On detecting open collars

Notice that b W need not be compact. In the proof, the fundamental group condition (77! condition) permits imbedded surgery with handles of dimension ^ 2 to find large compacta A<= W such that 2-connectivity of (W, W—K) can be proved. Poincare duality in the universal covering seems essential to show this. A well-known engulfing argument of Stallings completes the proof. (We first encountered a proof of this result for n =i 6 based entirely on handlebody theory cf. [40].) Although the ttx condition requires careful statement, our applications show that it is pleasant to use. Indeed the main contention of this paper is that it considerably extends the usefulness of Stallings' conception of engulfing. That the nx condition is necessary at all (i.e. not redundant) is at first sight surprising. This was first noticed in the wake of J. H. C. Whitehead's misproof of the 3-dimensional Poincare conjecture in 1934. Whitehead then constructed a

[1]  R. Kirby,et al.  On the triangulation of manifolds and the Hauptvermutung , 1969 .

[2]  W. Lickorish,et al.  REGULAR NEIGHBOURHOODS AND THE STABLE RANGE , 1969 .

[3]  Marshall M. Cohen.  A GENERAL THEORY OF RELATIVE REGULAR NEIGHBORHOODS , 1969 .

[4]  L. Siebenmann.  On detecting euclidean space homotopically among topological manifolds , 1968 .

[5]  J. L. Bryant,et al.  LOCALLY NICE EMBEDDINGS IN CODIMENSION THREE , 1968 .

[6]  Charles Terence Clegg Wall,et al.  Finiteness conditions for CW complexes. Il , 1966, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[7]  Klaus Jänich.  Differenzierbare Mannigfaltigkeiten mit rand als orbiträume differenzierbarer G-Mannigfaltigkeiten ohne rand☆ , 1966 .

[8]  M. Hirsch.  ON NORMAL MICROBUNDLES , 1966 .

[9]  B. Mazur.  The Method of Infinite Repetition in Pure Topology: II. Stable Applications , 1966 .

[10]  L. Siebenmann,et al.  Some homeomorphic sphere pairs that are combinatorially distinct , 1966 .

[11]  M. Hirsch.  On tangential equivalence of manifolds , 1966 .

[12]  M. Kervaire.  Le théorème deBarden-Mazur-Stallings , 1965 .

[13]  Charles Terence Clegg Wall,et al.  Piecewise linear bundles in the stable range , 1965 .

[14]  W. Browder,et al.  Finding a Boundary for an Open Manifold , 1965 .

[15]  M. Irwin.  Embeddings of Polyhedral Manifolds , 1965 .

[16]  E. Zeeman,et al.  On combinatorial isotopy , 1964 .

[17]  E. C. Zeeman,et al.  On Regular Neighbourhoods , 1964 .

[18]  B. Mazur.  The Method of Infinite Repetition in Pure Topology: I , 1964 .

[19]  D. Montgomery,et al.  Compact Groups in E n , 1964 .

[20]  D. Mcmillan.  A Criterion for Cellularity in a Manifold , 1964 .

[21]  J. Munkres.  Obstructions to extending diffeomorphisms , 1964 .

[22]  T. E. Stewart.  Fixed point sets and equivalence of differentiable transformation groups , 1963 .

[23]  E. C. Zeeman,et al.  Unknotting Combinatorial Balls , 1963 .

[24]  M. Hirsch.  Obstruction theories for smoothing manifolds and maps , 1963 .

[25]  J. Stallings.  ON TOPOLOGICALLY UNKNOTTED SPHERES , 1963 .

[26]  B. Mazur.  Relative Neighborhoods and the Theorems of Smale , 1963 .

[27]  M. Hirsch.  On combinatorial submanifolds of differentiable manifolds , 1962 .

[28]  M. Hirsch.  SMOOTH REGULAR NEIGHBORHOODS , 1962 .

[29]  J. Stallings.  The Piecewise-Linear Structure of Euclidean Space , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  J. Milnor.  Two Complexes Which are Homeomorphic But Combinatorially Distinct , 1961 .

[31]  B. Mazur.  Stable equivalence of differentiable manifolds , 1961 .

[32]  R. Bing.  A surface is tame if its complement is 1-ULC , 1961 .

[33]  Armand Borel,et al.  Seminar on Transformation Groups. , 1961 .

[34]  N. Steenrod.  Topology of Fibre Bundles , 1951 .

[35]  E. Artin,et al.  Some Wild Cells and Spheres in Three-Dimensional Space , 1948 .

[36]  M. Newman,et al.  Boundaries of ULC Sets in Euclidean n-Space. , 1948, Proceedings of the National Academy of Sciences of the United States of America.

[37]  M. Kervaire.  Le théorème de Barden-Mazur-Stallings , 1967 .

[38]  A. Scott.  Infinite Regular Neighbourhoods , 1967 .

[39]  J. Hudson.  Piecewise linear topology , 1966 .

[40]  J. Hempel.  LOCALLY NICE EMBEDDINGS OF MANIFOLDS , 1966 .

[41]  C. Wall.  Finiteness conditions for CW-complex I , 1965 .

[42]  E. C. Zeeman,et al.  On the dunce hat , 1963 .

[43]  James R. Munkres,et al.  Differentiable isotopies on the $2$-sphere. , 1960 .

[44]  R. Wilder,et al.  Uniform Local Connectedness and Contractibility , 1942 .

[45]  M. Newman,et al.  ON THE GROUP OF A CERTAIN LINKAGE , 1937 .

[46]  P. Alexandroff.  On Local Properties of Closed Sets , 1935 .