 2$ and indicate some applications to the topology of algebraic surfaces.

Regular Neighbourhoods of Graphs in Manifolds

Regular Neighbourhoods of Graphs in Manifolds

Regular neighborhoods in topological manifolds.

Regular neighborhoods have proved to be a very useful tool in the theory of PL manifolds. In this paper we want to make a very easy construction of regular neighborhoods in the topological category. F. E. A. Johnson [6] has constructs regular neighborhoods in the topological category, but only in the case of nonintersection with the boundary. R. D. Edwards [2] has announced a very general construction of regular neighborhoods; see also [3]. The present construction has the advantage of allowing a “relative” version , (Theorem 13), in the sense that if L is a complex, K is a subcomplex, and L is locally tamely embedded in a topological manifold V , then one may find a regular neighborhood ofK in V , intersecting L in a regular neighborhood of K in L, in the usual PL sense. This is used in [10] to prove embedding theorems for topological manifolds. In [11] we have a proof that the opposite procedure is possible; namely a spine of a topological manifold. We should emphasize that the regular neighborhoods we obtain are mapping cylinder neighborhoods; i. e. if K ⊂ N where N is a regular neighborhood of K, then there is a map π : ∂N → K such that N is homeomorphic to the mapping cylinder of π (Theorem 15). Let K be a compact topological space with a given simple homotopy structure; i. e. of the homotopy type of a finite CW-complex, with the homotopy equivalence specified up to torsion.

Longitude surgery on genus 1 knots

Let l ( K ) l(K) be the closed 3-manifold obtained by longitude surgery on the knot manifold K. Let C be the cube with holes obtained by removing an open regular neighborhood of a minimal spanning surface in K. The main result of this paper is that if K is of genus 1 and the longitude of K is in each term of the lower central series for Π 1 ( C ) {\Pi _1}(C) , then l ( K ) l(K) is not homeomorphic to the connected sum of S 1 × S 2 {S^1} \times {S^2} and a homotopy 3-sphere. In particular, this implies we cannot obtain the connected sum of S 1 × S 2 {S^1} \times {S^2} and a homotopy 3-sphere by longitude surgery on any pretzel knot of genus 1.

Singular Regular Neighborhoods and Local Flatness in Codimension One

Singular Regular Neighborhoods and Local Flatness in Codimension One

Mapping Cylinder Neighborhoods of Some ANR's

Let X be a closed subset of a manifold without boundary Q. A codimension 0 submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: AM-P X such that M is homeomorphic to the mapping cylinder of r, fixing AM and X in the natural way. For example, regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods. We do not insist that M be collared in Q since this can always be arranged by an isotopy of M into itself along the structure lines. Edwards ([5]) contains a lot of information about mapping cylinder neighborhoods. It follows immediately from the definition that subsets possessing mapping cylinder neighborhoods are retracts of the neighborhoods, hence are finite dimensional ANR's. The converse question, whether finite dimensional ANR's have mapping cylinder neighborhoods, at least for some embeddings in manifolds, is interesting. If they do, then compact, finite dimensional ANR's have finite homotopy type, since these neighborhoods are compact manifolds, and so have finite homotopy type by Kirby-Siebenmann (17]). We prove a partial result in the direction of the converse, namely, if X is a locally compact ANR, embedded as a closed, 1-LC codimension 4 subset of a finite dimensional manifold without boundary Q, then X x M, has a mapping cylinder neighborhood in Q x M, where either M. M= SI, or (M, Mj) = (R', [0, ao)). Our proof also works in infinite dimensions and yields the fact that if X is a locally compact ANR which is embedded as a closed Z-set in a Hilbert cube manifold Q, then X x M. has a cylinder neighborhood in Q x M. Taking suitable compactifications in the case (M, Mj) = (Ri, [0, ao)), one finds that the cone on X is the image of the cone on Q by a map with contractible point inverses. J. West ([10]) has taken this last form of our result and shown that in the infinite dimensional setting it implies X actually has a mapping cylinder