The l 1 -Completion of a Metric Combinatorial ∞-Manifold

1987 ◽  
Vol 100 (3) ◽  
pp. 574
Author(s):  
Katsuro Sakai
Keyword(s):  
Combinatorial Manifold

A Q-Manifold Local-Compactification of a Metric Combinatorial ∞- Manifold

1987 ◽  
Vol 100 (4) ◽  
pp. 775
Author(s):  
Katsuro Sakai
Keyword(s):  
Combinatorial Manifold

scholarly journals A geometric condition for smoothability of bounded combinatorial manifold

1969 ◽  
Vol 45 (4) ◽  
pp. 266-268
Author(s):  
Kazuaki Kobayashi
Keyword(s):  
Geometric Condition ◽  
Combinatorial Manifold

Structures on M × R

1965 ◽  
Vol 61 (2) ◽  
pp. 337-345 ◽  
Author(s):  
William Browder
Keyword(s):  
Weak Form ◽  
Combinatorial Structure ◽  
Differentiable Structure ◽  
Combinatorial Manifold

A central role in the theory of smoothing combinatorial manifolds is played by the Cairns–Hirsch Theorem, which may be expressed (in a weak form) as follows:If M is a combinatorial manifold and if M × R has a differentiable structure a, compatible with its combinatorial structure then M has a differentiable structure λ, such that (M × R)α is diffeomorphic with Mλ × R.

Construction of Transverse Fields

1969 ◽  
Vol 21 ◽  
pp. 1146-1159
Author(s):  
H. Putz
Keyword(s):  
Euclidean Space ◽  
Existence Theorem ◽  
Transverse Field ◽  
Local Conditions ◽  
Combinatorial Manifold ◽  
Transverse Fields ◽  
Special Case

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold,Mn, in Euclidean space, which are sufficient to prove thatMnhas a transverse field (see 1.1 and 1.2, definitions).In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifoldMn.Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).After the research for this paper was completed, a paper of Davis (2) came to our attention.

Regular neighbourhoods and mapping cylinders

1971 ◽  
Vol 70 (1) ◽  
pp. 15-18
Author(s):  
C. Kearton
Keyword(s):  
Piecewise Linear ◽  
Linear Manifold ◽  
Mapping Cylinder ◽  
Simple Characterization ◽  
Regular Neighbourhood ◽  
Simplicial Map ◽  
Combinatorial Manifold ◽  
Piecewise Linear Manifold

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

On Contractible Open Manifolds

1962 ◽  
Vol 58 (2) ◽  
pp. 221-224 ◽  
Author(s):  
D. R. McMillan ◽  
E. C. Zeeman
Keyword(s):  
Compact Space ◽  
Open Manifold ◽  
Open Manifolds ◽  
Combinatorial Manifold

By an open manifold we mean a non-compact space, that is triangulable by a countable complex which is a combinatorial manifold without boundary (see next section).

scholarly journals UNUSUAL FORMULAE FOR THE EULER CHARACTERISTIC

2002 ◽  
Vol 11 (05) ◽  
pp. 793-796 ◽  
Author(s):  
JUSTIN ROBERTS
Keyword(s):  
Euler Characteristic ◽  
Linear Combinations ◽  
Combinatorial Manifold

Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristic.

scholarly journals On a Vertex-Minimal Triangulation of $\mathbb R \mathrm P ^4$

10.37236/5956 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Sonia Balagopalan
Keyword(s):  
Projective Space ◽  
Double Cover ◽  
Dimensional Sphere ◽  
Simple Construction ◽  
Real Projective Space ◽  
Minimal Triangulation ◽  
Combinatorial Manifold

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}\mathrm{P}^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated $\mathbb{R}\mathrm{P}^4$ and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate $n$-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known $16$-vertex triangulation of $\mathbb{R}\mathrm{P}^4$. We also give a short, simple construction of the $22$-point Witt design, which is closely related to the complex we construct.

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